Chapter 3: Radian Measure and the Unit Circle – Study Notes

Section 3.1: Radian Measure

Definition and Conversion

Radian measure is a way of expressing angles based on the radius of a circle. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

• Degrees to Radians: To convert degrees to radians, use the formula: $\theta_{\text{radians}} = \theta_{\text{degrees}} \cdot \frac{\pi}{180}$

• Radians to Degrees: To convert radians to degrees, use the formula: $\theta_{\text{degrees}} = \theta_{\text{radians}} \cdot \frac{180}{\pi}$

• Examples:

  • $108^\circ = 108 \cdot \frac{\pi}{180} = \frac{3\pi}{5}$ radians

  • $135^\circ = 135 \cdot \frac{\pi}{180} = \frac{3\pi}{4}$ radians

  • $325.7^\circ = 325.7 \cdot \frac{\pi}{180} \approx 5.685$ radians

Key Point: Radian measure is essential for calculus and higher mathematics because it relates angles directly to arc length and is dimensionless.

Section 3.2: Applications of Radian Measure

Arc Length and Sector Area

Radian measure allows for straightforward calculation of arc lengths and areas of sectors in circles.

• Arc Length: $s = r\theta$, where $s$ is the arc length, $r$ is the radius, and $\theta$ is the angle in radians.

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

• Examples:

  • Given $r = 25.60$ cm, $\theta = \frac{7\pi}{8}$, $s = 25.60 \cdot \frac{7\pi}{8} \approx 70.37$ cm

  • Given $r = 25.60$ cm, $\theta = 54^\circ = \frac{3\pi}{10}$ radians, $s = 25.60 \cdot \frac{3\pi}{10} \approx 24.13$ cm

  • Distance between two cities on Earth (radius $= 6400$ km), $\theta = \frac{\pi}{45}$ radians, $s = 6400 \cdot \frac{\pi}{45} \approx 890$ km

Key Point: Always convert angles to radians before using these formulas.

Section 3.3: The Unit Circle and Circular Functions

The Unit Circle

The unit circle is a circle of radius 1 centered at the origin of the coordinate plane. It is fundamental in defining the trigonometric functions for all real numbers.

• Coordinates: Any point on the unit circle corresponding to an angle $\theta$ has coordinates $(\cos \theta, \sin \theta)$.

• Reference Angles: The values of trigonometric functions for any angle can be determined using reference angles and the unit circle.

• Example: An angle of $-3\pi$ intersects the unit circle at $(-1, 0)$. Thus, $\sin(-3\pi) = 0$, $\cos(-3\pi) = -1$, $\cot(-3\pi)$ is undefined.

Key Points on the Unit Circle

The unit circle contains several important angles and their corresponding coordinates, which are used to evaluate trigonometric functions.

• Common Angles: $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \pi$, etc.

• Coordinates: For example, at $\frac{\pi}{4}$ (45°), the coordinates are $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

• Symmetry: The unit circle is symmetric about the x- and y-axes, which helps in determining the signs of trigonometric functions in different quadrants.

Evaluating Trigonometric Functions Using the Unit Circle

Trigonometric functions can be evaluated for any angle using the unit circle. The sine of an angle is the y-coordinate, and the cosine is the x-coordinate of the corresponding point on the unit circle.

• Example: $\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}$, $\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

• Example: $\tan\left(\frac{7\pi}{4}\right) = -1$

• Reference Angles: Use the reference angle and the quadrant to determine the sign of the function value.

Special Right Triangles and Trigonometric Ratios

Special right triangles, such as the 45°-45°-90° and 30°-60°-90° triangles, are used to derive the values of trigonometric functions for common angles.

• 45°-45°-90° Triangle: The legs are equal, and the hypotenuse is $\sqrt{2}$ times a leg.

• 30°-60°-90° Triangle: The sides are in the ratio $1 : \sqrt{3} : 2$.

• Application: These triangles help in finding the coordinates on the unit circle for $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$, etc.

Geometric Representation of Trigonometric Functions

Trigonometric functions can be represented geometrically using the unit circle and associated right triangles.

• Cosine and Sine: The x- and y-coordinates of a point on the unit circle.

• Secant and Cosecant: Represented as line segments from the origin to the circle or extended lines.

• Tangent and Cotangent: Represented as lengths of segments tangent to the circle at specific points.

• Example: For $\theta = 45^\circ$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$, $\sin 45^\circ = \frac{\sqrt{2}}{2}$, $\tan 45^\circ = 1$.

Section 3.4: Linear and Angular Speed

Definitions and Formulas

Linear and angular speed describe the motion of objects along circular paths.

• Angular Speed ($\omega$): $\omega = \frac{\theta}{t}$, where $\theta$ is in radians and $t$ is time.

• Linear Speed ($v$): $v = r\omega$, where $r$ is the radius.

• Examples:

  • Given $r = 15$ in, $\omega = \frac{\pi}{12}$ rad/s, $t = 10$ s, $\theta = \omega t = \frac{5\pi}{6}$ radians

  • Arc length: $s = r\theta = 15 \cdot \frac{5\pi}{6} \approx 39.3$ in

  • Linear speed: $v = r\omega = 15 \cdot \frac{\pi}{12} = \frac{5\pi}{4}$ in/s

Key Point: Linear speed is the rate at which distance is covered along the arc, while angular speed is the rate at which the angle changes.

Summary Table: Key Formulas

Concept	Formula	Variables
Degrees to Radians	$\theta_{\text{radians}} = \theta_{\text{degrees}} \cdot \frac{\pi}{180}$	$\theta_{\text{degrees}}$
Radians to Degrees	$\theta_{\text{degrees}} = \theta_{\text{radians}} \cdot \frac{180}{\pi}$	$\theta_{\text{radians}}$
Arc Length	$s = r\theta$	$r$ = radius, $\theta$ = radians
Sector Area	$A = \frac{1}{2} r^2 \theta$	$r$ = radius, $\theta$ = radians
Angular Speed	$\omega = \frac{\theta}{t}$	$\theta$ = radians, $t$ = time
Linear Speed	$v = r\omega$	$r$ = radius, $\omega$ = angular speed