Trigonometry (B) Missing Angles and Lengths Worksheet | Cazoom ... - Free Printable

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
- Adjacent = 9 cm
- Angle $ x $ is at the bottom right

Use $ \tan(x) = \frac{4}{9} $

$$
x = \tan^{-1}\left(\frac{4}{9}\right) \approx \tan^{-1}(0.4444) \approx 23.9^\circ
$$

✔ $ x = 23.9^\circ $

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#### 2)
- Opposite = 14 cm
- Adjacent = 10 cm
- $ x $ is at bottom right

$ \tan(x) = \frac{14}{10} = 1.4 $

$$
x = \tan^{-1}(1.4) \approx 54.5^\circ
$$

✔ $ x = 54.5^\circ $

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

$ \tan(x) = \frac{7}{12} \approx 0.5833 $

$$
x = \tan^{-1}(0.5833) \approx 30.2^\circ
$$

✔ $ x = 30.2^\circ $

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#### 4)
- Opposite = 15 cm
- Hypotenuse = 18 cm
- Right angle at bottom right → $ x $ is top left

So $ \sin(x) = \frac{15}{18} = 0.8333 $

$$
x = \sin^{-1}(0.8333) \approx 56.4^\circ
$$

✔ $ x = 56.4^\circ $

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#### 5)
- Opposite = 0.83 cm
- Adjacent = 0.99 cm
- $ x $ is at top

$ \tan(x) = \frac{0.83}{0.99} \approx 0.8384 $

$$
x = \tan^{-1}(0.8384) \approx 40.2^\circ
$$

✔ $ x = 40.2^\circ $

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#### 6)
- Opposite = 410 mm
- Hypotenuse = 972 mm
- $ x $ is top angle

$ \sin(x) = \frac{410}{972} \approx 0.4218 $

$$
x = \sin^{-1}(0.4218) \approx 25.0^\circ
$$

✔ $ x = 25.0^\circ $

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✔ Section A Answers:

1) $ 23.9^\circ $
2) $ 54.5^\circ $
3) $ 30.2^\circ $
4) $ 56.4^\circ $
5) $ 40.2^\circ $
6) $ 25.0^\circ $

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

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#### 1)
- Opposite = 5 cm
- Adjacent = 11 cm
- $ x $ is at bottom right

$ \tan(x) = \frac{5}{11} \approx 0.4545 $

$$
x = \tan^{-1}(0.4545) \approx 24.4^\circ
$$

✔ $ x = 24.4^\circ $

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#### 2)
- Opposite = 0.21 cm
- Hypotenuse = 0.7 cm
- $ x $ is at bottom right

$ \sin(x) = \frac{0.21}{0.7} = 0.3 $

$$
x = \sin^{-1}(0.3) \approx 17.5^\circ
$$

✔ $ x = 17.5^\circ $

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

Use $ \cos(62^\circ) = \frac{x}{6.4} $

$$
x = 6.4 \times \cos(62^\circ) \approx 6.4 \times 0.4695 \approx 3.00 \text{ cm}
$$

✔ $ x = 3.00 \text{ cm} $

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#### 4)
- Hypotenuse = 27 mm
- Opposite = 1.5 cm = 15 mm
- $ x $ is top left angle

Convert units: 1.5 cm = 15 mm

$ \sin(x) = \frac{15}{27} \approx 0.5556 $

$$
x = \sin^{-1}(0.5556) \approx 33.7^\circ
$$

✔ $ x = 33.7^\circ $

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

Use $ \tan(13^\circ) = \frac{x}{2.3} $

$$
x = 2.3 \times \tan(13^\circ) \approx 2.3 \times 0.2309 \approx 0.531 \text{ cm}
$$

✔ $ x = 0.531 \text{ cm} $

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#### 6)
- Adjacent = 620 cm
- Hypotenuse = 8 m = 800 cm
- $ x $ is top angle

$ \cos(x) = \frac{620}{800} = 0.775 $

$$
x = \cos^{-1}(0.775) \approx 39.1^\circ
$$

✔ $ x = 39.1^\circ $

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✔ Section B Answers:

1) $ 24.4^\circ $
2) $ 17.5^\circ $
3) $ 3.00 \text{ cm} $
4) $ 33.7^\circ $
5) $ 0.531 \text{ cm} $
6) $ 39.1^\circ $

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🟩 Final Answers Summary:

#### Section A – Missing Angles:
1) $ 23.9^\circ $
2) $ 54.5^\circ $
3) $ 30.2^\circ $
4) $ 56.4^\circ $
5) $ 40.2^\circ $
6) $ 25.0^\circ $

#### Section B – Missing Angles or Lengths:
1) $ 24.4^\circ $
2) $ 17.5^\circ $
3) $ 3.00 \text{ cm} $
4) $ 33.7^\circ $
5) $ 0.531 \text{ cm} $
6) $ 39.1^\circ $

All answers are rounded to 3 significant figures, as instructed.

Let me know if you'd like a diagram explanation or steps in another format!