Lesson 1: Trig Functions of Any Angle Using Reference Angles
Estimated time: 30-35 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Define trig functions using the coordinate definition (x, y, r)
- Find trig values when given a point on the terminal side
- Use reference angles to evaluate trig functions in any quadrant
- Determine the quadrant of an angle given sign information about its trig values
The General Coordinate Definition
Let θ be an angle in standard position, and let (x, y) be any point on its terminal side (other than the origin). Let r = √(x² + y²) be the distance from the origin to that point.
General Definitions:
sin θ = y/r cos θ = x/r tan θ = y/x (x ≠ 0)
csc θ = r/y (y ≠ 0) sec θ = r/x (x ≠ 0) cot θ = x/y (y ≠ 0)
Note that r is always positive. The signs of the trig functions depend on the signs of x and y, which depend on the quadrant.
Evaluating from a Point on the Terminal Side
Example 1: Given a Point
The point (−3, 4) lies on the terminal side of angle θ. Find all six trig functions.
Solution:
x = −3, y = 4, r = √(9 + 16) = √25 = 5
sin θ = 4/5 cos θ = −3/5 tan θ = 4/(−3) = −4/3
csc θ = 5/4 sec θ = 5/(−3) = −5/3 cot θ = −3/4
Example 2: Point in Quadrant III
The point (−5, −12) lies on the terminal side. Find sin θ and cos θ.
Solution: r = √(25 + 144) = 13. sin θ = −12/13, cos θ = −5/13.
Using Reference Angles
Recall from Module 2: the reference angle is the acute angle between the terminal side and the x-axis. To evaluate any trig function:
- Find the reference angle θ′
- Evaluate the trig function at θ′ (using special values or a calculator)
- Apply the correct sign based on the quadrant
Example 3: Evaluating sin(225°)
Step 1: 225° is in QIII. Reference angle: 225° − 180° = 45°
Step 2: sin 45° = √2/2
Step 3: Sin is negative in QIII: sin 225° = −√2/2
Example 4: Evaluating tan(300°)
Step 1: 300° is in QIV. Reference angle: 360° − 300° = 60°
Step 2: tan 60° = √3
Step 3: Tan is negative in QIV: tan 300° = −√3
Determining the Quadrant from Sign Information
Example 5: Finding the Quadrant
If sin θ < 0 and cos θ > 0, in which quadrant is θ?
Solution: Sin negative means y < 0 (below x-axis). Cos positive means x > 0 (right of y-axis). This is Quadrant IV.
Undefined Trig Values
Some trig functions are undefined at certain angles because of division by zero:
- tan θ and sec θ are undefined when cos θ = 0, i.e., at θ = π/2, 3π/2, etc.
- cot θ and csc θ are undefined when sin θ = 0, i.e., at θ = 0, π, 2π, etc.
Check Your Understanding
1. The point (8, −6) lies on the terminal side of θ. Find sin θ.
2. Evaluate cos(150°).
3. If tan θ > 0 and sin θ < 0, name the quadrant.
Key Takeaways
- The coordinate definition uses a point (x, y) on the terminal side and r = √(x²+y²).
- Reference angles reduce any evaluation to a first-quadrant problem plus a sign.
- The sign is determined by the quadrant (ASTC rule).
- Trig functions are undefined when the denominator (x or y) is zero.