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AQA GCSE · Question 22 · Geometry and Measures
Use trigonometry to work out the value of x.
Use trigonometry to work out the value of x.
How to approach this question
1. Identify the sides of the right-angled triangle in relation to the 40° angle.
- The side of length x is Opposite the angle.
- The side of length 21 cm is the Hypotenuse (opposite the right angle).
- The third side is the Adjacent.
2. Choose the correct trigonometric ratio that relates the Opposite and Hypotenuse. This is the sine ratio: SOH CAH TOA -> SOH (Sine = Opposite / Hypotenuse).
3. Write down the equation: sin(40°) = x / 21.
4. Rearrange the equation to solve for x: x = 21 * sin(40°).
5. Use a calculator to find the value of sin(40°) and then multiply by 21. Make sure your calculator is in degrees mode.
6. sin(40°) ≈ 0.64278...
7. x = 21 * 0.64278... ≈ 13.498...
8. Round the answer to a suitable degree of accuracy, for example, one decimal place: 13.5 cm.
Full Answer
13.5 cm (to 1 d.p.)
Step 1: Label the sides of the triangle relative to the given angle (40°).
- The side opposite the 40° angle is `x`. This is the **Opposite** (O).
- The side opposite the right angle is the longest side, 21 cm. This is the **Hypotenuse** (H).
- The remaining side is the **Adjacent** (A).
Step 2: Choose the correct trigonometric ratio using SOH CAH TOA.
- We know the Hypotenuse (H) and we want to find the Opposite (O).
- The ratio that connects O and H is **SOH**: Sine = Opposite / Hypotenuse.
Step 3: Set up and solve the equation.
sin(angle) = O / H
sin(40°) = x / 21
To find x, multiply both sides by 21:
x = 21 × sin(40°)
Step 4: Calculate the value.
Using a calculator (make sure it is in degrees mode):
x ≈ 21 × 0.6428
x ≈ 13.4985...
Rounding to one decimal place, we get:
x = 13.5 cm.
Common mistakes
✗ Choosing the wrong trigonometric ratio (cos or tan).
✗ Setting up the ratio incorrectly (e.g., x/21 instead of 21/x, or hypotenuse over opposite).
✗ Having the calculator in radians or grads mode instead of degrees.
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