Extending Trigonometry Beyond Right Triangles

The Unit Circle: Foundation of Circular Functions

Definition: A circle with radius 1 centered at the origin

Equation: $$x^2+y^2=1$$

Circular Function Definitions:

For any angle θ in standard position, where the terminal side intersects the unit circle at point (x, y):

$$x=\cos\theta , y=\sin\theta , \tan\theta=\frac{y}{x} (x\neq0)$$

This extends trig functions to ALL angles, not just acute ones!

The Complete Unit Circle 

                        (0,1)
                           90° (π/2)
                           |
            (√2/2, √2/2)  |  (√2/2, √2/2)
           45° (π/4)      |      45° (π/4)
              \           |           /
               \          |          /
                \         |         /
                 \        |        /
                  \       |       /
                   \      |      /
                    \     |     /
                     \    |    /
                      \   |   /
                       \  |  /
                        \ | /
(-1,0) 180° (π) ---------(0,0)--------- 0° (0) (1,0)
                        / | \
                       /  |  \
                      /   |   \
                     /    |    \
                    /     |     \
                   /      |      \
                  /       |       \
                 /        |        \
                /         |         \
               /          |          \
   (√2/2, -√2/2)  |  (√2/2, -√2/2)
  45° (π/4)       |       45° (π/4)
                  |
                       270° (3π/2)
                       (0,-1)

Key Points to Remember:

• Quadrant I: (cos θ, sin θ) both positive

• Quadrant II: cos negative, sin positive

• Quadrant III: both negative

• Quadrant IV: cos positive, sin negative

Trigonometric Functions of Any Angle

Step-by-Step Method:

1. Find reference angle θ’ (acute angle to x-axis)

2. Determine sign based on quadrant (ASTC)

3. Use known values for θ’

4. Apply sign

Example: Find sin 210°

1. 210° is in QIII (180° < 210° < 270°)

2. Reference angle = 210° – 180° = 30°

3. sin 30° = 1/2

4. In QIII, sin is negative

5. sin 210° = -1/2

Periodicity: The Key to Understanding Circular Functions

Sine and Cosine:

$$\sin(\theta+2\pi{n})=\sin\theta , \cos(\theta+2\pi{n})=\cos\theta$$

for any integer $n$

Period: 2π (360°)

Tangent:

$$\tan(\theta+\pi{n})=\tan\theta$$

for any integer n

Period: π (180°)

Symmetry Properties

Even/Odd Functions:

• Cosine is EVEN: cos⁡(−θ)=cos⁡θ

• Sine is ODD: sin⁡(−θ)=−sin⁡θ

• Tangent is ODD: tan⁡(−θ)=−tan⁡θ

Cofunction Identities:

$$\sin\left(\frac{\pi}{2}-\theta\right)=\cos\theta , \cos\left(\frac{\pi}{2}-\theta\right)=\sin\theta , \tan\left(\frac{\pi}{2}-\theta\right)=\cot\theta$$

Graphs of Circular Functions

Sine Function: f(x)=sin⁡x

• Amplitude: 1 (height from midline)

• Period: 2π

• Domain: All real numbers

• Range: [−1,1]

• Zeros: x=nπ , $n\in \mathbb{Z}$

• Maxima: $$\frac{\pi}{2}+2\pi{n}$$

• Minima: $$\frac{3\pi}{2}+2\pi{n}$$

Cosine Function: $f(x)=\cos x$

• Amplitude: 1

• Period: 2π

• Domain: All real numbers

• Range: [−1,1]

• Zeros: $$\frac{\pi}{2}+\pi{n}$$

• Maxima: 2πn

• Minima: π+2πn

Tangent Function: f(x)=tan⁡x

• Amplitude: None (unbounded)

• Period: π

• Domain: $$x\neq\frac{\pi}{2}+\pi{n}$$

• Range: All real numbers

• Asymptotes: $$x=\frac{\pi}{2}+\pi{n}$$

• Zeros: x=πn