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Trigonometry in the Right Triangle · Ages 15–16

Special Angles and the Fundamental Identity Practice

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Designed for 10th grade geometry and trigonometry students. Solve each problem using exact values and algebraic identities without using a calculator.

  1. Find the exact value of sin(30°).

  2. Find the exact value of cos(45°).

  3. Find the exact value of tan(45°).

  4. In a right triangle with a 30° angle, the hypotenuse is 12 units long. What is the length of the side opposite the 30° angle?

  5. Find the exact value of cos(30°).

  6. Use the identity sin²(θ) + cos²(θ) = 1 to find cos(θ) if θ is an acute angle and sin(θ) = 5/13.

  7. If an acute angle has cos(θ) = 0.6, determine sin(θ).

  8. Evaluate the expression: (sin(60°))² + (cos(60°))².

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Special Angles and the Fundamental Identity Practice

Answer key

  1. Find the exact value of sin(30°).

    1/2

  2. Find the exact value of cos(45°).

    √2/2 (or 1/√2)

  3. Find the exact value of tan(45°).

    1

  4. In a right triangle with a 30° angle, the hypotenuse is 12 units long. What is the length of the side opposite the 30° angle?

    6 units (sin(30°) = opposite/hypotenuse = 1/2; 1/2 * 12 = 6)

  5. Find the exact value of cos(30°).

    √3/2

  6. Use the identity sin²(θ) + cos²(θ) = 1 to find cos(θ) if θ is an acute angle and sin(θ) = 5/13.

    12/13 (cos²(θ) = 1 - 25/169 = 144/169; cos(θ) = 12/13)

  7. If an acute angle has cos(θ) = 0.6, determine sin(θ).

    0.8 (sin²(θ) = 1 - 0.36 = 0.64; sin(θ) = 0.8)

  8. Evaluate the expression: (sin(60°))² + (cos(60°))².

    1 (by the fundamental identity, or (√3/2)² + (1/2)² = 3/4 + 1/4 = 1)