Q6. Evaluate the exact lengths of the sides labeled a, b, c, and d in the given triangle diagram.

Background

Topic: Right Triangle Trigonometry

This question tests your understanding of how to use trigonometric ratios (sine, cosine, tangent) and the Pythagorean Theorem to find missing side lengths in right triangles. You may need to use special triangle properties (like 45-45-90 or 30-60-90 triangles) or set up equations using the given angles and sides.

Key Terms and Formulas

• Pythagorean Theorem: (for a right triangle with hypotenuse )

• Sine:

• Cosine:

• Tangent:

• Special Triangles: 45-45-90 and 30-60-90 triangles have predictable side ratios.

Step-by-Step Guidance

1. Identify which sides and angles are given. For example, you know one side is 30 units, and the triangle contains 45°, 60°, and 90° angles.

2. Use the properties of special triangles: In a 30-60-90 triangle, the sides are in the ratio . In a 45-45-90 triangle, the sides are in the ratio .

3. Set up equations using the sine, cosine, or tangent of the known angles to relate the unknown sides to the known side (30 units).

4. For example, for the side adjacent to the 60° angle, use .

5. Solve for one unknown side at a time, expressing each in terms of the given side or using the ratios from special triangles.

Try solving on your own before revealing the answer!