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.