Sine Ratio (SOH)

Show your work for each problem.

1.
In a right triangle, the sine of an angle equals:
1. a)Hypotenuse / Opposite
2. b)Opposite / Hypotenuse
3. c)Opposite / Adjacent
4. d)Adjacent / Hypotenuse
Answer: Opposite / Hypotenuse
SOH stands for Sine = Opposite / Hypotenuse. The sine of an angle is always the ratio of the opposite side to the hypotenuse.
2.
What is ?
1. a)
2. b)
3. c)
4. d)
Answer:
. This is a special angle value derived from a 30-60-90 triangle where the opposite side is half the hypotenuse.
3.
What is ?
Answer: 1
. This is the maximum value of sine because at 90 degrees, the opposite side is equal to the hypotenuse, giving us .
4.
In a right triangle, the opposite side is and the hypotenuse is . What is ?
Answer: 0.5
. This corresponds to a 30° angle.
5.
What is ?
1. a)
2. b)
3. c)
4. d)
Answer:
. In a 30-60-90 triangle, the side opposite the 60° angle is times the short leg.
6.
A ladder is meters long and makes a angle with the ground. How high does it reach on the wall? (Round to 2 decimal places)
Answer: 5.66
Height = hypotenuse × sin(45°) = 8 × 0.707 ≈ 5.66 meters. Since sin(45°) = , the opposite side is meters.
7.
A rope attached to the top of a meter pole makes a angle with the ground. How long is the rope? (Use sin(50°) ≈ 0.766)
Answer: 19.58
1. What is the opposite side in this problem? 15
2. What is the unknown we're looking for? hypotenuse
3. Write the formula: hypotenuse = opposite ÷ sin(θ). What is 15 ÷ 0.766? 19.58
8.
An airplane climbs at a angle. After traveling meters along its flight path, how much altitude has it gained? (Use sin(20°) ≈ 0.342)
Answer: 171
1. What is the hypotenuse (flight path distance)? 500
2. What are we finding? (the altitude gained) opposite
3. Calculate: opposite = 500 × 0.342 = ? 171
9.
Which formula would you use to find the hypotenuse when you know the opposite side and the angle?
1. a)hypotenuse = opposite + sin(θ)
2. b)hypotenuse = sin(θ) ÷ opposite
3. c)hypotenuse = opposite × sin(θ)
4. d)hypotenuse = opposite ÷ sin(θ)
Answer: hypotenuse = opposite ÷ sin(θ)
Starting with sin(θ) = opposite/hypotenuse, multiply both sides by hypotenuse to get hypotenuse × sin(θ) = opposite, then divide both sides by sin(θ) to get hypotenuse = opposite ÷ sin(θ).
10.
A ramp has a height of meters and a length (hypotenuse) of meters. What angle does the ramp make with the ground? (Round to the nearest degree)
Answer: 14
1. Calculate sin(θ) = opposite/hypotenuse = 2/8 = ? 0.25
2. To find the angle, use inverse sine: θ = sin⁻¹(0.25). What is the result in degrees? 14
11.
A kite string is meters long and makes a angle with the ground. How high is the kite flying? (Use sin(35°) ≈ 0.574, round to 1 decimal place)
Answer: 28.7
Height = 50 × sin(35°) = 50 × 0.574 = 28.7 meters. The kite is flying at approximately 28.7 meters above the ground.
12.
A surveyor measures an angle of elevation of to the top of a building. The surveyor is standing meters from the base. Using sin and the Pythagorean theorem, find the height of the building. (Use sin(28°) ≈ 0.469, cos(28°) ≈ 0.883)
Answer: 21.2
1. The 40 meters is the adjacent side. What is the hypotenuse? Use hyp = adjacent / cos(28°) = 40 / 0.883 45.3
2. Now find the height using sin(28°): height = hypotenuse × sin(28°) = 45.3 × 0.469 21.2
13.
In a right triangle, the opposite side is cm and the hypotenuse is cm. Find the angle θ. (Round to the nearest degree)
Answer: 37
1. Calculate sin(θ) = opposite/hypotenuse = 6/10 = ? 0.6
2. Find θ = sin⁻¹(0.6). What is the angle in degrees? 37
14.
What is the value of ? (Give your answer as a decimal rounded to 3 decimal places)
Answer: 0.707
. This value comes from a 45-45-90 isosceles right triangle.
15.
If , which of the following could be the value of θ?
1. a)Approximately 60°
2. b)Approximately 53°
3. c)Approximately 37°
4. d)Approximately 45°
Answer: Approximately 53°
. Since sin(45°) ≈ 0.707 and sin(60°) ≈ 0.866, a sine of 0.8 falls between these, closer to 60°, which is approximately 53°.