Finding Missing Angles Worksheet with Answer Key (Free, Printable)

1.

If

\sin(\theta) = 0.5

, what is

\theta

?

a) $90°$
b) $30°$
c) $60°$
d) $45°$

Answer: $30°$

$\sin^{-1}(0.5) = 30°$ . This is one of the special angle values that you should memorize. The other options are common mistakes: $\sin(45°) \approx 0.707$ , $\sin(60°) \approx 0.866$ , $\sin(90°) = 1$ .

2.

Which inverse function should you use if you know the opposite and adjacent sides?

a) $\cos^{-1}$
b) $\tan^{-1}$
c) $\sin^{-1}$
d)Pythagorean Theorem

Answer: $\tan^{-1}$

When you know opposite and adjacent, use $\tan^{-1}$ because $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ . Sin uses opposite/hypotenuse, cos uses adjacent/hypotenuse.

3.

Calculate:

\tan^{-1}(1)

= ?° (Give your answer in degrees)

Answer: 45

$\tan^{-1}(1) = 45°$ because in a 45-45-90 triangle, the two legs are equal, so opposite/adjacent = 1.

4.

If

\cos(\theta) = 0.5

, what is

\theta

? (Answer in degrees)

Answer: 60

$\cos^{-1}(0.5) = 60°$ . Note that $\cos(30°) = \frac{\sqrt{3}}{2} \approx 0.866$ , while $\cos(60°) = 0.5$ .

5.

In a right triangle, the opposite side is 4 and the hypotenuse is 5. What is the angle? (Round to the nearest degree)

a) $45°$
b) $37°$
c) $53°$
d) $39°$

Answer: $53°$

$\sin(\theta) = \frac{4}{5} = 0.8$ , so $\theta = \sin^{-1}(0.8) \approx 53°$ . The 3-4-5 triangle is a classic! The other acute angle is $37°$ .

6.

A ramp rises 3 meters over a horizontal distance of 8 meters. Find the angle of inclination.

Answer: 20.6

What is the vertical rise (opposite)? 3
What is the horizontal distance (adjacent)? 8
Which trig function uses opposite and adjacent? tangent
Calculate $\tan^{-1}(3/8)$ to the nearest tenth of a degree. 20.6

7.

A ladder 10 meters long leans against a wall. The foot of the ladder is 6 meters from the wall. Find the angle between the ladder and the ground.

Answer: 53.1

What is the length of the ladder (hypotenuse)? 10
What is the distance from wall to ladder foot (adjacent)? 6
Which trig function uses adjacent and hypotenuse? cosine
Calculate $\cos^{-1}(6/10)$ to the nearest tenth of a degree. 53.1

8.

A building 21 meters tall casts a shadow 25 meters long on level ground. What is the angle of elevation of the sun? (Round to the nearest degree)

Answer: 40

$\tan(\theta) = \frac{21}{25} = 0.84$ , so $\theta = \tan^{-1}(0.84) \approx 40.0°$ . The sun is about $40°$ above the horizon.

9.

What is

\sin^{-1}(\sin(150°))

?

a) $-30°$
b) $30°$
c) $210°$
d) $150°$

Answer: $30°$

$\sin(150°) = 0.5$ , and $\sin^{-1}(0.5) = 30°$ . The inverse function returns the principal value in $[-90°, 90°]$ , not the original angle!

10.

A plane is at altitude 8000 feet and must descend to land, starting 30000 feet horizontally from the runway. Find the descent angle to the nearest tenth of a degree.

Answer: 14.9

What is the altitude (opposite)? 8000
What is the horizontal distance (adjacent)? 30000
Set up the fraction opposite/adjacent. 8000/30000
Calculate $\tan^{-1}(8000/30000)$ to the nearest tenth of a degree. 14.9

11.

In a right triangle, one leg is 7 and the other is 24. Find both acute angles (to the nearest tenth of a degree).

Answer: 16.3 and 73.7

Set up $\tan(\alpha)$ with angle $\alpha$ opposite the side of 7. 7/24
Calculate $\alpha = \tan^{-1}(7/24)$ to the nearest tenth of a degree. 16.3
How do you find the other acute angle $\beta$ ? 90 - 16.3
What is $\beta$ ? 73.7

12.

A surveyor stands 150 meters from the foot of a cliff that is 121.5 meters high. Her measuring instrument is 1.5 meters above the ground. What is the angle of elevation from the instrument to the top of the cliff? (Round to the nearest tenth of a degree)

Answer: 38.7

Opposite $= 121.5 - 1.5 = 120$ m, adjacent $= 150$ m. $\tan(\theta) = \frac{120}{150} = 0.8$ , so $\theta = \tan^{-1}(0.8) \approx 38.7°$ . If you forget the instrument height, you get $\tan^{-1}\left(\frac{121.5}{150}\right) \approx 39.0°$ , which is wrong.

13.

A guy wire is attached to a pole 12 meters up from the ground and anchored 5 meters from the base. Find the angle the wire makes with the ground.

Answer: 67.4

What is the height (opposite to ground angle)? 12
What is the ground distance (adjacent)? 5
Set up the tangent ratio. 12/5
Calculate $\tan^{-1}(12/5)$ to the nearest tenth. 67.4

14.

Why does

\sin^{-1}(1.5)

give an error on your calculator?

a)Sine values only range from $-1$ to $1$
b)The angle is too large
c)You need to use arcsin instead
d)The calculator is in radian mode

Answer: Sine values only range from $-1$ to $1$

The sine function only outputs values between $-1$ and $1$ . Since $1.5 > 1$ , there is no angle whose sine equals $1.5$ . The domain of $\sin^{-1}$ is $[-1, 1]$ .

15.

A roof has a pitch of 8/12 (rises 8 feet for every 12 feet horizontal). What is the roof angle to the nearest degree?

Answer: 34

What is the rise (opposite)? 8
What is the run (adjacent)? 12
The pitch is rise/run. What is this value, rounded to three decimal places? 0.667
Calculate $\tan^{-1}(8/12)$ and round to the nearest degree. 34