Applications

Problem 1

Given forces $A=10.2 \mathrm{kN}, B=4.08 \mathrm{kN}, \theta=30^{\circ}$ , find angle $\beta$ and force $C$ for resultant $6.5 \mathrm{kN}$ .

See Solution

Problem 2

Find the force $S$ (in kN) so that $P = 35$ kN, $Q = 45$ kN, and the resultant is $55$ kN to the right.

See Solution

Problem 3

Find (a) the complement and (b) the supplement of an angle measuring $17^{\circ} 15^{\prime}$ .

See Solution

Problem 4

1. A boat sees a lighthouse foot at $34^{\circ}$ elevation; cliff height is 150 m. Find the boat's distance from the cliff base.
2. Flying at $037^{\circ}$ , turn left $45^{\circ}$ . What is the new heading?

See Solution

Problem 5

Convert the angle $-51^{\circ} 30^{\prime}$ to decimal degree notation.

See Solution

Problem 6

Convert the angle $-75^{\circ} 3^{\prime}$ to decimal degrees.

See Solution

Problem 7

Convert the angle $-41^{\circ} 54^{\prime}$ to decimal degrees.

See Solution

Problem 8

Convert the angle $-76^{\circ} 12^{\prime}$ to decimal degree notation.

See Solution

Problem 9

Convert the angle $-81^{\circ} 33^{\prime}$ to decimal degrees.

See Solution

Problem 10

Sketch the least positive angle $\theta$ from the line $-7x - 3y = 0$ where $x \leq 0$ , and find the six trig functions of $\theta$ .

See Solution

Problem 11

Convert the angle $-37^{\circ} 30^{\prime}$ to decimal degrees.

See Solution

Problem 12

John, at $6 \mathrm{ft}$ high, measures a $35^{\circ}$ angle to a tree. After moving 30 ft closer, it's $47^{\circ}$ . Find the tree's height.

See Solution

Problem 13

Find the distance between two cars below a 1000-foot cliff with angles of depression $21^{\circ}$ and $28^{\circ}$ .

See Solution

Problem 14

Find the resultant velocity when crossing a river with a current of $15 \mathrm{~m/s}$ downstream and crossing at $2 \mathrm{~m/s}$ north.

See Solution

Problem 15

A car on a $1.7^{\circ}$ incline has a grade resistance of $123 \, \mathrm{lb}$ . Find the car's weight in hundreds using: Grade Resistance = Weight * sin(incline). Consider other forces too.

See Solution

Problem 16

Find the grade resistance for a 2100-pound car on a $0.5^{\circ}$ uphill grade using $F=W \sin \theta$ . Round to the nearest pound.

See Solution

Problem 17

Light changes speed and direction when moving between media. Use Snell's Law: $\frac{c_{1}}{c_{2}}=\frac{\sin \theta_{1}}{\sin \theta_{2}}$ . Given $c_{1}=3 \times 10^{8} \mathrm{~m/s}$ , find $c_{2}$ for: (a) $\theta_{1}=47^{\circ}$ , $\theta_{2}=34^{\circ}$ ; (b) $\theta_{1}=39^{\circ}$ , $\theta_{2}=25^{\circ}$ .

See Solution

Problem 18

A light ray changes speed and direction when moving between media. Use Snell's Law: $\frac{c_{1}}{c_{2}}=\frac{\sin \theta_{1}}{\sin \theta_{2}}$ , with $c_{1}=3 \times 10^{8} \mathrm{~m/s}$ .
(a) Find $c_{2}$ for $\theta_{1}=41^{\circ}$ , $\theta_{2}=25^{\circ}$ . (b) Find $c_{2}$ for $\theta_{1}=38^{\circ}$ , $\theta_{2}=24^{\circ}$ .

See Solution

Problem 19

A plane is $3.8^{\circ}$ off course after flying 127 miles. How far off course is it? Round to 1 decimal place.

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Problem 20

How far is a plane off course after being $3.0^{\circ}$ off track for 187 miles? Round to 1 decimal place.

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Problem 21

A ship is anchored off a shore with two points 13 miles apart. Find the shortest distance to the shore, rounding to the nearest tenth.

See Solution

Problem 22

Carmen sees two hot air balloons. One is 670 m away at $42^{\circ}$ and the other 945 m away at $36^{\circ}$ . Find the height difference.

See Solution

Problem 23

Find angle $\theta_{2}$ using Snell's law: $\frac{c_{1}}{c_{2}}=\frac{\sin \theta_{1}}{\sin \theta_{2}}$ given $c_{1}=3.00 \times 10^{8}$ m/s and $c_{2}=2.251 \times 10^{8}$ m/s.

See Solution

Problem 24

Two ships leave a port simultaneously. One sails at $56^{\circ}$ , 14 knots; the other at $146^{\circ}$ , 26 knots. Distance apart after 1.5 hours?

See Solution

Problem 25

Ein Aussichtsturm ist $30,75 \mathrm{~m}$ hoch. Die Felswand hat einen Tiefenwinkel von $2,5^{\circ}$ und einen Höhenwinkel von $10,3^{\circ}$ . Berechne die Entfernung zur Felswand und deren Höhe.

See Solution

Problem 26

Amy jogs 3 miles straight, then turns 50 degrees right and runs $x$ miles. Find her distance from the start. Round to 0.1.

