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
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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