Finding Missing Angles

Show your work for each problem.

1.
If , what is ?
1. a)
2. b)
3. c)
4. d)
Answer:
. This is one of the special angle values that you should memorize. The other options are common mistakes: , , .
2.
Which inverse function should you use if you know the opposite and adjacent sides?
1. a)
2. b)
3. c)
4. d)Pythagorean Theorem
Answer:
When you know opposite and adjacent, use because . Sin uses opposite/hypotenuse, cos uses adjacent/hypotenuse.
3.
Calculate: = ?° (Give your answer in degrees)
Answer: 45
because in a 45-45-90 triangle, the two legs are equal, so opposite/adjacent = 1.
4.
If , what is ? (Answer in degrees)
Answer: 60
. Note that , while .
5.
In a right triangle, the opposite side is 4 and the hypotenuse is 5. What is the angle? (Round to the nearest degree)
1. a)
2. b)
3. c)
4. d)
Answer:
, so . The 3-4-5 triangle is a classic! The other acute angle is .
6.
A ramp rises 3 meters over a horizontal distance of 8 meters. Find the angle of inclination.
Answer: 20.6
1. What is the vertical rise (opposite)? 3
2. What is the horizontal distance (adjacent)? 8
3. Which trig function uses opposite and adjacent? tangent
4. Calculate 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
1. What is the length of the ladder (hypotenuse)? 10
2. What is the distance from wall to ladder foot (adjacent)? 6
3. Which trig function uses adjacent and hypotenuse? cosine
4. Calculate to the nearest tenth of a degree. 53.1
8.
A building casts a shadow 25 meters long. If the angle of elevation of the sun is , how tall is the building? (Round to the nearest meter)
Answer: 21
Height meters. This uses forward trig (not inverse), but understanding angles helps!
9.
What is ?
1. a)
2. b)
3. c)
4. d)
Answer:
, and . The inverse function returns the principal value in , 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
1. What is the altitude (opposite)? 8000
2. What is the horizontal distance (adjacent)? 30000
3. Set up the fraction opposite/adjacent. 8000/30000
4. Calculate in degrees. 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
1. Set up with angle opposite the side of 7. 7/24
2. Calculate . 16.3
3. How do you find the other acute angle ? 90 - 16.3
4. What is ? 73.7
12.
A surveyor measures that a cliff is 150 meters away horizontally and the angle of elevation to the top is . How high is the cliff? (Round to the nearest meter)
Answer: 105
Height meters. This uses forward trig but reinforces the relationship between angles and sides.
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
1. What is the height (opposite to ground angle)? 12
2. What is the ground distance (adjacent)? 5
3. Set up the tangent ratio. 12/5
4. Calculate to the nearest tenth. 67.4
14.
Why does give an error on your calculator?
1. a)Sine values only range from to
2. b)The angle is too large
3. c)You need to use arcsin instead
4. d)The calculator is in radian mode
Answer: Sine values only range from to
The sine function only outputs values between and . Since , there is no angle whose sine equals . The domain of is .
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
1. What is the rise (opposite)? 8
2. What is the run (adjacent)? 12
3. The pitch is expressed as rise/run. What is this value? 0.667
4. Calculate and round to the nearest degree. 34