See Solution

Problem 27

Karl sieht ein Flugzeug in $10 \mathrm{~km}$ Höhe. a) Finde die Flugdistanz bei $\alpha=60^{\circ}$ und $\beta=25^{\circ}$ . b) Bestimme die Geschwindigkeit.

See Solution

Problem 28

Berechne die Steigung in Prozent für die Winkel: c) $\gamma=27,5^{\circ}$ , a) $a=35^{\circ}$ , b) $\beta=70^{\circ}$ .

See Solution

Problem 29

A soccer player kicks a ball at $27 \mathrm{~m/s}$ and $20^{\circ}$ . Find: (a) air time, (b) impact speed, (c) range.

See Solution

Problem 30

An airplane flies at $600 \mathrm{~km/hr}$ with a $50 \mathrm{~km/hr}$ southwest wind. What angle should it head to go west?

See Solution

Problem 31

What bearing must the aeroplane fly to return from city G to city F after flying at a bearing of $055^{\circ}$ ?

See Solution

Problem 32

Ein Sendemast ist $150 \mathrm{~m}$ hoch. Berechne den Neigungswinkel der $160 \mathrm{~m}$ langen Seile bei $125 \mathrm{~m}$ . Dann, bestimme die Länge der Seile, die bei $25^{\circ}$ in $60 \mathrm{~m}$ Höhe befestigt sind.

See Solution

Problem 33

Find the max weight $W$ a block can have if cords support up to $80 \, \mathrm{lb}$ at a $30^\circ$ angle from vertical.

See Solution

Problem 34

A crate on an incline has a static friction coefficient of 0.29. Find $\theta$ when it slips and its acceleration with kinetic friction 0.26.

See Solution

Problem 35

A 40.2 kg sign is held by two wires at angles 43.8° and 52.7°. Find the tension in wire 1 and the y-direction acceleration.

See Solution

Problem 36

A football is kicked at $18 \mathrm{~m/s}$ and $65^{\circ}$ . How long is it in the air? Options: A. $1.1 \mathrm{~s}$ B. $4.0 \mathrm{~s}$ C. $3.3 \mathrm{~s}$ D. $2.0 \mathrm{~s}$ E. $1.6 \mathrm{~s}$

See Solution

Problem 37

A man rows at $8 \mathrm{~km/h}$ across a $1.5 \mathrm{~km}$ river with a $5 \mathrm{~km/h}$ current. Find the heading, time, and downstream distance.

See Solution

Problem 38

Projectile motion problem: A football is kicked with a velocity of $25 \mathrm{~m/s}$ at a 50-degree angle.
a. Draw velocity and acceleration motion maps. b. Calculate $v_x$ and $v_y$ . c. Determine the time in the air.

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Problem 39

A projectile launched at 45 degrees takes 2.4 seconds to land. What was its launch speed? Options: $12 \mathrm{~m/s}$ , $17 \mathrm{~m/s}$ , $34 \mathrm{~m/s}$ , $24 \mathrm{~m/s}$ .

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Problem 40

Find the right angle in gradians and the radian measure of 150 grads.

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Problem 41

London Eye problem:
a) Find the angle in radians for 5 min. b) Calculate distance traveled in 5 min. c) Time for 2 radians? d) Angular velocity in radians/sec? e) Angular velocity in degrees/sec?

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Problem 42

A cyclist's bike wheels (2.1 ft diameter) turn at 270 RPM. Find angular speed in radians/min and cyclist speed in ft/min.

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Problem 43

Ceiling fan with 14-inch blades spins at 46 RPM.
(a) Calculate angular speed in radians/min. (b) Calculate linear speed of blade tip in feet/sec.

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Problem 44

Berechne den Schnittwinkel der Graphen $f$ und $g$ für die Anstiege: a) $m_{f}=3$ , $m_{g}=1$ ; b) $m_{f}=3$ , $m_{g}=-1$ ; c) $m_{f}=-2$ , $m_{g}=-0,5$ ; d) $m_{f}=\sqrt{3}$ , $m_{g}=1$ . Runde auf Zehntel.

See Solution

Problem 45

Two planes leave an airport simultaneously. Plane 1: $660 \mathrm{~m/h}$ at $12.3^{\circ}$ , Plane 2: $580 \mathrm{~m/h}$ at $148^{\circ}$ . Find distance apart after $3.3 \mathrm{~h}$ in $\mathrm{m}$ .

See Solution

Problem 46

Find the complement: $90^{\circ} - 27^{\circ} 35^{\prime} 15^{\prime \prime} = 62^{\circ} 24^{\prime} 45^{\prime \prime}$

See Solution

Problem 47

Find the supplement of an angle measuring $27^{\circ} 35^{\prime} 15^{\prime \prime}$ .

See Solution

Problem 48

Find the expression for an angle coterminal with a $300^{\circ}$ angle: $300^{\circ}-860^{\circ}$ , $300^{\circ}-840^{\circ}$ , $300^{\circ}-740^{\circ}$ , $300^{\circ}-720^{\circ}$ .

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Problem 49

Bestimme die Modellfunktion für eine Kugel, die aus $1,80 \mathrm{~m}$ Höhe mit $42^{\circ}$ abgeworfen wird, bei $8,40 \mathrm{~m}$ Stoßweite. Finde die maximale Höhe und den Aufprallwinkel.

