Deck 8: Triangle Trigonometry and Polar Coordinates

Question 1

Find the following exactly using the figure given if $a=4$ and $b=5$ . Express your answers as unsimplified radicals when appropriate.

A) $\sin A^{\circ}$
B) $\cos A^{\circ}$
C) $\tan A^{\circ}$
D) $\sin B^{\circ}$
E) $\cos B^{\circ}$
F) $\tan B^{\circ}$

Answer 1

Part A: $\sin A^{\circ}=\frac{4}{\sqrt{41}}$
Part B: $\cos A^{\circ}=\frac{5}{\sqrt{41}}$
Part C: $\tan A^{\circ}=\frac{4}{5}$
Part D: $\sin B^{\circ}=\frac{5}{\sqrt{41}}$
Part E: $\cos B^{\circ}=\frac{4}{\sqrt{41}}$
Part F: $\tan B^{\circ}=\frac{5}{4}$

Question 2

Find the following exactly using the figure given if $a=2$ and $c=13$ . Express your answers as unsimplified radicals when appropriate.
A) $\sin A^{\circ}$
B) $\cos A^{\circ}$
C) $\tan A^{\circ}$
D) $\sin B^{\circ}$
E) $\cos B^{\circ}$
F) $\tan B^{\circ}$

Answer 2

Part A: $\sin A^{\circ}=\frac{2}{13}$
Part B: $\cos A^{\circ}=\frac{\sqrt{165}}{13}$
Part C: $\tan A^{\circ}=\frac{2}{\sqrt{165}}$
Part D: $\sin B^{\circ}=\frac{\sqrt{165}}{13}$
Part E: $\cos B^{\circ}=\frac{2}{13}$
Part F: $\tan B^{\circ}=\frac{\sqrt{165}}{2}$

Question 3

If $A=29^{\circ}, B=61^{\circ}$ , and $c=7$ , find exact values for $a$ and $b$ . Express your answers in terms of $\sin$ and $\cos$ .

Answer 3

Part A: $a=7 \sin \left(29^{\circ}\right)$
Part B: $b=7 \cos \left(29^{\circ}\right)$

Question 4

If $A=29^{\circ}$ and $a=2$ , find exact values for $b$ and $c$ .

Question 5

A rope is attached to the top of a tree and to a stake in the ground. The angle of the rope with the ground is $33^{\circ}$ and the rope comes out 150 feet from the tree. How tall is the tree? Give your answer correct to 2 decimal places.

Question 6

Find the following exactly using the figure given if $A=B$ and $b=3$ . Express your answers as unsimplified radicals when appropriate.
A) $\sin A^{\circ}$
B) $\cos A^{\circ}$
C) $\tan A^{\circ}$
D) $\sin B^{\circ}$
E) $\cos B^{\circ}$
F) $\tan B^{\circ}$

Question 7

If $A=22^{\circ}$ and $c=7$ , find two different expressions for the exact value of $a$ .

Question 8

If $a=4$ and $c=7$ , find the following exactly. Express your answers as unsimplified radicals when appropriate.
A) $\sin A^{\circ}$
B) $\cos A^{\circ}$
C) $\tan A^{\circ}$

Question 9

Find the area of the triangle given if $c=9$ and $A=37^{\circ}$ . Give your answer correct to 3 decimal places.

Question 10

An ant walks up a ramp at an incline of $11^{\circ}$ . If, after 2 minutes, his vertical distance from the ground is 14 inches, how far has the ant traveled in the horizontal direction? Round your answer to 2 decimal places.

Question 11

An ant walks up a ramp at an incline of $12^{\circ}$ . If, after 2 minutes, his vertical distance from the ground is 12 inches, how fast was the ant travelling in inches per minute? Round your answer to 2 decimal places.

Question 12

If $a=2$ and $c=7$ , find the following exactly.
A) $\sin A^{\circ}$
B) $\cos A^{\circ}$
C) $\tan A^{\circ}$

Question 13

If $a=8$ and $b=2$ , find the following. Give your answers to 3 decimal places.
A)c
B)A
C)B

Question 14

Let $a=7$ and $A=25^{\circ}$ , find the following. Give your answers to 3 decimal places.
A) $c$
B) $b$
C) $B$

Question 15

Let $c=7$ and $A=28$ , find the following. Give your answers to 3 decimal places.
A) $a$
B) $b$
C) $B$

Question 16

A 69 inch tall man casts a 101 inch long shadow. What is the angle $\theta$ of the sun? Give your answer to 3 decimal places.

