Lecture Notes: Finding Exact Values of Trig Functions

Special Right Triangles

30-60-90 Triangle

• Sides:
  • Across 30°: 1
  • Across 60°: ( \sqrt{3} )
  • Hypotenuse: 2

45-45-90 Triangle

• Sides:
  • Across 45°: 1 (both sides)
  • Across 90°: ( \sqrt{2} )

SOHCAHTOA

• Sine Ratio: Opposite / Hypotenuse
• Cosine Ratio: Adjacent / Hypotenuse
• Tangent Ratio: Opposite / Adjacent

Examples of Finding Exact Values

Example 1: ( \sin(30^\circ) )

• Formula: Opposite / Hypotenuse
• ( \sin(30^\circ) = \frac{1}{2} )

Example 2: ( \cos(\frac{5\pi}{6}) )

• Convert radians to degrees:
  • ( \frac{5\pi}{6} \times \frac{180}{\pi} = 150^\circ )
• 150° is in quadrant 2, reference angle is 30°.
• Cosine is adjacent/hypotenuse:
  • ( \cos(150^\circ) = \frac{-\sqrt{3}}{2} )

Example 3: ( \tan(\frac{\pi}{6}) )

• Convert radians to degrees:
  • ( \frac{180}{6} = 30^\circ )
• ( \tan(30^\circ) = \frac{1}{\sqrt{3}} )
• Rationalized: ( \frac{\sqrt{3}}{3} )

Example 4: ( \cos(240^\circ) )

• Reference angle is 60° in quadrant 3.
• Cosine is adjacent/hypotenuse:
  • ( \cos(240^\circ) = \frac{-1}{2} )

Example 5: ( \tan(-45^\circ) )

• Negative angle, reference angle 45° in quadrant 4.
• ( \tan(-45^\circ) = -1 )

Example 6: ( \sin(\frac{10\pi}{3}) )

• Convert radians to degrees:
  • ( \frac{10\pi}{3} \times \frac{180}{\pi} = 600^\circ )
• Coterminal angle: 600° - 360° = 240°
• ( \sin(240^\circ) = \frac{-\sqrt{3}}{2} )

Handling Large or Negative Angles

• For angles > 360°, subtract 360° until 0° to 360°.
• For negative angles, add 360° until 0° to 360°.

Conclusion

• Use special triangles and reference angles to find exact trig values.
• Apply SOHCAHTOA to determine sine, cosine, and tangent values using triangle relationships.