Introduction to Trigonometric Ratios

Show your work for each problem.

1.
In SOH-CAH-TOA, what does 'SOH' stand for?
1. a)Sine = Opposite / Hypotenuse
2. b)Sine = Hypotenuse / Opposite
3. c)Sine = Opposite / Adjacent
4. d)Sine = Adjacent / Hypotenuse
Answer: Sine = Opposite / Hypotenuse
SOH means Sine = Opposite / Hypotenuse. The mnemonic helps remember that sine is the ratio of the opposite side to the hypotenuse.
2.
Which side of a right triangle is always the longest?
1. a)Adjacent
2. b)Opposite
3. c)It depends on the angle
4. d)Hypotenuse
Answer: Hypotenuse
The hypotenuse is always the longest side of a right triangle. It is located opposite the right angle (90 degrees).
3.
In a right triangle, if the opposite side is 3 and the hypotenuse is 5, what is ? Express as a decimal.
Answer: 0.6
4.
In a right triangle, if the adjacent side is 4 and the hypotenuse is 5, what is ? Express as a decimal.
Answer: 0.8
5.
Which trigonometric ratio uses the opposite and adjacent sides (not the hypotenuse)?
1. a)Tangent
2. b)Secant
3. c)Cosine
4. d)Sine
Answer: Tangent
Tangent (TOA) = Opposite / Adjacent. It's the only primary ratio that doesn't include the hypotenuse.
6.
In a right triangle, if the opposite side is 5 and the adjacent side is 12, what is ? Express as a fraction (e.g., 5/12).
Answer: 5/12
7.
A right triangle has sides 5, 12, and 13. Find all three trigonometric ratios for the angle opposite the side of length 5.
Answer: sin = 5/13, cos = 12/13, tan = 5/12
1. Which side is the hypotenuse? 13
2. Which side is opposite to our angle? 5
3. Which side is adjacent to our angle? 12
4. Calculate sin (opposite/hypotenuse) 5/13
8.
If , what can you conclude about the opposite side and hypotenuse?
1. a)The opposite is twice the hypotenuse
2. b)The opposite equals the hypotenuse
3. c)The adjacent is half the hypotenuse
4. d)The opposite is half the hypotenuse
Answer: The opposite is half the hypotenuse
Since , the opposite side is half the length of the hypotenuse.
9.
In a right triangle, the opposite side is 8 and the hypotenuse is 10. Find all three trigonometric ratios.
Answer: sin = 0.8, cos = 0.6, tan = 4/3
1. Calculate sin (opposite/hypotenuse) as a decimal 0.8
2. Use Pythagorean theorem to find adjacent: sqrt(10^2 - 8^2) 6
3. Calculate cos (adjacent/hypotenuse) as a decimal 0.6
4. Calculate tan (opposite/adjacent) as a fraction 8/6
10.
If , what must be true about the opposite and adjacent sides?
Answer: equal
If , then opposite = adjacent. The sides must be equal in length.
11.
A ladder 10 meters long leans against a wall. The base is 6 meters from the wall. Find the sine of the angle the ladder makes with the ground.
Answer: 0.8
1. What is the hypotenuse in this problem? 10
2. What is the adjacent side (distance from wall)? 6
3. Use Pythagorean theorem to find height: sqrt(10^2 - 6^2) 8
4. Calculate sin (height/ladder length) 0.8
12.
For which angle is ?
1. a)45 degrees
2. b)90 degrees
3. c)60 degrees
4. d)30 degrees
Answer: 45 degrees
At 45 degrees, the opposite and adjacent sides are equal, so , meaning
13.
In a right triangle, . Find and .
Answer: cos = 12/13, tan = 5/12
1. From sin = 5/13, what is the opposite side? 5
2. What is the hypotenuse? 13
3. Find adjacent using Pythagorean theorem: sqrt(13^2 - 5^2) 12
4. Calculate cos (adjacent/hypotenuse) 12/13
14.
If and the hypotenuse is 15, what is the length of the adjacent side?
Answer: 9
Adjacent =
15.
A ramp rises 3 meters over a horizontal distance of 4 meters. Find the sine, cosine, and tangent of the angle the ramp makes with the ground.
Answer: sin = 0.6, cos = 0.8, tan = 0.75
1. What is the rise (opposite side)? 3
2. What is the run (adjacent side)? 4
3. Find hypotenuse: sqrt(3^2 + 4^2) 5
4. Calculate tan (rise/run) 0.75