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Problem 50

Find the height of a building 40 feet away, with angles of elevation $62^{\circ}$ and depression $55^{\circ}$ . Round to nearest foot.

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Problem 51

Hikers observe a mountain with a $35^{\circ}$ elevation from 1250 m away. Find slant heights using slopes of $48^{\circ}$ and $65^{\circ}$ .

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Problem 52

A projectile is launched at $11 \mathrm{~m/s}$ at $30^{\circ}$ from vertical. Find time to max height. Options: (a) $0.97 \mathrm{~s}$ (b) $0.56 \mathrm{~s}$ (c) $0.79 \mathrm{~s}$ (d) $1.12 \mathrm{~s}$ .

See Solution

Problem 53

A jet traveling at $651 \mathrm{mph}$ east encounters a $516 \mathrm{mph}$ wind at $35^{\circ}$ north of east. Find its new speed in mph.

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Problem 54

A ball is launched at $38.0 \, \text{m/s}$ at $60.0^{\circ}$ . Find: a) horizontal velocity, b) horizontal distance, c) max height.

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Problem 55

What force is needed to pull a $56.5 \mathrm{~N}$ weight up a $12.6^{\circ}$ slope, ignoring friction?

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Problem 56

Two ropes pull a crate: one with $994 \mathrm{~N}$ at $15.5^{\circ}$ and another with $624 \mathrm{~N}$ at $25.2^{\circ}$ . Find the crate's weight in newtons.

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Problem 57

Find the resistance and reactance of a circuit with an impedance of $975 \mathrm{ohm}$ and a phase angle of 28.0 degrees (3 SIG FIGS).

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Problem 58

In a snowball fight, if one snowball is thrown at $61^{\circ}$ with speed $11 \mathrm{~m/s}$ , at what angle should the second be thrown?

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Problem 59

A boat moves east at $6.16 \mathrm{~m/s}$ while the river flows north at $2.34 \mathrm{~m/s}$ . Find the resultant speed and angle.

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Problem 60

In a snowball fight, if one snowball is thrown at $69^{\circ}$ with speed $18 \mathrm{~m/s}$ , what angle should the second one be thrown at to hit the same point? Use $g = 9.8 \mathrm{~m/s}^{2}$ .

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Problem 61

In a snowball fight, if one snowball is thrown at $60^{\circ}$ with speed $34.4 \mathrm{~m/s}$ , what angle should the second snowball be thrown at to hit the same point? Use $g = 9.8 \mathrm{~m/s}^{2}$ .

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Problem 62

Find the angle between vectors $v = 2i + 5j$ and $w = -7i + 8j$ . Round to one decimal place if needed.

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Problem 63

Austin and Bill are $103 \mathrm{~m}$ apart. A roller coaster car moves at $5 \mathrm{~m/s}$ at $33^{\circ}$ . Find distance $r$ after 16 seconds. Round to the nearest tenth.

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Problem 64

Austin and Bill are $103 \mathrm{~m}$ apart. A roller coaster car moves at $5 \mathrm{~m/s}$ at a $33^{\circ}$ angle. Find remaining distance $r$ to the highest point after 16 seconds. Round to the nearest tenth of a meter.

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Problem 65

Calculate the arc length of a basketball shot with an initial velocity of 17 ft/s and a launch angle of 60 degrees.

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Problem 66

已知单位向量 $\vec{a}, \vec{b}$ ， $|2 \vec{a}+\vec{b}|=\sqrt{3}$ ，求向量 $\vec{a}, \vec{b}$ 的夹角。

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Problem 67

Ein Segelflugzeug ist in $1250 \mathrm{~m}$ Höhe, $42 \mathrm{~km}$ vom Ziel entfernt und hat einen Gleitwinkel von $1,5^{\circ}$ . Bestimme die Höhe, in der es das Ziel überfliegt.

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Problem 68

Find the angle of the resultant force with the horizontal when two forces of 17.3 lbs and 42.5 lbs act at 21.9°. Options: $6.3^{\circ}$ , $21.9^{\circ}$ , $15.7^{\circ}$ , $10.9^{\circ}$ .

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Problem 69

A plane flies at $290 \mathrm{mph}$ , 24.5° South of West, with a $30.5 \mathrm{mph}$ wind, 21° East of North. Find ground speed.

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Problem 70

Find the angle between vectors $v = 3i + 2j + k$ and $w = 6i + 8j + 2k$ . Round to one decimal place.

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Problem 71

Find the angle between vectors $v = 3i + 2j + k$ and $w = 6i + 8j + 2k$ . Round to one decimal place.

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Problem 72

A sphere hangs from a string in a van. If $\theta=10.7^{\circ}$ , find the van's acceleration.

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Problem 73

A climber weighing $515 \mathrm{~N}$ rests on a rope between cliffs. Angles are $65^\circ$ (left) and $80^\circ$ (right). Find tensions.

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Problem 74

Jahi "Tornado" liigub põhjasuunas, hoides nurka $30^{\circ}$ . Leidke läänetuule kiirus, kui kiirus on 18,52 $\frac{km}{h}$ . Vastus $\frac{m}{s}$ .

See Solution

Problem 75

Find one positive and one negative coterminal angle for each given angle in degrees: (a) $135^{\circ}$ , (b) $-315^{\circ}$ .

