Trigonometry Unit 3 Review: Radian Measure and the Unit Circle

Key Concepts and Definitions

• Radian measure expresses angles in terms of the radius of a circle
• One radian is the angle subtended by an arc length equal to the radius of the circle
• The circumference of a circle is $2\pi r$, where $r$ is the radius
• The unit circle has a radius of 1 and is centered at the origin (0, 0)
• Trigonometric functions (sine, cosine, tangent) relate angles to the coordinates of points on the unit circle
  • Sine is the y-coordinate of a point on the unit circle
  • Cosine is the x-coordinate of a point on the unit circle
  • Tangent is the ratio of the y-coordinate to the x-coordinate
• Special angles include 0°, 30°, 45°, 60°, 90°, and their radian equivalents

Radian Measure Basics

• Radian measure is a way to express the size of an angle using the radius of a circle
• One radian is defined as the angle formed when the arc length is equal to the radius
• To find the radian measure of an angle, divide the arc length by the radius: $\theta = \frac{s}{r}$
• The circumference of a circle is $2\pi r$, so a full circle has $2\pi$ radians
• Radian measure is often preferred in calculus and physics because it simplifies many equations
  • For example, the derivative of $\sin x$ is $\cos x$ when $x$ is in radians
• Radians are dimensionless, meaning they have no units
• To convert from radians to degrees, multiply by $\frac{180}{\pi}$

The Unit Circle: Structure and Properties

• The unit circle is a circle with a radius of 1 centered at the origin (0, 0)
• The equation of the unit circle is $x^2 + y^2 = 1$
• Angles are measured counterclockwise from the positive x-axis
• The coordinates of a point on the unit circle are $(\cos \theta, \sin \theta)$
  • $\cos \theta$ is the x-coordinate and represents the horizontal distance from the origin
  • $\sin \theta$ is the y-coordinate and represents the vertical distance from the origin
• The unit circle is symmetric about both the x-axis and y-axis
• Trigonometric functions have periodicities on the unit circle
  • Sine and cosine have a period of $2\pi$
  • Tangent has a period of $\pi$

Converting Between Degrees and Radians

• To convert from degrees to radians, multiply by $\frac{\pi}{180}$
  • For example, 90° = $90 \cdot \frac{\pi}{180} = \frac{\pi}{2}$ radians
• To convert from radians to degrees, multiply by $\frac{180}{\pi}$
  • For example, $\frac{\pi}{3}$ radians = $\frac{\pi}{3} \cdot \frac{180}{\pi} = 60$°
• Memorize common angle conversions, such as 30°, 45°, 60°, and 90°
  • 30° = $\frac{\pi}{6}$ radians
  • 45° = $\frac{\pi}{4}$ radians
  • 60° = $\frac{\pi}{3}$ radians
  • 90° = $\frac{\pi}{2}$ radians
• When solving problems, be consistent with the angle measure (degrees or radians)

Trigonometric Functions on the Unit Circle

• Sine, cosine, and tangent can be defined using the unit circle
• $\sin \theta$ is the y-coordinate of the point on the unit circle at angle $\theta$
• $\cos \theta$ is the x-coordinate of the point on the unit circle at angle $\theta$
• $\tan \theta$ is the ratio of the y-coordinate to the x-coordinate: $\tan \theta = \frac{\sin \theta}{\cos \theta}$
• Trigonometric functions have symmetries on the unit circle
  • $\sin(-\theta) = -\sin \theta$
  • $\cos(-\theta) = \cos \theta$
  • $\tan(-\theta) = -\tan \theta$
• Trigonometric identities can be derived from the unit circle
  • Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$
  • Reciprocal identities: $\csc \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\cot \theta = \frac{1}{\tan \theta}$

Special Angles and Their Values

• Memorize the sine, cosine, and tangent values for special angles (0°, 30°, 45°, 60°, 90°)
  • 0°: $\sin 0 = 0$, $\cos 0 = 1$, $\tan 0 = 0$
  • 30°: $\sin \frac{\pi}{6} = \frac{1}{2}$, $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$, $\tan \frac{\pi}{6} = \frac{\sqrt{3}}{3}$
  • 45°: $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, $\tan \frac{\pi}{4} = 1$
  • 60°: $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$, $\cos \frac{\pi}{3} = \frac{1}{2}$, $\tan \frac{\pi}{3} = \sqrt{3}$
  • 90°: $\sin \frac{\pi}{2} = 1$, $\cos \frac{\pi}{2} = 0$, $\tan \frac{\pi}{2}$ is undefined
• Use the unit circle to visualize these special angles and their trigonometric values
• Apply the symmetries of trigonometric functions to find values for angles in other quadrants

Applications and Problem-Solving

• Radian measure and the unit circle have numerous applications in mathematics, physics, and engineering
• Harmonic motion can be modeled using trigonometric functions on the unit circle
  • For example, the motion of a pendulum or a spring-mass system
• Trigonometric functions are used to analyze periodic phenomena, such as sound waves and electrical signals
• In navigation, angles measured in radians are used to calculate distances and bearings
• When solving problems, first identify the given information and the desired quantity
• Sketch the unit circle and label the given angle and coordinates, if applicable
• Use trigonometric identities and the properties of the unit circle to solve for the unknown values

Common Mistakes and How to Avoid Them

• Confusing degrees and radians
  • Always check the angle measure and convert if necessary
• Mixing up the x and y coordinates on the unit circle
  • Remember: cosine is the x-coordinate, and sine is the y-coordinate
• Forgetting to apply the negative sign when working with angles in the 3rd and 4th quadrants
  • Pay attention to the signs of the trigonometric functions in each quadrant
• Misusing trigonometric identities
  • Understand the conditions under which each identity is valid
• Rounding too early in calculations
  • Keep at least 4 decimal places throughout the problem and round only at the end
• Not checking the reasonableness of the answer
  • Estimate the expected range of the solution and verify that the answer makes sense in the context of the problem