Trigonometry worksheet for finding missing angles and lengths in right-angled triangles.

Let's solve each problem step by step using trigonometry (SOHCAHTOA), which applies to right-angled triangles.

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🔷 Section A: Find the missing angle $ x $

We use:
- $ \sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} $
- $ \cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} $
- $ \tan(x) = \frac{\text{opposite}}{\text{adjacent}} $

Then, use inverse functions: $ x = \sin^{-1}, \cos^{-1}, \tan^{-1} $

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#### 1)
Right triangle:
- Opposite = 4 cm
- Hypotenuse = 9 cm

Use sine:
$$
\sin(x) = \frac{4}{9} \Rightarrow x = \sin^{-1}\left(\frac{4}{9}\right) \approx \sin^{-1}(0.4444) \approx 26.4^\circ
$$

✔ x ≈ 26.4°

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#### 2)
- Adjacent = 10 cm
- Hypotenuse = 14 cm

Use cosine:
$$
\cos(x) = \frac{10}{14} = \frac{5}{7} \approx 0.7143 \Rightarrow x = \cos^{-1}(0.7143) \approx 44.4^\circ
$$

✔ x ≈ 44.4°

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#### 3)
- Opposite = 7 cm
- Adjacent = 12 cm

Use tangent:
$$
\tan(x) = \frac{7}{12} \approx 0.5833 \Rightarrow x = \tan^{-1}(0.5833) \approx 30.2^\circ
$$

✔ x ≈ 30.2°

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#### 4)
- Opposite = 15 cm
- Hypotenuse = 18 cm

Use sine:
$$
\sin(x) = \frac{15}{18} = \frac{5}{6} \approx 0.8333 \Rightarrow x = \sin^{-1}(0.8333) \approx 56.4^\circ
$$

✔ x ≈ 56.4°

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#### 5)
- Opposite = 0.83 cm
- Adjacent = 0.99 cm

Use tangent:
$$
\tan(x) = \frac{0.83}{0.99} \approx 0.8384 \Rightarrow x = \tan^{-1}(0.8384) \approx 40.1^\circ
$$

✔ x ≈ 40.1°

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#### 6)
- Opposite = 410 mm
- Hypotenuse = 972 mm

Use sine:
$$
\sin(x) = \frac{410}{972} \approx 0.4218 \Rightarrow x = \sin^{-1}(0.4218) \approx 25.0^\circ
$$

✔ x ≈ 25.0°

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✔ Section A Answers:

1) 26.4°
2) 44.4°
3) 30.2°
4) 56.4°
5) 40.1°
6) 25.0°

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🔷 Section B: Find missing angles or lengths $ x $

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#### 1)
- Opposite = 5 cm
- Adjacent = 11 cm

Use tangent:
$$
\tan(x) = \frac{5}{11} \approx 0.4545 \Rightarrow x = \tan^{-1}(0.4545) \approx 24.4^\circ
$$

✔ x ≈ 24.4°

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#### 2)
- Opposite = 0.21 cm
- Hypotenuse = 0.7 cm

Use sine:
$$
\sin(x) = \frac{0.21}{0.7} = 0.3 \Rightarrow x = \sin^{-1}(0.3) \approx 17.5^\circ
$$

✔ x ≈ 17.5°

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#### 3)
- Hypotenuse = 6.4 cm
- Angle = 62°
- Find adjacent side $ x $

Use cosine:
$$
\cos(62^\circ) = \frac{x}{6.4} \Rightarrow x = 6.4 \times \cos(62^\circ)
$$
$$
\cos(62^\circ) \approx 0.4695 \Rightarrow x \approx 6.4 \times 0.4695 \approx 3.00 \text{ cm}
$$

✔ x ≈ 3.00 cm

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#### 4)
- Opposite = 1.5 cm
- Hypotenuse = 27 mm = 2.7 cm (convert to same units)

Use sine:
$$
\sin(x) = \frac{1.5}{2.7} \approx 0.5556 \Rightarrow x = \sin^{-1}(0.5556) \approx 33.7^\circ
$$

✔ x ≈ 33.7°

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#### 5)
- Adjacent = 2.3 cm
- Angle = 13°
- Find opposite side $ x $

Use tangent:
$$
\tan(13^\circ) = \frac{x}{2.3} \Rightarrow x = 2.3 \times \tan(13^\circ)
$$
$$
\tan(13^\circ) \approx 0.2309 \Rightarrow x \approx 2.3 \times 0.2309 \approx 0.531 \text{ cm}
$$

✔ x ≈ 0.531 cm

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#### 6)
- Opposite = 8 m
- Adjacent = 620 cm = 6.2 m (convert)

Use tangent:
$$
\tan(x) = \frac{8}{6.2} \approx 1.2903 \Rightarrow x = \tan^{-1}(1.2903) \approx 52.2^\circ
$$

✔ x ≈ 52.2°

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✔ Section B Answers:

1) 24.4°
2) 17.5°
3) 3.00 cm
4) 33.7°
5) 0.531 cm
6) 52.2°

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📌 Final Answer Summary:

#### Section A – Missing Angles:
1) 26.4°
2) 44.4°
3) 30.2°
4) 56.4°
5) 40.1°
6) 25.0°

#### Section B – Missing Angles or Lengths:
1) 24.4°
2) 17.5°
3) 3.00 cm
4) 33.7°
5) 0.531 cm
6) 52.2°

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💡 Notes:

- All answers are rounded to 3 significant figures as instructed.
- Make sure your calculator is in degree mode.
- Always identify which sides you have (opposite, adjacent, hypotenuse) before choosing SOHCAHTOA.

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