Trigonometric Functions, Reference Angles, and Quadrant Analysis

Trigonometric Functions and Their Evaluation

Six Trigonometric Functions

The six fundamental trigonometric functions relate the angles of a right triangle to the ratios of its sides. They are defined for any angle using the unit circle or coordinates of a point on the terminal side of the angle.

• Sine (sin): $\sin \theta = \dfrac{y}{r}$

• Cosine (cos): $\cos \theta = \dfrac{x}{r}$

• Tangent (tan): $\tan \theta = \dfrac{y}{x}$

• Cosecant (csc): $\csc \theta = \dfrac{r}{y}$

• Secant (sec): $\sec \theta = \dfrac{r}{x}$

• Cotangent (cot): $\cot \theta = \dfrac{x}{y}$

Where $(x, y)$ is a point on the terminal side of angle $\theta$, and $r = \sqrt{x^2 + y^2}$.

Example: For the point $(-12, 5)$, $r = \sqrt{(-12)^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13$.

• $\sin \theta = \dfrac{5}{13}$

• $\cos \theta = \dfrac{-12}{13}$

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

• $\csc \theta = \dfrac{13}{5}$

• $\sec \theta = -\dfrac{13}{12}$

• $\cot \theta = -\dfrac{12}{5}$

Trigonometric Functions at Quadrantal Angles

Quadrantal angles are angles whose terminal sides lie along the axes (e.g., $0$,$\dfrac{\pi}{2}$,$\pi$,$\dfrac{3\pi}{2}$, $2\pi$). Some trigonometric functions are undefined at these angles due to division by zero.

• Example: $\sec\left(\dfrac{\pi}{2}\right)$ is undefined because $\cos\left(\dfrac{\pi}{2}\right) = 0$.

• Example: $\cot\left(\dfrac{3\pi}{2}\right)$ is also undefined because $\tan\left(\dfrac{3\pi}{2}\right)$ is undefined.

Quadrant Analysis and Signs of Trigonometric Functions

Determining the Quadrant

The sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies:

• Quadrant I: All functions are positive.

• Quadrant II: Sine and cosecant are positive.

• Quadrant III: Tangent and cotangent are positive.

• Quadrant IV: Cosine and secant are positive.

Example: If $\sec \theta > 0$ and $\tan \theta < 0$, $\theta$ is in Quadrant IV.

Example: If $\csc \theta < 0$ and $\cot \theta > 0$, $\theta$ is in Quadrant III.

Reference Angles

Definition and Calculation

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 $90^\circ$ or $\dfrac{\pi}{2}$ radians.

• For angles in Quadrant I: Reference angle is the angle itself.

• Quadrant II: $\text{Reference angle} = 180^\circ - \theta$ or $\pi - \theta$

• Quadrant III: $\text{Reference angle} = \theta - 180^\circ$ or $\theta - \pi$

• Quadrant IV: $\text{Reference angle} = 360^\circ - \theta$ or $2\pi - \theta$

Example: The reference angle for $240^\circ$ is $240^\circ - 180^\circ = 60^\circ$.

Example: The reference angle for $\dfrac{11\pi}{3}$ is $\dfrac{11\pi}{3} - 2\pi = \dfrac{11\pi}{3} - \dfrac{6\pi}{3} = \dfrac{5\pi}{3}$, and then $2\pi - \dfrac{5\pi}{3} = \dfrac{6\pi}{3} - \dfrac{5\pi}{3} = \dfrac{\pi}{3}$.

Exact Values of Trigonometric Functions

Using Reference Angles

To find the exact value of a trigonometric function for any angle, use the reference angle and the sign determined by the quadrant.

• Example: $\csc(-210^\circ)$: Reference angle is $30^\circ$, and $-210^\circ$ is in Quadrant III, where sine is negative. $\csc(-210^\circ) = -2$.

• Example: $\cot\left(\dfrac{5\pi}{3}\right)$: Reference angle is $\dfrac{\pi}{3}$, and $\dfrac{5\pi}{3}$ is in Quadrant IV, where cotangent is negative.

Special Angles and Their Values

Common angles ($30^\circ$, $45^\circ$, $60^\circ$ or $\dfrac{\pi}{6}$, $\dfrac{\pi}{4}$, $\dfrac{\pi}{3}$) have well-known trigonometric values:

• $\sin \dfrac{\pi}{6} = \dfrac{1}{2}$, $\cos \dfrac{\pi}{6} = \dfrac{\sqrt{3}}{2}$

• $\sin \dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}$, $\cos \dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}$

• $\sin \dfrac{\pi}{3} = \dfrac{\sqrt{3}}{2}$, $\cos \dfrac{\pi}{3} = \dfrac{1}{2}$

Evaluating Expressions Involving Trigonometric Functions

To evaluate expressions such as $\sin \dfrac{\pi}{3} \cos \dfrac{\pi}{4} - \cos \dfrac{\pi}{3} \sin \dfrac{\pi}{4}$, substitute the exact values and simplify:

• $\sin \dfrac{\pi}{3} = \dfrac{\sqrt{3}}{2}$, $\cos \dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}$

• $\cos \dfrac{\pi}{3} = \dfrac{1}{2}$, $\sin \dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}$

• So, $\dfrac{\sqrt{3}}{2} \cdot \dfrac{\sqrt{2}}{2} - \dfrac{1}{2} \cdot \dfrac{\sqrt{2}}{2} = \dfrac{\sqrt{6}}{4} - \dfrac{\sqrt{2}}{4} = \dfrac{\sqrt{6} - \sqrt{2}}{4}$

Summary Table: Signs of Trigonometric Functions by Quadrant

Quadrant	sin, csc	cos, sec	tan, cot
I	+	+	+
II	+	-	-
III	-	-	+
IV	-	+	-

Additional info:

• Some context and examples were inferred to provide a self-contained study guide, as the original file consisted of questions only.

• All formulas and values are standard for Precalculus trigonometry.