Angles, Trigonometric Functions, and Their Applications in Precalculus

Angles and Their Measurement

Degrees and Radians

Angles are fundamental in precalculus, measured in degrees or radians. Understanding how to convert between these units is essential for solving trigonometric problems.

• Degree: A full rotation is 360°, and degrees are commonly used in geometry.

• Radian: One radian is the angle subtended by an arc equal in length to the radius of the circle. A full rotation is $2\pi$ radians.

• Conversion Formula: $\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$ $\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$

Example: Convert 180° to radians: $180 \times \frac{\pi}{180} = \pi$ radians.

Trigonometric Functions

Definitions and Properties

Trigonometric functions relate the angles of a triangle to the lengths of its sides. They are essential for analyzing periodic phenomena and solving geometric problems.

• Sine ($\sin$): Ratio of the opposite side to the hypotenuse in a right triangle.

• Cosine ($\cos$): Ratio of the adjacent side to the hypotenuse.

• Tangent ($\tan$): Ratio of the opposite side to the adjacent side.

• Cotangent ($\cot$): Reciprocal of tangent: $\cot \theta = \frac{1}{\tan \theta}$

• Secant ($\sec$): Reciprocal of cosine: $\sec \theta = \frac{1}{\cos \theta}$

• Cosecant ($\csc$): Reciprocal of sine: $\csc \theta = \frac{1}{\sin \theta}$

• Example: For $\theta = 30^\circ$, $\sin 30^\circ = 0.5$, $\cos 30^\circ = 0.866$, $\tan 30^\circ = 0.577$.

Inverse Trigonometric Functions

Inverse trigonometric functions allow you to determine the angle when given a trigonometric ratio.

• Arcsin ($\sin^{-1}$): If $\sin \theta = x$, then $\theta = \sin^{-1} x$.

• Arccos ($\cos^{-1}$): If $\cos \theta = x$, then $\theta = \cos^{-1} x$.

• Arctan ($\tan^{-1}$): If $\tan \theta = x$, then $\theta = \tan^{-1} x$.

• Example: If $\tan \theta = -1.281$, then $\theta = \tan^{-1}(-1.281) \approx -52.023^\circ$.

Reference Angles and Coterminal Angles

Reference Angles

A reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. It is always positive and less than or equal to 90°.

• Finding Reference Angles: - For angles in Quadrant II: $\text{Reference Angle} = 180^\circ - \theta$ - For Quadrant III: $\text{Reference Angle} = \theta - 180^\circ$ - For Quadrant IV: $\text{Reference Angle} = 360^\circ - \theta$

• Example: For $\theta = 230^\circ$ (Quadrant III), reference angle is $230^\circ - 180^\circ = 50^\circ$.

Coterminal Angles

Coterminal angles share the same initial and terminal sides but may differ by multiples of 360° (or $2\pi$ radians).

• Formula: $\text{Coterminal Angle} = \theta + 360^\circ \times n$, where $n$ is any integer.

• Example: $45^\circ$ and $405^\circ$ are coterminal because $405^\circ = 45^\circ + 360^\circ$.

Solving Right Triangles

Pythagorean Theorem

The Pythagorean Theorem relates the sides of a right triangle.

• Formula: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

• Example: If $a = 3$, $b = 4$, then $c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Using Trigonometric Ratios

Trigonometric ratios can be used to find unknown sides or angles in right triangles.

• Example: Given $\theta$ and one side, use $\sin \theta$, $\cos \theta$, or $\tan \theta$ to solve for other sides.

Summary Table: Common Trigonometric Values

This table summarizes the values of sine, cosine, and tangent for key angles.

Angle (°)	Angle (rad)	$\sin \theta$	$\cos \theta$	$\tan \theta$
0	0	0	1	0
30	$\frac{\pi}{6}$	0.5	0.866	0.577
45	$\frac{\pi}{4}$	0.707	0.707	1
60	$\frac{\pi}{3}$	0.866	0.5	1.732
90	$\frac{\pi}{2}$	1	0	undefined

Applications and Problem Solving

Finding Angles and Sides

Problems often require finding unknown angles or sides using trigonometric functions and their inverses.

• Example: If $\tan \theta = -1.281$, then $\theta = \tan^{-1}(-1.281) \approx -52.023^\circ$.

• Example: To find $\theta$ for $\cot \theta = 0.540$, use $\theta = \cot^{-1}(0.540)$.

Graphical Representation

Graphs of trigonometric functions help visualize their periodic nature and key properties such as amplitude, period, and phase shift.

• Sine and Cosine Graphs: Both have a period of $2\pi$ radians (360°).

• Tangent Graph: Has a period of $\pi$ radians (180°) and vertical asymptotes where $\cos \theta = 0$.

Additional info:

• Some content was inferred from context and standard precalculus curriculum, as handwriting and layout were partially unclear.

• Key formulas and examples were expanded for clarity and completeness.