Question 17

The grade of the road is $8.3 \%$ . What angle does the road make with the horizontal? Round your answer to 3 decimal places.

Question 18

A ramp is 109 feet long and the change in height of the ramp is 4 feet. What angle does the ramp form with the horizontal? Give your answer correct to 3 decimal places.

Question 19

If $\sin \theta=0.993$ , find $\theta$ in degrees. Round to the nearest degree.

Question 20

If $\cos \theta=0.407$ , find $\theta$ in degrees. Round to the nearest degree.

Question 21

If $\tan \theta=3.271$ , find $\theta$ in degrees. Round to the nearest degree.

Question 22

Find $\sin \theta=60^{\circ}$ exactly.

Question 23

Find $\cos \theta=150^{\circ}$ exactly.

Question 24

Find the area of the following triangle if $a=13, b=10$ , and $A=87^{\circ}$ . Round to two decimal places.

Question 25

Find the area of a triangle with angles $110^{\circ}, 21^{\circ}$ , and $49^{\circ}$ if the length of the side opposite the $110^{\circ}$ angle is 8 . Round to 2 decimal places.

Question 26

Two airplanes are headed toward the same airport. The first plane is flying $151 \mathrm{mph}$ and the second plane is flying $134 \mathrm{mph}$ . The angle between the planes with its vertex at the airport is $59^{\circ}$ . If both planes reach the airport in 2 hours, how many miles apart are they now? Round to 2 decimal places.

Question 27

Use the figure below to find the missing side, $c$ , when $a=7, b=4$ , and $C=42^{\circ}$ . Round to 4 decimal places.

Question 28

In the figure below, when $a=12, b=9$ , and $C=43^{\circ}$ , we have (A= ---------°). Round to 2 decimal places. If the triangle is not possible, enter "not possible".

Question 29

Use the figure below to find side $b$ when $A=65^{\circ}, B=49^{\circ}$ , and $c=12$ . Round to 2 decimal places.

Question 30

Use the figure below to find angle $C$ when $b=13$ , and $c=14$ , and $B=95^{\circ}$ . Round to 2 decimal places. If the triangle is not possible, enter "not possible".

Question 31

In the revolving door in the figure below, each panel is 1 meter long. How many meters is the distance along a straight line between $A$ and $D$ ? Round to 2 decimal places.

Question 32

The Pentagon proposes to install a 5 panel revolving door with a panel width of 0.9 meters. A 5-star general wants to enter with a chart that is as wide as possible. How many meters wide is the widest chart that he can fit in the opening without bending it? In other words, how many meters is the distance across the opening? Round to 2 decimal places.

Question 33

Find $A$ and $c$ if $a=7, b=13$ , and $B=12^{\circ}$ . Round to two decimal places.

Question 34

Find the area of the triangle if $a=6, b=12$ , and $B=13^{\circ}$ . Round to two decimal places.

Question 35

To measure the height of a mountain, a person stands a certain distance from the base and measures the angle of elevation to the top of the mountain. This turns out to be $69^{\circ}$ . Moving $100 \mathrm{ft}$ closer, the angle of elevation to the top of the mountain is $74^{\circ}$ . How tall is the mountain? Round your answer to 2 decimal places.

Question 36

Two ships leave port and head out to sea in straight lines at $1 \mathrm{pm}$ . It is noted that they are travelling away from each other at an angle of $41^{\circ}$ . Ship A travels at 15 miles per hour and Ship B travels at 16 miles per hour. After how many hours are they 82 miles apart?

Question 37

Consider the triangle formed by the points $A(1,0), B(0,0)$ , and $C(7,4)$ . Find the measure of angle $A B C$ . Give your answer to 2 decimal places.

Question 38

Let $A(1,0), B(0,0)$ , and $C(8,3)$ . Find the area of triangle $A B C$ .

Question 39

Let $A(1,0), B(3,5)$ , and $C(0,-1)$ . Find the measure of angle $A B C$ . Give your answer to 2 decimal places.

Question 40

Find the area of the triangle if $a=6, b=9$ , and $c=5$ . Round to two decimal places.

Question 41

Find the measures of the angles of the triangle if $a=6, b=8$ , and $c=4$ . Round to two decimal places.

Question 42

In the following figure, suppose $c=1.2, b=2.2$ , and $C=22^{\circ}$ . Find $a$ , and round your answer to two decimal places. If there are two possible answers, list both.

Question 43

In the following figure, suppose $c=1.3, b=2.3$ , and $C=23^{\circ}$ . Find $A$ , and round your answer to two decimal places. If there are two possible answers, list both.