1.3 Special Right Triangles

Goals of the Lesson

• Focus on finding values of specific angles, specifically for:

  • 30-60-90 Triangle

  • 45-45-90 Triangle

• Tasks include:

  • Finding the exact values of trigonometric functions for 30°, 45°, and 60°.

  • Utilizing a calculator for approximating trigonometric function values.

  • Solving applications using trigonometric functions.

Exact Values of Trigonometric Functions

• Determine the exact six trig values for the following angle measures:

  • 60° (𝜋/3)

  • 30° (𝜋/6)

  • 45° (𝜋/4)

• Specific Functions:

  • sin 60°, cos 60°, tan 60°, cot 60°, csc 30°, sec 30°, etc.

Examples

• Use pre-calculated values to simplify the following expressions:

  • a. sin 45° cos 30°

  • b. tan 𝜋/4 − sin 𝜋/3

  • c. tan² 𝜋/6 + sin² 𝜋/4

  • d. 4 cos 45° − 2 sin 45°

  • e. 1 − cos² 30° − cos² 60°

  • f. csc² 𝜋/3 − 5

  • g. sin 𝜋/4 + cos 𝜋/4 − 2

  • h. sec 60° − cos 60° + tan 60°

Using a Calculator

• Ensure that the calculator is set to the correct mode (Radians or Degrees):

  • a. cos 48°

  • b. csc 21°

  • c. tan 𝜋/12

  • d. sin 35°

  • e. cot 103°

  • f. sec (4/5)𝜋

Applications of Trigonometry

Angle of Elevation & Depression:

1. Example 1:

   • The angle of elevation of the Sun is 35.1°. The shadow of the Washington Monument measures 789 feet long. Calculate the height of the monument.

2. Example 2:

   • A pilot in a helicopter observes a landing pad below. If the angle of depression is 73° and the horizontal distance to the pad is 1,200 feet, determine the helicopter's altitude.

   • Draw a diagram and label the components to assist in calculation.