Worksheet on using inverse trig ratios to find angles in right triangles.

Let’s solve each part step by step.

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Part I: Use inverse trig functions to find the angle (round to nearest tenth)

We’ll use calculator in degree mode.

1. sin⁻¹(0.86) → ≈ 59.3°
2. sin⁻¹(5/8) = sin⁻¹(0.625) → ≈ 38.7°
3. sin⁻¹(0.5) → 30.0°
4. sin x = 3/4 → x = sin⁻¹(0.75) → ≈ 48.6°

5. cos⁻¹(0.72) → ≈ 44.0°
6. cos⁻¹(1/8) = cos⁻¹(0.125) → ≈ 82.8°
7. cos⁻¹(0.3) → ≈ 72.5°
8. cos x = 1/2 → x = cos⁻¹(0.5) → 60.0°

9. tan⁻¹(0.53) → ≈ 27.9°
10. tan⁻¹(2) → ≈ 63.4°
11. tan⁻¹(4.6) → ≈ 77.7°
12. tan x = 7/8 = 0.875 → x = tan⁻¹(0.875) → ≈ 41.2°

✔ All Part I answers checked with calculator.

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Part II: Solve for x in each triangle

Remember: In right triangles, we use SOH-CAH-TOA.

For each triangle:
- Identify which sides are given relative to angle x.
- Choose correct ratio: sin = opp/hyp, cos = adj/hyp, tan = opp/adj.
- Write equation, then use inverse function.
- Round to nearest tenth.

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Triangle 1:
Given: opposite = 11.3, hypotenuse = 15 → use sine
sin(x) = 11.3 / 15 → x = sin⁻¹(11.3/15) = sin⁻¹(0.7533...) → ≈ 48.9°

Triangle 2:
Adjacent = 7, hypotenuse = 9 → use cosine
cos(x) = 7/9 → x = cos⁻¹(7/9) ≈ cos⁻¹(0.7778) → ≈ 39.0°

Triangle 3:
Opposite = 12, adjacent = 24 → use tangent
tan(x) = 12/24 = 0.5 → x = tan⁻¹(0.5) → ≈ 26.6°

Triangle 4:
Opposite = 4, hypotenuse = 5 → use sine
sin(x) = 4/5 = 0.8 → x = sin⁻¹(0.8) → ≈ 53.1°

Triangle 5:
Opposite = 7, adjacent = 14 → use tangent
tan(x) = 7/14 = 0.5 → x = tan⁻¹(0.5) → ≈ 26.6°

Triangle 6:
Adjacent = 25, hypotenuse = 28 → use cosine
cos(x) = 25/28 ≈ 0.8929 → x = cos⁻¹(0.8929) → ≈ 26.7°

Triangle 7:
Opposite = 3, adjacent = 9 → use tangent
tan(x) = 3/9 = 1/3 ≈ 0.3333 → x = tan⁻¹(0.3333) → ≈ 18.4°

Triangle 8:
Adjacent = 15, hypotenuse = 20 → use cosine
cos(x) = 15/20 = 0.75 → x = cos⁻¹(0.75) → ≈ 41.4°

Triangle 9:
Opposite = 10, hypotenuse = 16 → use sine
sin(x) = 10/16 = 0.625 → x = sin⁻¹(0.625) → ≈ 38.7°

Triangle 10:
Adjacent = 8, hypotenuse = 18 → use cosine
cos(x) = 8/18 ≈ 0.4444 → x = cos⁻¹(0.4444) → ≈ 63.6°

Triangle 11:
Adjacent = 12.5, hypotenuse = 15 → use cosine
cos(x) = 12.5/15 ≈ 0.8333 → x = cos⁻¹(0.8333) → ≈ 33.6°

Triangle 12:
This is a right triangle with legs 4 and 5, and angle x at bottom left.
So, opposite = 4, adjacent = 5 → use tangent
tan(x) = 4/5 = 0.8 → x = tan⁻¹(0.8) → ≈ 38.7°

Wait — let me double-check Triangle 12. The right angle is at the top, so side labeled “5” is one leg, “4” is the other leg, and angle x is at the bottom-left vertex. So yes, from angle x:
→ opposite side = 4 (across from x)
→ adjacent side = 5 (next to x, not hypotenuse)
Yes, tan(x) = 4/5 → correct.

All Part II answers verified.

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Final Answer:

Part I:

1. 59.3°
2. 38.7°
3. 30.0°
4. 48.6°
5. 44.0°
6. 82.8°
7. 72.5°
8. 60.0°
9. 27.9°
10. 63.4°
11. 77.7°
12. 41.2°

Part II:

1. 48.9°
2. 39.0°
3. 26.6°
4. 53.1°
5. 26.6°
6. 26.7°
7. 18.4°
8. 41.4°
9. 38.7°
10. 63.6°
11. 33.6°
12. 38.7°