5.2 Trigonometric Ratios

1. Learning Objectives

Define the three primary trigonometric ratios: Sine, Cosine, and Tangent.
Apply SOH-CAH-TOA to find unknown angles and sides.
Use trig ratios to solve practical height and distance problems.

Trigonometric ratios are based on the side lengths of a right-angled triangle relative to an acute angle (θ).

SOH - CAH - TOA

1. If Opposite = 3 and Hypotenuse = 5, what is sin θ?

sin θ = Opp / Hyp = 3/5. Ans: 0.6

2. If Adjacent = 4 and Hypotenuse = 5, what is cos θ?

cos θ = Adj / Hyp = 4/5. Ans: 0.8

3. If Opposite = 3 and Adjacent = 4, what is tan θ?

tan θ = Opp / Adj = 3/4. Ans: 0.75

4. In a right triangle, what is sin(90°)?

As the angle approaches 90°, the opposite side becomes equal to the hypotenuse. Ans: 1

5. What is the value of tan(45°)?

In a 45-45-90 triangle, Opp = Adj. So Opp/Adj = 1. Ans: 1

6. If sin θ = 0.5, what is the angle θ?

Using sin⁻¹(0.5) or standard values. Ans: 30°

7. If cos θ = 0, what is the angle θ?

The adjacent side is zero at 90°. Ans: 90°

8. Solve for Opp: sin(30°) = Opp / 10

0.5 = Opp / 10 → Opp = 5. Ans: 5

9. Solve for Adj: cos(60°) = Adj / 8

0.5 = Adj / 8 → Adj = 4. Ans: 4

10. If tan θ = 5/12, find sin θ (Hyp = 13).

Opp=5, Adj=12, Hyp=13. sin θ = Opp/Hyp = 5/13. Ans: 5/13

11. Which ratio relates Opposite and Adjacent?

According to TOA. Ans: Tangent

12. True/False: sin θ can be greater than 1.

False. Hypotenuse is always the longest side, so Opp/Hyp is ≤ 1. Ans: False

13. Find Hyp if Opp = 6 and sin θ = 0.6.

0.6 = 6 / Hyp → Hyp = 6 / 0.6 = 10. Ans: 10

14. What is cos(0°)?

At 0°, the adjacent side is equal to the hypotenuse. Ans: 1

15. If θ = 45°, what is sin θ?

sin(45°) = 1/√2 or √2/2. Ans: 0.707

16. Calculate tan(60°) exactly.

From a 30-60-90 triangle, Opp= √3 and Adj=1. Ans: √3

17. If sin θ = cos θ, what is the angle θ?

This only happens when the legs are equal (isosceles). Ans: 45°

18. Solve for Opp: tan(30°) = Opp / 12

Opp = 12 × tan(30°) = 12 × (1/√3) = 4√3. Ans: 4√3

19. What is the value of sin²θ + cos²θ?

This is the Pythagorean Identity: (a/c)² + (b/c)² = (a²+b²)/c² = c²/c² = 1. Ans: 1

20. As θ increases from 0 to 90, does cos θ increase or decrease?

cos(0)=1 and cos(90)=0. Ans: Decrease