Arc Length and Unit Circle Study Guide – Trigonometry

Q1. Compute the arc length for a circle with radius 11 ft and central angle 315°.

Background

Topic: Arc Length in Circles

This question tests your ability to use the arc length formula for circles, which relates the radius and the central angle (in radians) to the length of the arc.

Key Terms and Formulas

• Arc length ($s$): The distance along the curved part of the circle.

• Radius ($r$): The distance from the center to the edge of the circle.

• Central angle ($\theta$): The angle subtended by the arc at the center, measured in radians.

Arc Length Formula:

$s = r \theta$

Remember: The angle $\theta$ must be in radians for this formula.

Step-by-Step Guidance

1. Identify the given values: $r = 11$ ft, $\theta = 315^\circ$.

2. Convert the central angle from degrees to radians using $\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}$.

3. Plug the radius and the radian measure of the angle into the arc length formula: $s = r \theta$.

Try solving on your own before revealing the answer!

Q2. Compute the arc length for a circle with radius 13 ft and central angle 270°.

Background

Topic: Arc Length in Circles

This question also tests your ability to use the arc length formula, requiring conversion of degrees to radians.

Key Terms and Formulas

• Arc length ($s$)

• Radius ($r$)

• Central angle ($\theta$)

Arc Length Formula:

$s = r \theta$

Step-by-Step Guidance

1. Identify the given values: $r = 13$ ft, $\theta = 270^\circ$.

2. Convert $270^\circ$ to radians: $\theta_{\text{rad}} = 270 \times \frac{\pi}{180}$.

3. Plug the radius and radian angle into $s = r \theta$.

Try solving on your own before revealing the answer!

Q3. Compute the arc length for a circle with radius 16 ft and central angle $\frac{3\pi}{2}$ radians.

Background

Topic: Arc Length in Circles

This question tests your ability to use the arc length formula when the angle is already given in radians.

Key Terms and Formulas

• Arc length ($s$)

• Radius ($r$)

• Central angle ($\theta$) in radians

Arc Length Formula:

$s = r \theta$

Step-by-Step Guidance

1. Identify the given values: $r = 16$ ft, $\theta = \frac{3\pi}{2}$ radians.

2. Since the angle is already in radians, you can directly use the formula $s = r \theta$.

3. Multiply the radius by the radian measure to set up the calculation.

Try solving on your own before revealing the answer!

Q4. Use the unit circle to find exact values of trigonometric functions for common angles.

Background

Topic: Unit Circle and Trigonometric Functions

This question tests your ability to use the unit circle to determine sine, cosine, tangent, and other trig function values for specific angles.

Key Terms and Formulas

• Unit circle: A circle with radius 1 centered at the origin.

• Coordinates $(x, y)$ on the unit circle correspond to $(\cos \theta, \sin \theta)$ for angle $\theta$.

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

Key relationships:

$\sin \theta = y$

$\cos \theta = x$

$\tan \theta = \frac{y}{x}$

Step-by-Step Guidance

1. Locate the angle on the unit circle diagram.

2. Read the coordinates $(x, y)$ for the angle.

3. Use the coordinates to find $\sin \theta$, $\cos \theta$, and $\tan \theta$.

Try solving on your own before revealing the answer!