MC

1.4 Trig Functions for General Angles

Goals of the Lesson

  • Extend the concept of trigonometric ratios to all four quadrants of the coordinate plane.

  • Specific objectives include:

    • Determine the signs of trigonometric functions in each quadrant.

    • Use co-terminal angles to find exact values of trigonometric functions.

    • Find reference angles and use them for exact values of trig functions.

  • Key concepts:

    • Given

      • Angle 𝜃

      • Point P(x, y)

      • Radius r = √(x² + y²)

Signs of Trigonometric Functions

  • The signs of trig functions change based on the quadrant location:

    • Quadrant I: All positive

    • Quadrant II: sin positive, cos and tan negative

    • Quadrant III: tan positive, sin and cos negative

    • Quadrant IV: cos positive, sin and tan negative

  • Importance of knowing quadrant for determining signs.

Finding the Six Trigonometric Functions

  • Example coordinates to analyze:

    • P(−3, −5)

    • P(1, −3)

    • P(12, 15)

    • P(−4, 6)

    • P(3, 0)

  • Process involves rationalizing and simplifying trig functions derived from these points.

Finding Trig Values from Given Ratios

  • Examples of calculating trig functions given certain conditions:

    • tan 𝜃 = −2/3 where cos 𝜃 > 0

    • cos 𝜃 = √2/5 where sin 𝜃 > 0

    • tan 𝜃 = −4 within a specific range of angles

  • Understanding the quadrant where each ratio exists helps determine corresponding trig function values.

Co-terminal Angles

  • Definition: Two angles are co-terminal if they have the same terminal side.

  • Examples include: 60° and 240°; −40° and 320°.

  • Finding co-terminal angles expressed as:

    • For degrees: 𝜃 ± 360°k

    • For radians: 𝜃 ± 2𝜋k where k is an integer.

  • Process to find co-terminal angles involves adding or subtracting 360° or 2𝜋 until the angle falls within the desired range.

  • Exercises require finding trig values for angles such as sin 390°, csc (−270°), and cos 420°.

Reference Angle

  • Definition: Positive acute angle 𝛼 formed by the terminal side of 𝜃 and the x-axis.

  • Reference angles help derive trig function values from Quadrant I for angles in other quadrants.

  • Process for identifying reference angle includes:

    1. Finding co-terminal angle, if applicable.

    2. Determining the quadrant based on given angle 𝜃.

    3. Applying the correct formula to find 𝛼.

  • Examples of finding reference angles include those for angles such as 345°, −135°, and 3π/5.

Using Reference Angles to Find Exact Values of Trig Functions

  • Theorem: For an angle 𝜃 in a quadrat and its reference angle 𝛼, the trig values can be derived using reference angles.

  • Procedures involve:

    • sin 𝜃 = sin 𝛼, cos 𝜃 = ±cos 𝛼 depending on quadrant, tan 𝜃 = ±tan 𝛼 based on quadrant.

  • Examples demonstrated:

    • Calculate sin 135°, cos 600°, and tan (−π/3).

Recap: Finding Trigonometric Values of Any Angle

  • If 𝜃 is a quadrantal angle:

    • Draw angle and apply definition of trig functions based on point placement.

  • For angles in quadrants:

    1. Find reference angle 𝛼.

    2. Compute trig function value for 𝛼.

    3. Adjust sign based on quadrant of 𝜃.