Trigonometry

Introduction

This tutorial provides a comprehensive introduction to trigonometry, focusing on evaluating trigonometric functions such as sine (sin), cosine (cos), and tangent (tan) using right triangle trigonometry. It also covers the concepts of reference angles and coterminal angles, which are essential for evaluating other trig functions like secant (sec), cosecant (csc), and cotangent (cot). Understanding these concepts will help you solve various problems in mathematics and physics that involve angles and triangles.

Step 1: Understanding Right Triangle Trigonometry

• Identify the sides of a right triangle:

  • Hypotenuse: The longest side opposite the right angle.
  • Opposite side: The side opposite the angle of interest.
  • Adjacent side: The side next to the angle of interest.

• Learn the trigonometric ratios:

  • Sine (sin):
    • sin(θ) = opposite / hypotenuse
  • Cosine (cos):
    • cos(θ) = adjacent / hypotenuse
  • Tangent (tan):
    • tan(θ) = opposite / adjacent

• Use the mnemonic SOH CAH TOA to remember these ratios:

  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent

Step 2: Evaluating Trigonometric Functions

• Finding values using angles:

  • Use known angles (like 30°, 45°, and 60°) to find exact values of sin, cos, and tan.
  • Example:
    • For 30°:
      • sin(30°) = 1/2
      • cos(30°) = √3/2
      • tan(30°) = 1/√3

• Practice evaluating the functions using a calculator or unit circle for other angles.

Step 3: Understanding Reference Angles

• Define reference angles:

  • The reference angle is the acute angle formed by the terminal side of an angle and the x-axis.

• Determine the reference angle:

  • For angles between 0° and 90°: The reference angle is the angle itself.
  • For angles between 90° and 180°: Subtract from 180°.
  • For angles between 180° and 270°: Subtract from 180°.
  • For angles between 270° and 360°: Subtract from 360°.

• Use the reference angle to evaluate trig functions:

  • Trig functions of angles greater than 90° can be found using their reference angles.

Step 4: Exploring Coterminal Angles

• Define coterminal angles:

  • Coterminal angles are angles that share the same terminal side. They can be found by adding or subtracting full rotations (360° or 2π radians).

• Calculate coterminal angles:

  • To find a coterminal angle for an angle θ:
    • coterminal angle = θ + 360n (where n is an integer)

• Evaluate trig functions for coterminal angles:

  • The trig values for coterminal angles are the same.

Step 5: Evaluating Other Trigonometric Functions

• Understand the reciprocal functions:

  • Secant (sec):
    • sec(θ) = 1/cos(θ)
  • Cosecant (csc):
    • csc(θ) = 1/sin(θ)
  • Cotangent (cot):
    • cot(θ) = 1/tan(θ)

• Use these definitions to find values:

  • If you have the value of cos(θ), you can easily find sec(θ) by taking its reciprocal.

Conclusion

Trigonometry is a vital mathematical tool that helps in understanding relationships in triangles and angles. By mastering basic concepts such as right triangle trigonometry, reference angles, and coterminal angles, you can effectively evaluate various trigonometric functions.

To deepen your understanding, practice problems involving these concepts, and consider reviewing additional resources or formula sheets for more complex scenarios. For further learning, check out the free formula sheet and additional video lessons provided in the video description.