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Problem 76

Ein Turm ist $34 \mathrm{~m}$ hoch. Unter einem Winkel von $18^{\circ}$ sieht man das hintere Ufer und $61^{\circ}$ das vordere Ufer.
a) Wie weit ist das vordere Ufer vom Turm? b) Wie weit ist das hintere Ufer vom Turm? c) Wie breit ist der Fluss?

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Problem 77

Find the diagonal length AG of a cuboid with length 33, width 21, and angle 72° with the base using trigonometry.

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Problem 78

Find the tension in a support cable inclined at $71.5^{\circ}$ to the horizontal, with a horizontal pull of $875 \mathrm{~N}$ .

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Problem 79

A shell is fired at $50.0 \mathrm{~m/s}$ and $53.1^{\circ}$ . Find vertical/horizontal/resultant velocities at $25 \mathrm{~m}$ up, times to reach, and max height velocity.

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Problem 80

Find the gravitational force component down a $40.5^{\circ}$ incline for a box weighing $365 \mathrm{~N}$ . Answer in $\mathrm{N}$ .

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Problem 81

A cable car goes from point $P$ to restaurant $R$ . Angles of elevation from $P$ and point $Q$ (1.8 km closer) are $22^{\circ}$ and $63^{\circ}$ . Find the height of the mountain.

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Problem 82

A hot-air balloon has wires at points $A$ and $C$ . The wire from $A$ is $84 \mathrm{~m}$ long with angles $58^{\circ}$ and $67^{\circ}$ . Find the balloon's height and wire length from $C$ .

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Problem 83

A golfer is $480 \mathrm{ft}$ away and $50 \mathrm{ft}$ above the hole. With an initial speed of $120 \mathrm{ft/s}$ , what angle(s) should the ball be hit?

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Problem 84

Find the components of vector $A$ with magnitude 8.6 in the fourth quadrant at a 37-degree angle to the $x$ -axis.

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Problem 85

A rocket moves up at $900 \mathrm{~km/hr}$ . An observer is $2 \mathrm{~km}$ away. Find the angle's increase rate in degrees/min after lift-off.

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Problem 86

A golfer hits a ball at $42 \mathrm{~m/s}$ at $32^{\circ}$ . Find the horizontal range ignoring air resistance.

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Problem 87

A jet flying at $148 \mathrm{mph}$ east encounters a $445 \mathrm{mph}$ wind at $77^{\circ}$ north of east. Find its new speed in $\mathrm{mph}$ .

See Solution

Problem 88

A worker on a 15 ft pole sees angles of depression of $26^{\circ}$ and $40^{\circ}$ . Find the distance across the yard.

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Problem 89

Aufgabe 1: Eine Tanne wirft einen Schatten von $38 \mathrm{~m}$ . Bei einem Winkel von $28^{\circ}$ , wie hoch ist die Tanne?
Aufgabe 2: Eine Rampe mit $19^{\circ}$ Neigungswinkel soll an eine 1,2 m hohe LKW-Ladefläche. Wie lang muss die Rampe sein?
Aufgabe 3: Eine Seilbahn überwindet $160 \mathrm{~m}$ auf $500 \mathrm{~m}$ . Wie groß ist der Steigungswinkel?

See Solution

Problem 90

A fish is reeled in at 1 ft/sec from 10 ft above water. Find the angle's change rate when 25 ft of line is out.

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Problem 91

1. Find $k$ if $1$ is a root of $3x^{2}+kx^{2}-2x-2k+2$ .
2. A ladder at $30^{\circ}$ is $8$ ft long. What is the height of the tree?
3. Is $f(x)=2x^{3}-4x$ odd, even, or neither?
4. In triangle $ABC$ , $\csc A=\frac{13}{5}$ . Find $\tan A$ .
5. What is the period of $y=-4\sin\left(\frac{2}{3}x\right)$ ?
6. A $10$ ft guy wire makes a $60^{\circ}$ angle. Find the distance to the tree base.
7. Compute $(\tan \beta)(\cot \beta \cos \beta+\sin \beta)$ .
8. For $f(x)=\frac{5}{x}$ , find $f(x+2)-f(x)$ .
9. Given $g(x)=3x^{2}-5$ , find $g(x+2)$ .
10. From a $200$ ft cliff, the angle of depression to a boat is $60^{\circ}$ . Find the distance to the cliff base.
11. What is the amplitude of $y=2\sin x$ ?
12. A $20$ ft flagpole casts a $60$ ft shadow. Find the sun's angle of elevation.

See Solution

Problem 92

Gegeben ist eine Feuerwehrdrehleiter von $8 \mathrm{~m}$ bis $30 \mathrm{~m}$ . Bei $l=20 \mathrm{~m}$ berechne $h$ für $\alpha=0^{\circ}$ bis $80^{\circ}$ mit $h=l \cdot \sin(\alpha)$ .

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Problem 93

Find the distance the tip of a 3-foot pendulum swings at a $40^{\circ}$ angle in 1 second. Round to 1 decimal place.

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Problem 94

Find the distance from tracking station B to an airplane with angles of elevation $47^{\circ}$ and $53^{\circ}$ , 5.8 km apart.

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Problem 95

Black Panther slides down a rooftop at $65^{\circ}$ with a friction coefficient of 0.50. Find his acceleration. If he starts from rest, how fast is he sliding after $10 \mathrm{~m}$ ?

