Overview
This lecture explains how to recall and calculate exact values for sine, cosine, and tangent of key angles (0°, 30°, 45°, 60°, 90°) without a calculator, and shows how to apply these values in trigonometric problems.
Exact Trig Values Table
- For sine, use: sin(θ) = √(number)/2, where the number counts from 0 to 4 as θ increases from 0° to 90°.
- For cosine, use: cos(θ) = √(number)/2, but count from 4 to 0 as θ increases from 0° to 90°.
- For tangent, use: tan(θ) = sin(θ)/cos(θ) or equivalently, √(number increasing from 0 to 4)/√(number decreasing from 4 to 0).
Simplified Exact Values
- sin(0°) = 0, sin(30°) = 1/2, sin(45°) = √2/2, sin(60°) = √3/2, sin(90°) = 1
- cos(0°) = 1, cos(30°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, cos(90°) = 0
- tan(0°) = 0, tan(30°) = 1/√3 or √3/3, tan(45°) = 1, tan(60°) = √3, tan(90°) is undefined
Solving Triangles Using Exact Trig Values
- Use SOHCAHTOA to choose the correct trigonometric ratio based on side information.
- Substitute table values for angles like 30°, 45°, 60° to solve equations without a calculator.
- Example: If cos(60°) = 1/2 and adjacent/hypotenuse = x/10, then x = 5.
- Example: If sin(x) = 1/2 and opposite/hypotenuse = 11/22, then x = 30°.
Deriving Values from Special Triangles
- 45°: Isosceles right triangle with legs 1 and hypotenuse √2 yields sin(45°) = cos(45°) = √2/2, tan(45°) = 1.
- 30° and 60°: Halving an equilateral triangle of side 2 gives sides 1, √3, 2, leading to sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3.
- Switching opposite and adjacent sides gives the 60° values.
Trigonometric Limits as Angle Approaches 0° or 90°
- As θ → 90°, sin(θ) → 1, cos(θ) → 0, tan(θ) is undefined.
- As θ → 0°, sin(θ) → 0, cos(θ) → 1, tan(θ) → 0.
Key Terms & Definitions
- Exact Trig Values — Specific values of sine, cosine, and tangent for special angles requiring no calculator.
- SOHCAHTOA — Mnemonic for remembering sine (Opposite/Hypotenuse), cosine (Adjacent/Hypotenuse), tangent (Opposite/Adjacent).
Action Items / Next Steps
- Memorize the exact values for sine, cosine, and tangent at 0°, 30°, 45°, 60°, and 90°.
- Practice using these values to solve right triangle problems without a calculator.
- Try the exam questions provided in the video description.