Trigonometric Functions and Their Values

Trigonometric Functions

Introduction to Angles

In trigonometry, angles are measured in degrees. The commonly used angles are: 0°, 30°, 45°, 60°, and 90°.

Values of Trigonometric Functions for Common Angles

1. Sine Function: The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse.
      - For 0°:
   $\sin(0°) = 0$
      - For 30°:
   $\sin(30°) = \frac{1}{2}$
      - For 45°:
   $\sin(45°) = \frac{\sqrt{2}}{2}$
      - For 60°:
   $\sin(60°) = \frac{\sqrt{3}}{2}$
      - For 90°:
   $\sin(90°) = 1$

2. Cosine Function: The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
      - For 0°:
   $\cos(0°) = 1$
      - For 30°:
   $\cos(30°) = \frac{\sqrt{3}}{2}$
      - For 45°:
   $\cos(45°) = \frac{\sqrt{2}}{2}$
      - For 60°:
   $\cos(60°) = \frac{1}{2}$
      - For 90°:
   $\cos(90°) = 0$

3. Tangent Function: The tangent of an angle is the ratio of the sine to the cosine of that angle.
      - For 0°:
   $\tan(0°) = 0$
      - For 30°:
   $\tan(30°) = \frac{1}{\sqrt{3}}$
      - For 45°:
   $\tan(45°) = 1$
      - For 60°:
   $\tan(60°) = \sqrt{3}$
      - For 90°:
   $\tan(90°) = \text{undefined}$

Summary of Trigonometric Values

• 0°:
    - $\sin(0°) = 0$
    - $\cos(0°) = 1$
    - $\tan(0°) = 0$
• 30°:
    - $\sin(30°) = \frac{1}{2}$
    - $\cos(30°) = \frac{\sqrt{3}}{2}$
    - $\tan(30°) = \frac{1}{\sqrt{3}}$
• 45°:
    - $\sin(45°) = \frac{\sqrt{2}}{2}$
    - $\cos(45°) = \frac{\sqrt{2}}{2}$
    - $\tan(45°) = 1$
• 60°:
    - $\sin(60°) = \frac{\sqrt{3}}{2}$
    - $\cos(60°) = \frac{1}{2}$
    - $\tan(60°) = \sqrt{3}$
• 90°:
    - $\sin(90°) = 1$
    - $\cos(90°) = 0$
    - $\tan(90°) = \text{undefined}$

Conclusion

Understanding the values of sine, cosine, and tangent for these key angles is fundamental in trigonometry, and these basic ratios serve as foundational building blocks for more advanced studies in the subject.