Trigonometry basics: SOH CAH TOA

Mathematics · Geometry · ages 13-15

Name ______________________ Date ____________

In a right triangle, you pick one of the two acute angles. The side that touches your angle, but is not the hypotenuse, has which name?
- The opposite side
- The adjacent side
- The hypotenuse
Which fraction is tan of the angle?
- adjacent over hypotenuse
- opposite over adjacent
- opposite over hypotenuse
Two right triangles are similar. The smaller has opposite 3 and hypotenuse 5. The larger has hypotenuse 15. What is sin of the matching angle in the larger triangle?
- 0.6
- 0.2
- 0.9
Which of these can be the value of sin θ for an acute angle θ in a right triangle?
- 1.5
- 0.75
- 2
Two ramps meet level ground at the same angle θ, where sin θ = 0.5. Ramp A is 10 m long and ramp B is 6 m long. What is the difference in their rises, in metres?
Answer: ______________
A right triangle has an opposite side of 4 m and a hypotenuse of 8 m. Which calculation finds the angle?
- sin⁻¹(4/8)
- sin(4/8)
- 4 ÷ 8
A surveyor's eye is 1.6 m above level ground. She sights the top of a vertical pole at 30° above horizontal from a horizontal distance of 20 m. Since cos(30°) ≈ 0.866 and sin(30°) = 0.5, how tall is the pole, in metres? Give your answer to one decimal place.
Answer: ______________
In a right triangle, the opposite side is 6 m and the hypotenuse is 12 m, so sin θ = 0.5. Since sin⁻¹(0.5) = 30°, what is θ, in degrees?
Answer: ______________
Omar knows the adjacent side is 5 and the hypotenuse is 10. To find the angle, he wrote sin⁻¹(5/10). What should he fix?
- He should compute sin(5/10) instead
- He should use cos⁻¹, since cos connects adjacent and hypotenuse
- He should flip the fraction to 10/5 and keep sin⁻¹
A right triangle has opposite side 6 and adjacent side 8. Zara writes tan θ = 6/8. Omar finds the hypotenuse is 10, then writes cos θ = 6/10. Whose ratio is wrong, and what should it be?
- Zara is wrong; tan also needs the hypotenuse to find the angle
- Omar is wrong; cos needs the adjacent side, so cos θ = 8/10
- Both are right; the two ratios give the same angle

Answer key

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Trigonometry basics: SOH CAH TOA W1-mt_KB_Czd7RQH-s1

The adjacent side · Adjacent means next to. The adjacent side touches your angle, while the hypotenuse is the long side across from the right angle.
opposite over adjacent · TOA stands for Tan = Opposite over Adjacent.
0.6 · Similar triangles have equal matching angles, so the sine is unchanged. Sin = 3/5 = 0.6 in both triangles.
0.75 · Sin θ is opposite over hypotenuse. The hypotenuse is the longest side, so the ratio is always less than 1. Only 0.75 is under 1.
2 · Rise = hypotenuse × sin θ. Ramp A rises 10 × 0.5 = 5 m and ramp B rises 6 × 0.5 = 3 m. The difference is 5 − 3 = 2 m.
sin⁻¹(4/8) · First form the ratio opposite/hypotenuse, then use the inverse sin to get the angle in degree mode.
13.1 · The line of sight is 20 ÷ 0.866 ≈ 23.095 m. The height above eye level is 23.095 × 0.5 ≈ 11.547 m. Adding the eye height gives 11.547 + 1.6 = 13.147 ≈ 13.1 m.
30 · sin θ = 6/12 = 0.5, and the inverse sin gives θ = sin⁻¹(0.5) = 30°.
He should use cos⁻¹, since cos connects adjacent and hypotenuse · Adjacent over hypotenuse is the cos ratio, so the inverse must be cos⁻¹, not sin⁻¹.
Omar is wrong; cos needs the adjacent side, so cos θ = 8/10 · Zara's tan θ = 6/8 is correct for opposite over adjacent. Omar's cos θ = 6/10 uses opposite over hypotenuse, which is sin. Cos should be adjacent over hypotenuse, 8/10.