Comprehensive Study Guide: Core Topics in College Trigonometry

Trigonometric Angles and Radian Measure

Degree and Radian Conversion

Understanding how to convert between degrees and radians is fundamental in trigonometry, as both units are used to measure angles.

• Degrees to Radians: Multiply the degree measure by $\frac{\pi}{180}$.

• Radians to Degrees: Multiply the radian measure by $\frac{180}{\pi}$.

• Example: Convert $36^\circ$ to radians: $36^\circ \times \frac{\pi}{180} = \frac{\pi}{5}$.

Evaluating Trigonometric Functions at Special Angles

Exact values of trigonometric functions at standard angles (such as $30^\circ$, $45^\circ$, $60^\circ$, and their radian equivalents) are essential for solving many problems.

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

• Example: $\sin \frac{\pi}{6} = \frac{1}{2}$, $\cos \frac{\pi}{3} = \frac{1}{2}$.

Arc Length and Sector Area

Applications of Radian Measure

Radian measure is used to calculate arc lengths and areas of sectors in circles.

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

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

• Example: For $r = 12.3$ cm and $\theta = \frac{2\pi}{3}$, $s = 12.3 \times \frac{2\pi}{3}$.

The Unit Circle

Exact Values and Properties

The unit circle allows for the determination of trigonometric function values for any angle.

• Definition: A circle of radius 1 centered at the origin in the coordinate plane.

• Coordinates: For an angle $\theta$, the coordinates $(x, y)$ on the unit circle are $(\cos \theta, \sin \theta)$.

• Example: $\cos \frac{2\pi}{3} = -\frac{1}{2}$, $\sin \frac{7\pi}{6} = -\frac{1}{2}$.

Calculator Approximations

Decimal Values of Trigonometric Functions

Trigonometric functions can be evaluated using a calculator to obtain decimal approximations.

• Example: $\tan 5.09632 \approx -0.44067$ (rounded to five decimal places).

Linear and Angular Speed

Relating Rotational and Linear Motion

Linear speed and angular speed describe how fast a point moves along a circular path.

• Linear Speed ($v$): $v = r\omega$, where $\omega$ is angular speed in radians per unit time.

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

• Example: If a drum of diameter 16.0 in rotates 52 times per minute, the angular speed is $\omega = 52 \times 2\pi$ radians/minute.

Trigonometric Identities

Fundamental Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values in their domains.

• Pythagorean Identities:

  • $\sin^2 \theta + \cos^2 \theta = 1$

  • $1 + \tan^2 \theta = \sec^2 \theta$

  • $1 + \cot^2 \theta = \csc^2 \theta$

• Reciprocal Identities:

  • $\sin \theta = \frac{1}{\csc \theta}$

  • $\cos \theta = \frac{1}{\sec \theta}$

  • $\tan \theta = \frac{1}{\cot \theta}$

• Quotient Identities:

  • $\tan \theta = \frac{\sin \theta}{\cos \theta}$

  • $\cot \theta = \frac{\cos \theta}{\sin \theta}$

Verifying Trigonometric Identities

To verify an identity, manipulate one or both sides using algebraic and trigonometric properties until both sides are identical.

• Example: Verify $\tan \theta \cdot \csc \theta = \sec \theta$.

Sum, Difference, Double, and Half-Angle Identities

Sum and Difference Identities

These identities allow the calculation of trigonometric functions for sums or differences of angles.

• Sine: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

• Cosine: $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

• Tangent: $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

• Example: $\cos(165^\circ) = \cos(120^\circ + 45^\circ)$

Double-Angle Identities

Double-angle identities express trigonometric functions of $2\theta$ in terms of $\theta$.

• $\sin 2\theta = 2 \sin \theta \cos \theta$

• $\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$

• $\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}$

Half-Angle Identities

Half-angle identities are used to find the sine, cosine, or tangent of half an angle.

• $\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}$

• $\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}$

• $\tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}$

Product-to-Sum and Sum-to-Product Identities

These identities convert products of sines and cosines into sums or differences, and vice versa.

• Product-to-Sum: $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$

• Sum-to-Product: $\sin A + \sin B = 2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$

Solving Trigonometric Equations

Trigonometric equations can be solved for all solutions in a given interval, often using identities and algebraic manipulation.

• Example: Solve $\sin x = \frac{1}{2}$ for $x$ in $[0, 2\pi]$.

Table: Common Trigonometric Values

The following table summarizes the exact values of sine and cosine for common angles:

Angle	$\sin \theta$	$\cos \theta$
$0^\circ$ ($0$)	$0$	$1$
$30^\circ$ ($\frac{\pi}{6}$)	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$
$45^\circ$ ($\frac{\pi}{4}$)	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$
$60^\circ$ ($\frac{\pi}{3}$)	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$
$90^\circ$ ($\frac{\pi}{2}$)	$1$	$0$

Summary

• Trigonometry involves the study of angles, their measurement, and the relationships between the sides and angles of triangles.

• Key skills include converting between degrees and radians, evaluating trigonometric functions, applying identities, and solving equations.

• Applications include circular motion, arc length, sector area, and modeling periodic phenomena.