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.

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