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Problem 96

Die Zugbrücke ist $12 \mathrm{~m}$ lang und bildet einen Winkel von $142^{\circ}$ . Wie lang muss die Kette sein, wenn sie $3,45 \mathrm{~m}$ in der Mauer benötigt?

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Problem 97

A balloon rises while tracked by an observer 2 miles away. If the angle is $\frac{\pi}{6}$ and changing at $0.1 \mathrm{rad/min}$ , find the rise rate. Answer: miles/min

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Problem 98

What was the launch angle of a projectile with an initial speed of $12 \mathrm{~m/s}$ and a speed of $6 \mathrm{~m/s}$ at its highest point?

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Problem 99

Convert the angle $24^{\circ} 43^{\prime} 29^{\prime \prime}$ to decimal degrees.

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Problem 100

A plane flies at 6 km altitude and 600 km/h. Find the rate of change of angle $\theta$ 24 min after passing the radar.

See Solution

1 2 3 4 5

Applications

Problem 1

Given forces A=10.2kN,B=4.08kN,θ=30∘A=10.2 \mathrm{kN}, B=4.08 \mathrm{kN}, \theta=30^{\circ}A=10.2kN,B=4.08kN,θ=30∘, find angle β\betaβ and force CCC for resultant 6.5kN6.5 \mathrm{kN}6.5kN.

Problem 2

Find the force SSS (in kN) so that P=35P = 35P=35 kN, Q=45Q = 45Q=45 kN, and the resultant is 555555 kN to the right.

Problem 3

Find (a) the complement and (b) the supplement of an angle measuring 17∘15′17^{\circ} 15^{\prime}17∘15′.

Problem 4

1. A boat sees a lighthouse foot at 34∘34^{\circ}34∘ elevation; cliff height is 150 m. Find the boat's distance from the cliff base. 2. Flying at 037∘037^{\circ}037∘, turn left 45∘45^{\circ}45∘. What is the new heading?

Problem 5

Convert the angle −51∘30′-51^{\circ} 30^{\prime}−51∘30′ to decimal degree notation.

Problem 6

Convert the angle −75∘3′-75^{\circ} 3^{\prime}−75∘3′ to decimal degrees.

Problem 7

Convert the angle −41∘54′-41^{\circ} 54^{\prime}−41∘54′ to decimal degrees.

Problem 8

Convert the angle −76∘12′-76^{\circ} 12^{\prime}−76∘12′ to decimal degree notation.

Problem 9

Convert the angle −81∘33′-81^{\circ} 33^{\prime}−81∘33′ to decimal degrees.

Problem 10

Sketch the least positive angle θ\thetaθ from the line −7x−3y=0-7x - 3y = 0−7x−3y=0 where x≤0x \leq 0x≤0, and find the six trig functions of θ\thetaθ.

Problem 11

Convert the angle −37∘30′-37^{\circ} 30^{\prime}−37∘30′ to decimal degrees.

Problem 12

John, at 6ft6 \mathrm{ft}6ft high, measures a 35∘35^{\circ}35∘ angle to a tree. After moving 30 ft closer, it's 47∘47^{\circ}47∘. Find the tree's height.

Problem 13

Find the distance between two cars below a 1000-foot cliff with angles of depression 21∘21^{\circ}21∘ and 28∘28^{\circ}28∘.

Problem 14

Find the resultant velocity when crossing a river with a current of 15 m/s15 \mathrm{~m/s}15 m/s downstream and crossing at 2 m/s2 \mathrm{~m/s}2 m/s north.

Problem 15

A car on a 1.7∘1.7^{\circ}1.7∘ incline has a grade resistance of 123 lb123 \, \mathrm{lb}123lb. Find the car's weight in hundreds using: Grade Resistance = Weight * sin(incline). Consider other forces too.

Problem 16

Find the grade resistance for a 2100-pound car on a 0.5∘0.5^{\circ}0.5∘ uphill grade using F=Wsin⁡θF=W \sin \thetaF=Wsinθ. Round to the nearest pound.

Problem 17

Problem 18

Problem 19

A plane is 3.8∘3.8^{\circ}3.8∘ off course after flying 127 miles. How far off course is it? Round to 1 decimal place.

Problem 20

How far is a plane off course after being 3.0∘3.0^{\circ}3.0∘ off track for 187 miles? Round to 1 decimal place.

Problem 21

A ship is anchored off a shore with two points 13 miles apart. Find the shortest distance to the shore, rounding to the nearest tenth.

Problem 22

Carmen sees two hot air balloons. One is 670 m away at 42∘42^{\circ}42∘ and the other 945 m away at 36∘36^{\circ}36∘. Find the height difference.

Problem 23

Find angle θ2\theta_{2}θ2​ using Snell's law: c1c2=sin⁡θ1sin⁡θ2\frac{c_{1}}{c_{2}}=\frac{\sin \theta_{1}}{\sin \theta_{2}}c2​c1​​=sinθ2​sinθ1​​ given c1=3.00×108c_{1}=3.00 \times 10^{8}c1​=3.00×108 m/s and c2=2.251×108c_{2}=2.251 \times 10^{8}c2​=2.251×108 m/s.

Problem 24

Two ships leave a port simultaneously. One sails at 56∘56^{\circ}56∘, 14 knots; the other at 146∘146^{\circ}146∘, 26 knots. Distance apart after 1.5 hours?

