Year 9: Evaluating trigonometric functions

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1. What are trigonometric functions?

Trigonometric functions are mathematical relationships between the angles and sides of a right-angled triangle. The three main functions are sine (sin), cosine (cos), and tangent (tan).

2. The SOH-CAH-TOA rule

To remember the relationships between sides and angles, use the acronym SOH-CAH-TOA:

3. Identifying the sides

When working with trigonometric functions, it's important to identify the sides correctly:

• Hypotenuse: The longest side opposite the right angle.
• Opposite: The side opposite the given angle.
• Adjacent: The side next to the given angle (not the hypotenuse).

4. Using a calculator

To evaluate trigonometric functions on a calculator:

1. Make sure your calculator is set to degrees (not radians).
2. Use the sin, cos, or tan button followed by the angle.
3. Example: sin(30°) = 0.5

5. Example problems

Given a right triangle with an angle θ = 45° and a hypotenuse of 10 cm:

• Find the opposite side:
  sin(45°) = opposite ÷ 10 ⇒ opposite = 10 × sin(45°) = 7.07 cm
• Find the adjacent side:
  cos(45°) = adjacent ÷ 10 ⇒ adjacent = 10 × cos(45°) = 7.07 cm

6. Inverse trigonometric functions

To find the angle when you know two sides of a right triangle, use inverse functions:

• sin^-1, cos^-1, and tan^-1 on your calculator

Example: If opposite = 3 and hypotenuse = 5, then:
θ = sin^-1(3 ÷ 5) = 36.87°

7. Common trig values

8. Practice questions

1. Evaluate sin(60°), cos(30°), and tan(45°).
2. Find the length of the opposite side if θ = 30° and hypotenuse = 12 cm.
3. If the opposite side is 6 cm and the adjacent side is 8 cm, find θ.

9. Solutions

1. sin(60°) = 0.87, cos(30°) = 0.87, tan(45°) = 1
2. sin(30°) = opposite ÷ 12 ⇒ opposite = 12 × 0.5 = 6 cm
3. θ = tan^-1(6 ÷ 8) = 36.87°