Trigonometric Functions and Their Properties: Study Notes for Precalculus

Trigonometric Functions on Right Triangles

Definition and Basic Ratios

Trigonometric functions relate the angles of a right triangle to the ratios of its sides. These functions are fundamental in precalculus and are used to solve problems involving triangles and periodic phenomena.

• Sine (sin): Ratio of the length of the opposite side to the hypotenuse.

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

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

• Cosecant (csc): Reciprocal of sine.

• Secant (sec): Reciprocal of cosine.

• Cotangent (cot): Reciprocal of tangent.

Formulas:

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

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

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

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

Mnemonic: SOH-CAH-TOA helps remember the definitions:

• SOH: Sine = Opposite / Hypotenuse

• CAH: Cosine = Adjacent / Hypotenuse

• TOA: Tangent = Opposite / Adjacent

Example: In a right triangle with an angle $\theta = 45^\circ$, both legs are equal, so:

• $\sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}}$

• $\tan 45^\circ = 1$

Unit Circle and Reference Angles

Special Angles and Their Trigonometric Values

The unit circle allows us to define trigonometric functions for all angles, not just those in right triangles. Reference angles help compute trig values for any angle using known values from the first quadrant.

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

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

• Quadrants: The coordinate plane is divided into four quadrants, each affecting the sign of trig functions.

Common Angles (in degrees and radians):

Angle (°)	Angle (rad)	sin	cos	tan
0	$0$	$0$	$1$	$0$
30	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
45	$\frac{\pi}{4}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}$	$1$
60	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
90	$\frac{\pi}{2}$	$1$	$0$	undefined

Signs of Trig Functions by Quadrant:

• Quadrant I: All functions positive

• Quadrant II: Sine and cosecant positive

• Quadrant III: Tangent and cotangent positive

• Quadrant IV: Cosine and secant positive

Mnemonic: "All Students Take Calculus" (A: All, S: Sine, T: Tangent, C: Cosine)

Example: $\sin 135^\circ = \sin (180^\circ - 45^\circ) = \sin 45^\circ = \frac{1}{\sqrt{2}}$ (since sine is positive in Quadrant II)

Reciprocal and Quotient Identities

Relationships Among Trigonometric Functions

Reciprocal and quotient identities express trigonometric functions in terms of one another.

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

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

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

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

Example: If $\sin \theta = \frac{3}{5}$ and $\theta$ is in Quadrant II, then $\cos \theta = -\frac{4}{5}$ and $\tan \theta = -\frac{3}{4}$.

Evaluating Trigonometric Functions for Any Angle

Using Reference Angles and Signs

To find the value of a trigonometric function for any angle:

1. Find the reference angle (the acute angle formed with the x-axis).

2. Evaluate the trig function for the reference angle.

3. Determine the correct sign based on the quadrant.

Example: $\cos \left( \frac{4\pi}{3} \right ) = -\frac{1}{2}$ (since $\frac{4\pi}{3}$ is in Quadrant III, where cosine is negative).

Solving Right Triangles

Finding Side Lengths and Angles

Given one side and one non-right angle in a right triangle, use trigonometric ratios to find unknown sides or angles.

• Example: If $\sin \theta = \frac{3}{5}$, then the opposite side is 3, hypotenuse is 5, and the adjacent side is $\sqrt{5^2 - 3^2} = 4$.

• Use Pythagorean theorem: $a^2 + b^2 = c^2$.

Summary Table: Trigonometric Values for Special Angles

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

Additional info: Some content was inferred and expanded for clarity, including the structure of the unit circle, the use of reference angles, and the summary tables for special angles. The notes also clarify the use of mnemonics and the process for evaluating trigonometric functions in all quadrants.