Trigonometry interactive worksheet for GCSE - Free Printable

Let's go through each part of the trigonometry worksheet step by step and solve it thoroughly.

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1) Label the sides of the three right angle triangles

Use:
- O = Opposite
- A = Adjacent
- H = Hypotenuse

We label based on the angle marked in each triangle.

#### Triangle 1 (Angle x):
- The side opposite to angle x → O
- The side adjacent to angle x → A
- The hypotenuse (longest side, opposite the right angle) → H

So:
- Opposite to x: O
- Adjacent to x: A
- Hypotenuse: H

#### Triangle 2 (Angle y):
- Opposite to y: O
- Adjacent to y: A
- Hypotenuse: H

#### Triangle 3 (Angle z):
- Opposite to z: O
- Adjacent to z: A
- Hypotenuse: H

✔ All triangles follow the same logic — just identify the side relative to the given angle.

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2) Complete the formula triangles for each trigonometric function

These are "formula triangles" used to remember SOHCAHTOA:

#### Sin(x) = O / H
- In the sin triangle:
- Top: O
- Bottom: H
- So: Sin(x) = O/H

#### Cos(y) = A / H
- Top: A
- Bottom: H
- Cos(y) = A/H

#### Tan(z) = O / A
- Top: O
- Bottom: A
- Tan(z) = O/A

✔ Fill in accordingly:
- Sin(x): O over H
- Cos(y): A over H
- Tan(z): O over A

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3) Order the steps for using trigonometry

Correct order is:

| Step No. | Steps |
|---------|-------|
| 1) | Label the sides O, H and A |
| 2) | Identify which of the SOHCAHTOA you would need to use |
| 3) | Write down your chosen formula e.g. Sin(x) = O/H |
| 4) | Substitute the values and calculate |
| 5) | Missing Angles only (Use the inverse function) e.g. Cos⁻¹(0.5) |
| 6) | Round to a suitable figure e.g. 3sf |

✔ This sequence ensures logical progression from labeling to calculation.

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4) Find missing side (Round to 3 significant figures)

#### a)
Given:
- Angle = 30°
- Hypotenuse = 10
- Find a (opposite to 30°)

Use:
Sin(30°) = O / H = a / 10

$$
\sin(30^\circ) = \frac{a}{10}
$$
$$
0.5 = \frac{a}{10} \Rightarrow a = 10 \times 0.5 = 5.00
$$

✔ a = 5.00

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#### b)
Given:
- Angle = 55°
- Hypotenuse = 7
- Find b (adjacent to 55°)

Use:
Cos(55°) = A / H = b / 7

$$
\cos(55^\circ) \approx 0.5736 = \frac{b}{7}
$$
$$
b = 7 \times 0.5736 \approx 4.0152
$$

✔ b ≈ 4.02 (to 3 sf)

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#### c)
Given:
- Angle = 32°
- Adjacent = 18
- Find c (hypotenuse)

Use:
Cos(32°) = A / H = 18 / c

$$
\cos(32^\circ) \approx 0.8480 = \frac{18}{c}
$$
$$
c = \frac{18}{0.8480} \approx 21.238
$$

✔ c ≈ 21.2 (to 3 sf)

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5) Find missing angle (Round to 3 significant figures)

#### a)
Given:
- Opposite = 7
- Hypotenuse = 21
- Find a°

Use:
Sin(a°) = O / H = 7 / 21 = 1/3 ≈ 0.3333

$$
a^\circ = \sin^{-1}(0.3333) \approx 19.47^\circ
$$

✔ a° ≈ 19.5° (to 3 sf)

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#### b)
Given:
- Adjacent = 10
- Hypotenuse = 24
- Find b°

Use:
Cos(b°) = A / H = 10 / 24 ≈ 0.4167

$$
b^\circ = \cos^{-1}(0.4167) \approx 65.53^\circ
$$

✔ b° ≈ 65.5° (to 3 sf)

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#### c)
Given:
- Opposite = 3
- Adjacent = 5
- Find c°

Use:
Tan(c°) = O / A = 3 / 5 = 0.6

$$
c^\circ = \tan^{-1}(0.6) \approx 30.96^\circ
$$

✔ c° ≈ 31.0° (to 3 sf)

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✔ Final Answers Summary:

#### 1) Side Labels:
- All labeled as O, A, H relative to the marked angle.

#### 2) Formula Triangles:
- Sin(x): O over H
- Cos(y): A over H
- Tan(z): O over A

#### 3) Correct Step Order:
1. Label sides O, H, A
2. Identify SOHCAHTOA
3. Write formula
4. Substitute and calculate
5. Use inverse if finding angle
6. Round

#### 4) Missing Sides:
- a) a = 5.00
- b) b = 4.02
- c) c = 21.2

#### 5) Missing Angles:
- a) a° = 19.5°
- b) b° = 65.5°
- c) c° = 31.0°

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Click Finish and then Check my Answers to verify! 🎉