Problem 25

Ein Aussichtsturm ist 30,75 m30,75 \mathrm{~m}30,75 m hoch. Die Felswand hat einen Tiefenwinkel von 2,5∘2,5^{\circ}2,5∘ und einen Höhenwinkel von 10,3∘10,3^{\circ}10,3∘. Berechne die Entfernung zur Felswand und deren Höhe.

Problem 26

Amy jogs 3 miles straight, then turns 50 degrees right and runs xxx miles. Find her distance from the start. Round to 0.1.

Problem 27

Karl sieht ein Flugzeug in 10 km10 \mathrm{~km}10 km Höhe. a) Finde die Flugdistanz bei α=60∘\alpha=60^{\circ}α=60∘ und β=25∘\beta=25^{\circ}β=25∘. b) Bestimme die Geschwindigkeit.

Problem 28

Berechne die Steigung in Prozent für die Winkel: c) γ=27,5∘\gamma=27,5^{\circ}γ=27,5∘, a) a=35∘a=35^{\circ}a=35∘, b) β=70∘\beta=70^{\circ}β=70∘.

Problem 29

A soccer player kicks a ball at 27 m/s27 \mathrm{~m/s}27 m/s and 20∘20^{\circ}20∘. Find: (a) air time, (b) impact speed, (c) range.

Problem 30

An airplane flies at 600 km/hr600 \mathrm{~km/hr}600 km/hr with a 50 km/hr50 \mathrm{~km/hr}50 km/hr southwest wind. What angle should it head to go west?

Problem 31

What bearing must the aeroplane fly to return from city G to city F after flying at a bearing of 055∘055^{\circ}055∘?

Problem 32

Ein Sendemast ist 150 m150 \mathrm{~m}150 m hoch. Berechne den Neigungswinkel der 160 m160 \mathrm{~m}160 m langen Seile bei 125 m125 \mathrm{~m}125 m. Dann, bestimme die Länge der Seile, die bei 25∘25^{\circ}25∘ in 60 m60 \mathrm{~m}60 m Höhe befestigt sind.

Problem 33

Find the max weight WWW a block can have if cords support up to 80 lb80 \, \mathrm{lb}80lb at a 30∘30^\circ30∘ angle from vertical.

Problem 34

A crate on an incline has a static friction coefficient of 0.29. Find θ\thetaθ when it slips and its acceleration with kinetic friction 0.26.

Problem 35

A 40.2 kg sign is held by two wires at angles 43.8° and 52.7°. Find the tension in wire 1 and the y-direction acceleration.

Problem 36

A football is kicked at 18 m/s18 \mathrm{~m/s}18 m/s and 65∘65^{\circ}65∘. How long is it in the air? Options: A. 1.1 s1.1 \mathrm{~s}1.1 s B. 4.0 s4.0 \mathrm{~s}4.0 s C. 3.3 s3.3 \mathrm{~s}3.3 s D. 2.0 s2.0 \mathrm{~s}2.0 s E. 1.6 s1.6 \mathrm{~s}1.6 s

Problem 37

A man rows at 8 km/h8 \mathrm{~km/h}8 km/h across a 1.5 km1.5 \mathrm{~km}1.5 km river with a 5 km/h5 \mathrm{~km/h}5 km/h current. Find the heading, time, and downstream distance.

Problem 38

Projectile motion problem: A football is kicked with a velocity of 25 m/s25 \mathrm{~m/s}25 m/s at a 50-degree angle. a. Draw velocity and acceleration motion maps. b. Calculate vxv_xvx​ and vyv_yvy​. c. Determine the time in the air.

Problem 39

A projectile launched at 45 degrees takes 2.4 seconds to land. What was its launch speed? Options: 12 m/s12 \mathrm{~m/s}12 m/s, 17 m/s17 \mathrm{~m/s}17 m/s, 34 m/s34 \mathrm{~m/s}34 m/s, 24 m/s24 \mathrm{~m/s}24 m/s.

Problem 40

Find the right angle in gradians and the radian measure of 150 grads.

Problem 41

Given forces $A=10.2 \mathrm{kN}, B=4.08 \mathrm{kN}, \theta=30^{\circ}$ , find angle $\beta$ and force $C$ for resultant $6.5 \mathrm{kN}$ .

Find the force $S$ (in kN) so that $P = 35$ kN, $Q = 45$ kN, and the resultant is $55$ kN to the right.

Find (a) the complement and (b) the supplement of an angle measuring $17^{\circ} 15^{\prime}$ .

1. A boat sees a lighthouse foot at $34^{\circ}$ elevation; cliff height is 150 m. Find the boat's distance from the cliff base.
2. Flying at $037^{\circ}$ , turn left $45^{\circ}$ . What is the new heading?

Convert the angle $-51^{\circ} 30^{\prime}$ to decimal degree notation.

Convert the angle $-75^{\circ} 3^{\prime}$ to decimal degrees.

Convert the angle $-41^{\circ} 54^{\prime}$ to decimal degrees.

Convert the angle $-76^{\circ} 12^{\prime}$ to decimal degree notation.

Convert the angle $-81^{\circ} 33^{\prime}$ to decimal degrees.

Sketch the least positive angle $\theta$ from the line $-7x - 3y = 0$ where $x \leq 0$ , and find the six trig functions of $\theta$ .

Convert the angle $-37^{\circ} 30^{\prime}$ to decimal degrees.

John, at $6 \mathrm{ft}$ high, measures a $35^{\circ}$ angle to a tree. After moving 30 ft closer, it's $47^{\circ}$ . Find the tree's height.

