The Unit Circle

The unit circle is a circle with:

• Center at the origin $(0, 0)$
• Radius of 1

Its equation is: $x^2 + y^2 = 1$

Why the Unit Circle Matters

The unit circle allows us to define trigonometric functions for all angles, not just acute angles in right triangles.

For any angle $\theta$ in standard position:

• $\cos(\theta) = x$-coordinate of the point on the unit circle
• $\sin(\theta) = y$-coordinate of the point on the unit circle
• $\tan(\theta) = \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)}$

Radian Measure

A radian is the angle formed when the arc length equals the radius.

Key Conversions

$180° = \pi \text{ radians}$

To convert:

• Degrees to radians: multiply by $\frac{\pi}{180}$
• Radians to degrees: multiply by $\frac{180}{\pi}$

Common Angle Conversions

Special Angles on the Unit Circle

First Quadrant ($0° \text{ to } 90°$ or $0 \text{ to } \frac{\pi}{2}$)

Reference Angles

A reference angle is the acute angle formed between the terminal side and the x-axis.

Finding Reference Angles

• Quadrant I: reference angle = $\theta$
• Quadrant II: reference angle = $180° - \theta$ or $\pi - \theta$
• Quadrant III: reference angle = $\theta - 180°$ or $\theta - \pi$
• Quadrant IV: reference angle = $360° - \theta$ or $2\pi - \theta$

CAST Rule (Signs in Each Quadrant)

Remembering which trig functions are positive in each quadrant:

• Quadrant I: All positive (sine, cosine, tangent)
• Quadrant II: Sine positive only
• Quadrant III: Tangent positive only
• Quadrant IV: Cosine positive only

Memory trick: "All Students Take Calculus"

Arc Length and Sector Area

For a circle with radius $r$ and central angle $\theta$ (in radians):

Arc length: $s = r\theta$

Sector area: $A = \frac{1}{2}r^2\theta$

Part b) Convert $\frac{5\pi}{6}$ radians to degrees

Multiply by $\frac{180}{\pi}$:

$\frac{5\pi}{6} \times \frac{180}{\pi} = \frac{5 \times 180}{6} = \frac{900}{6} = 150°$

Answers:

• a) $\frac{3\pi}{4}$ radians
• b) $150°$

a) In which quadrant is point $P$? b) If $P$ corresponds to angle $\theta$ in standard position, find $\sin\theta$, $\cos\theta$, and $\tan\theta$. c) Find $\sec\theta$ and $\csc\theta$.

Given:

• Radius: $r = 8$ cm
• Central angle: $\theta = \frac{2\pi}{3}$ radians

Arc Length:

Using $s = r\theta$:

$s = 8 \cdot \frac{2\pi}{3} = \frac{16\pi}{3} \text{ cm}$

$s \approx 16.76 \text{ cm}$

Sector Area:

Using $A = \frac{1}{2}r^2\theta$:

$A = \frac{1}{2}(8)^2 \cdot \frac{2\pi}{3}$

$A = \frac{1}{2} \cdot 64 \cdot \frac{2\pi}{3}$

$A = 32 \cdot \frac{2\pi}{3} = \frac{64\pi}{3} \text{ cm}^2$

$A \approx 67.02 \text{ cm}^2$

Answers:

• Arc length: $\frac{16\pi}{3} \approx 16.76$ cm
• Sector area: $\frac{64\pi}{3} \approx 67.02$ cm²