11.2 Trigonometric Functions and the Unit Circle

Trigonometric functions and the unit circle are the foundation for understanding periodic behavior in math. By connecting angle measures to coordinates on a circle of radius 1, you get precise definitions of sine, cosine, and the other trig functions that work for any angle, not just acute ones in a right triangle.

Trigonometric Functions on the Unit Circle

Defining Trigonometric Functions

The unit circle is a circle with radius 1 centered at the origin $(0, 0)$. To define trig functions, you start at the point $(1, 0)$ and rotate counterclockwise by an angle $\theta$. The point where the terminal side of $\theta$ hits the circle has coordinates $(x, y)$, and those coordinates are your trig values:

• $\cos\theta = x$ (the x-coordinate of the point)
• $\sin\theta = y$ (the y-coordinate of the point)

This connects directly to right triangle trig. On the unit circle, the radius is the hypotenuse and equals 1, so cosine simplifies to adjacent/hypotenuse = $x/1 = x$, and sine simplifies to opposite/hypotenuse = $y/1 = y$.

Reciprocal Trigonometric Functions

The remaining four trig functions are built from sine and cosine:

• Tangent: $\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}$ (opposite over adjacent in a right triangle)
• Secant: $\sec\theta = \frac{1}{\cos\theta}$ (hypotenuse over adjacent)
• Cosecant: $\csc\theta = \frac{1}{\sin\theta}$ (hypotenuse over opposite)
• Cotangent: $\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$ (adjacent over opposite)

Because these involve division, any time the denominator equals zero the function is undefined. For example, $\tan\theta$ is undefined whenever $\cos\theta = 0$ (at $90°$ and $270°$).

Evaluating Trigonometric Functions

Using the Unit Circle

To evaluate a trig function at angle $\theta$:

1. Draw (or visualize) the angle $\theta$ in standard position, starting from the positive x-axis.
2. Find the point $(x, y)$ where the terminal side meets the unit circle.
3. Read off the values: $\cos\theta = x$, $\sin\theta = y$.
4. Compute the other functions from those two values as needed.

When the terminal side lands on an axis (like $0°$, $90°$, $180°$, or $270°$), one or both coordinates are 0. That gives you clean values of 0 or 1 for sine and cosine, but it also makes some reciprocal functions undefined.

Quadrant Rules

The signs of $x$ and $y$ change depending on the quadrant, which determines which trig functions are positive. A common mnemonic is "All Students Take Calculus" (A-S-T-C), moving counterclockwise from Quadrant I:

• Quadrant I ($0° < \theta < 90°$): All trig functions are positive. Both $x$ and $y$ are positive.
• Quadrant II ($90° < \theta < 180°$): Only sin and csc are positive. Here $x < 0$ and $y > 0$.
• Quadrant III ($180° < \theta < 270°$): Only tan and cot are positive. Both $x$ and $y$ are negative, but their ratio is positive.
• Quadrant IV ($270° < \theta < 360°$): Only cos and sec are positive. Here $x > 0$ and $y < 0$.

Periodicity of Trigonometric Functions

Periodic Behavior

Trig functions are periodic, meaning they repeat their values at regular intervals. The period is the smallest positive value $p$ such that $f(\theta + p) = f(\theta)$ for all $\theta$.

• Sine, cosine, secant, and cosecant all have a period of $2\pi$ radians (360°). One full trip around the unit circle brings you back to the same $(x, y)$ point.
• Tangent and cotangent have a period of $\pi$ radians (180°). The ratio $y/x$ repeats after just a half rotation because both coordinates flip sign simultaneously.

Using Periodicity to Simplify

Periodicity lets you evaluate trig functions for any angle by finding a coterminal angle between $0°$ and $360°$ (or between $0$ and $2\pi$).

For example, to find $\sin(750°)$, subtract $360°$ twice: $750° - 720° = 30°$. So $\sin(750°) = \sin(30°) = \frac{1}{2}$. The same idea works for negative angles: add $360°$ until you land in the first rotation.

Trigonometric Functions for Special Angles

Exact Values without a Calculator

Certain angles produce coordinates you can determine exactly. These come from the 30-60-90 and 45-45-90 special right triangles. You should memorize these five key points on the unit circle:

Notice the pattern: the cosine values for $0°$ through $90°$ are $\frac{\sqrt{4}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{0}}{2}$. The sine values follow the same sequence in reverse. That pattern can help you reconstruct the table quickly on a test.

Reference Angles and Quadrant Rules

To find exact values for angles outside Quadrant I, use a reference angle, which is the acute angle between the terminal side and the x-axis.

1. Determine which quadrant the angle falls in.
2. Find the reference angle (the positive acute angle to the nearest part of the x-axis).
3. Look up the trig values for that reference angle from the table above.
4. Apply the correct sign using the quadrant rules (A-S-T-C).

For example, to evaluate $\sin(150°)$: the angle is in Quadrant II, and the reference angle is $180° - 150° = 30°$. Sine is positive in Quadrant II, so $\sin(150°) = +\sin(30°) = \frac{1}{2}$. For $\cos(150°)$, cosine is negative in Quadrant II, so $\cos(150°) = -\cos(30°) = -\frac{\sqrt{3}}{2}$.