Find the distance between two cars below a 1000-foot cliff with angles of depression $21^{\circ}$ and $28^{\circ}$ .

Find the resultant velocity when crossing a river with a current of $15 \mathrm{~m/s}$ downstream and crossing at $2 \mathrm{~m/s}$ north.

A car on a $1.7^{\circ}$ incline has a grade resistance of $123 \, \mathrm{lb}$ . Find the car's weight in hundreds using: Grade Resistance = Weight * sin(incline). Consider other forces too.

Find the grade resistance for a 2100-pound car on a $0.5^{\circ}$ uphill grade using $F=W \sin \theta$ . Round to the nearest pound.

A plane is $3.8^{\circ}$ off course after flying 127 miles. How far off course is it? Round to 1 decimal place.

How far is a plane off course after being $3.0^{\circ}$ off track for 187 miles? Round to 1 decimal place.

Carmen sees two hot air balloons. One is 670 m away at $42^{\circ}$ and the other 945 m away at $36^{\circ}$ . Find the height difference.

Find angle $\theta_{2}$ using Snell's law: $\frac{c_{1}}{c_{2}}=\frac{\sin \theta_{1}}{\sin \theta_{2}}$ given $c_{1}=3.00 \times 10^{8}$ m/s and $c_{2}=2.251 \times 10^{8}$ m/s.

Two ships leave a port simultaneously. One sails at $56^{\circ}$ , 14 knots; the other at $146^{\circ}$ , 26 knots. Distance apart after 1.5 hours?

Ein Aussichtsturm ist $30,75 \mathrm{~m}$ hoch. Die Felswand hat einen Tiefenwinkel von $2,5^{\circ}$ und einen Höhenwinkel von $10,3^{\circ}$ . Berechne die Entfernung zur Felswand und deren Höhe.

Amy jogs 3 miles straight, then turns 50 degrees right and runs $x$ miles. Find her distance from the start. Round to 0.1.

Karl sieht ein Flugzeug in $10 \mathrm{~km}$ Höhe. a) Finde die Flugdistanz bei $\alpha=60^{\circ}$ und $\beta=25^{\circ}$ . b) Bestimme die Geschwindigkeit.

Berechne die Steigung in Prozent für die Winkel: c) $\gamma=27,5^{\circ}$ , a) $a=35^{\circ}$ , b) $\beta=70^{\circ}$ .

A soccer player kicks a ball at $27 \mathrm{~m/s}$ and $20^{\circ}$ . Find: (a) air time, (b) impact speed, (c) range.

An airplane flies at $600 \mathrm{~km/hr}$ with a $50 \mathrm{~km/hr}$ southwest wind. What angle should it head to go west?

What bearing must the aeroplane fly to return from city G to city F after flying at a bearing of $055^{\circ}$ ?

Ein Sendemast ist $150 \mathrm{~m}$ hoch. Berechne den Neigungswinkel der $160 \mathrm{~m}$ langen Seile bei $125 \mathrm{~m}$ . Dann, bestimme die Länge der Seile, die bei $25^{\circ}$ in $60 \mathrm{~m}$ Höhe befestigt sind.

Find the max weight $W$ a block can have if cords support up to $80 \, \mathrm{lb}$ at a $30^\circ$ angle from vertical.

A crate on an incline has a static friction coefficient of 0.29. Find $\theta$ when it slips and its acceleration with kinetic friction 0.26.

A football is kicked at $18 \mathrm{~m/s}$ and $65^{\circ}$ . How long is it in the air? Options: A. $1.1 \mathrm{~s}$ B. $4.0 \mathrm{~s}$ C. $3.3 \mathrm{~s}$ D. $2.0 \mathrm{~s}$ E. $1.6 \mathrm{~s}$

A man rows at $8 \mathrm{~km/h}$ across a $1.5 \mathrm{~km}$ river with a $5 \mathrm{~km/h}$ current. Find the heading, time, and downstream distance.

Projectile motion problem: A football is kicked with a velocity of $25 \mathrm{~m/s}$ at a 50-degree angle.
a. Draw velocity and acceleration motion maps. b. Calculate $v_x$ and $v_y$ . c. Determine the time in the air.

A projectile launched at 45 degrees takes 2.4 seconds to land. What was its launch speed? Options: $12 \mathrm{~m/s}$ , $17 \mathrm{~m/s}$ , $34 \mathrm{~m/s}$ , $24 \mathrm{~m/s}$ .

London Eye problem:
a) Find the angle in radians for 5 min. b) Calculate distance traveled in 5 min. c) Time for 2 radians? d) Angular velocity in radians/sec? e) Angular velocity in degrees/sec?

Ceiling fan with 14-inch blades spins at 46 RPM.
(a) Calculate angular speed in radians/min. (b) Calculate linear speed of blade tip in feet/sec.

Berechne den Schnittwinkel der Graphen $f$ und $g$ für die Anstiege: a) $m_{f}=3$ , $m_{g}=1$ ; b) $m_{f}=3$ , $m_{g}=-1$ ; c) $m_{f}=-2$ , $m_{g}=-0,5$ ; d) $m_{f}=\sqrt{3}$ , $m_{g}=1$ . Runde auf Zehntel.

