Precalculus Trigonometry: Angles, Triangles, and Trigonometric Functions

Trigonometric Functions and Angles in Standard Position

Definition and Sketching of Angles

Angles in standard position have their vertex at the origin and their initial side along the positive x-axis. The terminal side is determined by the angle's measure, which can be positive (counterclockwise) or negative (clockwise).

• Reference Angle: The smallest angle between the terminal side and the x-axis.

• Coordinates on Terminal Side: For a point (x, y) on the terminal side, the distance from the origin is $r = \sqrt{x^2 + y^2}$.

• Trigonometric Functions:

  • $\sin \theta = \frac{y}{r}$

  • $\cos \theta = \frac{x}{r}$

  • $\tan \theta = \frac{y}{x}$

  • $\csc \theta = \frac{r}{y}$

  • $\sec \theta = \frac{r}{x}$

  • $\cot \theta = \frac{x}{y}$

• Coterminal Angles: Angles that share the same terminal side, differing by multiples of $360^\circ$ or $2\pi$ radians.

Example: For $120^\circ$ in standard position, using a 30-60-90 triangle, if $r=2$, $x=-1$, $y=\sqrt{3}$:

• $\cos(120^\circ) = \frac{-1}{2}$

• $\sin(120^\circ) = \frac{\sqrt{3}}{2}$

• $\tan(120^\circ) = -\sqrt{3}$

Special Right Triangles

Special right triangles are used to find exact trigonometric values for common angles.

• 30-60-90 Triangle: Side ratios are $1 : \sqrt{3} : 2$.

• 45-45-90 Triangle: Side ratios are $1 : 1 : \sqrt{2}$.

Example: For $-150^\circ$, reference angle is $30^\circ$, and in quadrant III, $x=-\sqrt{3}$, $y=-1$, $r=2$.

• $\cos(-150^\circ) = \frac{-\sqrt{3}}{2}$

• $\sin(-150^\circ) = \frac{-1}{2}$

• $\tan(-150^\circ) = \frac{1}{\sqrt{3}}$

Trigonometric Functions for Arbitrary Points

Finding Trig Values from Coordinates

Given a point $(x, y)$ on the terminal side of an angle $\theta$, calculate $r$ and then the six trigonometric functions.

• Example: For $(-12, -5)$, $r = \sqrt{(-12)^2 + (-5)^2} = 13$.

• $\sin \theta = \frac{-5}{13}$

• $\tan \theta = \frac{-5}{-12} = \frac{5}{12}$

• $\sec \theta = \frac{13}{-12}$

Exact Values and Reference Angles

Solving for Angles Given Trig Values

To find all angles $\theta$ in $[0^\circ, 360^\circ]$ or $[0, 2\pi]$ that satisfy a given trigonometric equation, use reference angles and quadrant information.

• Example: $\sin \theta = -0.71648873$

• Reference angle: $\theta = \sin^{-1}(0.71648873) = 45.8^\circ$

• Since sine is negative in QIII and QIV: $\theta = 180 + 45.8 = 225.8^\circ$, $\theta = 360 - 45.8 = 314.2^\circ$

Special Angles and Their Trig Values

Memorize the trigonometric values for $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$.

Angle	$\sin \theta$	$\cos \theta$	$\tan \theta$
$0^\circ$	$0$	$1$	$0$
$30^\circ$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$45^\circ$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$60^\circ$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$90^\circ$	$1$	$0$	undefined

Trigonometric Identities

Reciprocal and Pythagorean Identities

These identities are fundamental for simplifying and solving trigonometric equations.

• Reciprocal Identities:

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

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

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

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

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

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

• Pythagorean Identities:

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

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

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

Applications of Trigonometry

Solving Right Triangles

Use trigonometric ratios to solve for unknown sides or angles in right triangles.

• Example: Given $A = 26^\circ$, $a = 125$, $C = 90^\circ$:

• Use $\sin$, $\cos$, or $\tan$ to find missing sides.

Word Problems Involving Trigonometry

Trigonometry is used to solve real-world problems involving heights, distances, and angles.

• Shadow Problem: If a building casts a 69 ft shadow at $36^\circ$ elevation, $\tan 36^\circ = \frac{\text{building}}{69}$, so $\text{building} = 69 \times \tan 36^\circ = 50.13$ ft.

• Ladder Problem: For a 10 m wall and ladder 3.2 m from the wall, $\tan x = \frac{3.2}{10}$, $x = \tan^{-1}(0.32)$.

• Navigation Problems: Use the Law of Cosines or Pythagorean Theorem to find distances between moving objects.

Unit Circle and Trigonometric Values

Unit Circle Basics

The unit circle is a circle of radius 1 centered at the origin. It is used to define trigonometric functions for all real numbers.

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

• Common angles are marked in both degrees and radians.

Finding Exact Values Using the Unit Circle

• Example: $\sin \frac{\pi}{2} = 1$, $\cos \frac{5\pi}{3} = \frac{1}{2}$, $\tan \frac{2\pi}{3} = -\sqrt{3}$.

• Use reference angles and quadrant signs to determine values.

Graphing Trigonometric Functions

Identifying Key Features

Trigonometric functions can be transformed by changing amplitude, period, phase shift, and vertical translation.

• Amplitude: The height from the midline to the peak.

• Period: The length of one complete cycle.

• Phase Shift: Horizontal shift of the graph.

• Vertical Translation: Up or down shift of the graph.

Example: For $y = 3 + 5\sin\left(\pi x + \frac{\pi}{4}\right)$:

• Amplitude: $5$

• Period: $\frac{2\pi}{\pi} = 2$

• Phase Shift: $-\frac{1}{4}$

• Vertical Translation: $3$

Things to Memorize

• Ratios for 30-60-90 and 45-45-90 triangles

• Definitions of trigonometric functions in standard position

• Trigonometric identities

• Right triangle trigonometric functions

• Table of trigonometric values for special angles from $0^\circ$ to $90^\circ$

Additional info: Some context and explanations have been expanded for clarity and completeness, including the use of reference angles, quadrant analysis, and applications in word problems.