Two planes leave an airport simultaneously. Plane 1: $660 \mathrm{~m/h}$ at $12.3^{\circ}$ , Plane 2: $580 \mathrm{~m/h}$ at $148^{\circ}$ . Find distance apart after $3.3 \mathrm{~h}$ in $\mathrm{m}$ .

Find the complement: $90^{\circ} - 27^{\circ} 35^{\prime} 15^{\prime \prime} = 62^{\circ} 24^{\prime} 45^{\prime \prime}$

Find the supplement of an angle measuring $27^{\circ} 35^{\prime} 15^{\prime \prime}$ .

Bestimme die Modellfunktion für eine Kugel, die aus $1,80 \mathrm{~m}$ Höhe mit $42^{\circ}$ abgeworfen wird, bei $8,40 \mathrm{~m}$ Stoßweite. Finde die maximale Höhe und den Aufprallwinkel.

Find the height of a building 40 feet away, with angles of elevation $62^{\circ}$ and depression $55^{\circ}$ . Round to nearest foot.

Hikers observe a mountain with a $35^{\circ}$ elevation from 1250 m away. Find slant heights using slopes of $48^{\circ}$ and $65^{\circ}$ .

A projectile is launched at $11 \mathrm{~m/s}$ at $30^{\circ}$ from vertical. Find time to max height. Options: (a) $0.97 \mathrm{~s}$ (b) $0.56 \mathrm{~s}$ (c) $0.79 \mathrm{~s}$ (d) $1.12 \mathrm{~s}$ .

A jet traveling at $651 \mathrm{mph}$ east encounters a $516 \mathrm{mph}$ wind at $35^{\circ}$ north of east. Find its new speed in mph.

A ball is launched at $38.0 \, \text{m/s}$ at $60.0^{\circ}$ . Find: a) horizontal velocity, b) horizontal distance, c) max height.

What force is needed to pull a $56.5 \mathrm{~N}$ weight up a $12.6^{\circ}$ slope, ignoring friction?

Two ropes pull a crate: one with $994 \mathrm{~N}$ at $15.5^{\circ}$ and another with $624 \mathrm{~N}$ at $25.2^{\circ}$ . Find the crate's weight in newtons.

Find the resistance and reactance of a circuit with an impedance of $975 \mathrm{ohm}$ and a phase angle of 28.0 degrees (3 SIG FIGS).

In a snowball fight, if one snowball is thrown at $61^{\circ}$ with speed $11 \mathrm{~m/s}$ , at what angle should the second be thrown?

A boat moves east at $6.16 \mathrm{~m/s}$ while the river flows north at $2.34 \mathrm{~m/s}$ . Find the resultant speed and angle.

In a snowball fight, if one snowball is thrown at $69^{\circ}$ with speed $18 \mathrm{~m/s}$ , what angle should the second one be thrown at to hit the same point? Use $g = 9.8 \mathrm{~m/s}^{2}$ .

In a snowball fight, if one snowball is thrown at $60^{\circ}$ with speed $34.4 \mathrm{~m/s}$ , what angle should the second snowball be thrown at to hit the same point? Use $g = 9.8 \mathrm{~m/s}^{2}$ .

Find the angle between vectors $v = 2i + 5j$ and $w = -7i + 8j$ . Round to one decimal place if needed.

Austin and Bill are $103 \mathrm{~m}$ apart. A roller coaster car moves at $5 \mathrm{~m/s}$ at $33^{\circ}$ . Find distance $r$ after 16 seconds. Round to the nearest tenth.

Austin and Bill are $103 \mathrm{~m}$ apart. A roller coaster car moves at $5 \mathrm{~m/s}$ at a $33^{\circ}$ angle. Find remaining distance $r$ to the highest point after 16 seconds. Round to the nearest tenth of a meter.

已知单位向量 $\vec{a}, \vec{b}$ ， $|2 \vec{a}+\vec{b}|=\sqrt{3}$ ，求向量 $\vec{a}, \vec{b}$ 的夹角。

Ein Segelflugzeug ist in $1250 \mathrm{~m}$ Höhe, $42 \mathrm{~km}$ vom Ziel entfernt und hat einen Gleitwinkel von $1,5^{\circ}$ . Bestimme die Höhe, in der es das Ziel überfliegt.

Find the angle of the resultant force with the horizontal when two forces of 17.3 lbs and 42.5 lbs act at 21.9°. Options: $6.3^{\circ}$ , $21.9^{\circ}$ , $15.7^{\circ}$ , $10.9^{\circ}$ .

A plane flies at $290 \mathrm{mph}$ , 24.5° South of West, with a $30.5 \mathrm{mph}$ wind, 21° East of North. Find ground speed.

Find the angle between vectors $v = 3i + 2j + k$ and $w = 6i + 8j + 2k$ . Round to one decimal place.

Find the angle between vectors $v = 3i + 2j + k$ and $w = 6i + 8j + 2k$ . Round to one decimal place.

A sphere hangs from a string in a van. If $\theta=10.7^{\circ}$ , find the van's acceleration.

A climber weighing $515 \mathrm{~N}$ rests on a rope between cliffs. Angles are $65^\circ$ (left) and $80^\circ$ (right). Find tensions.

Jahi "Tornado" liigub põhjasuunas, hoides nurka $30^{\circ}$ . Leidke läänetuule kiirus, kui kiirus on 18,52 $\frac{km}{h}$ . Vastus $\frac{m}{s}$ .