Trigonometric Functions

You have probably heard of sine and cosine before, perhaps in a geometry class or while playing with a graphing calculator. Maybe someone once told you these functions have something to do with triangles, or you have seen them pop up in physics problems involving waves or circles. But what actually are these functions, and why do mathematicians treat them like old friends?

Here is the good news: if you have read the previous chapter on the unit circle, you already know what sine and cosine are. The x-coordinate of a point on the unit circle is cosine, and the y-coordinate is sine. That is it. In this chapter, we are going to build on that foundation by introducing four more trigonometric functions, exploring their properties, and discovering the elegant relationships that connect them all.

By the end, you will see that these six functions are not arbitrary inventions; they are natural ways to describe the geometry of circles and triangles, and they unlock a powerful toolkit for solving problems throughout mathematics, physics, and engineering.

Core Concepts

The Six Trigonometric Functions

In the previous chapter, we defined sine and cosine using the unit circle: for any angle $\theta$, if you draw the angle in standard position and find the point where the terminal side intersects the unit circle, that point has coordinates $(\cos\theta, \sin\theta)$.

From these two fundamental functions, we can define four more. Here is the complete family:

The Primary Pair:

• Sine: $\sin\theta = y$ (the y-coordinate on the unit circle)
• Cosine: $\cos\theta = x$ (the x-coordinate on the unit circle)

The Tangent Family:

• Tangent: $\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}$
• Cotangent: $\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{x}{y}$

The Reciprocal Functions:

• Secant: $\sec\theta = \frac{1}{\cos\theta} = \frac{1}{x}$
• Cosecant: $\csc\theta = \frac{1}{\sin\theta} = \frac{1}{y}$

Why do we need six functions when two seem to capture everything? The answer is convenience. Certain problems become much simpler when expressed using tangent or secant rather than sine and cosine. Over centuries of mathematical practice, these six functions emerged as the most useful combination for solving a wide variety of problems.

Understanding Tangent

The tangent function deserves special attention because it appears so frequently in applications. Since $\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}$, tangent represents the slope of the line from the origin to the point on the unit circle.

Think about what this means: as you rotate around the unit circle, the tangent tells you how steep that line is. At $\theta = 0$, the point is at $(1, 0)$, so the “slope” is $\frac{0}{1} = 0$. At $\theta = 45°$ (or $\frac{\pi}{4}$), the point is at $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, and the slope is $\frac{\sqrt{2}/2}{\sqrt{2}/2} = 1$.

What happens at $\theta = 90°$? The point is at $(0, 1)$, and we would need to compute $\frac{1}{0}$, which is undefined. This makes geometric sense: a vertical line has undefined slope. So $\tan 90°$ does not exist; we say it is undefined.

Exact Values for Special Angles

Certain angles appear so often in mathematics that knowing their exact trigonometric values is essential. Here is the complete table for the first quadrant:

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

Notice the beautiful pattern: the sine values at 0°, 30°, 45°, 60°, and 90° are $\frac{\sqrt{0}}{2}$, $\frac{\sqrt{1}}{2}$, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{3}}{2}$, and $\frac{\sqrt{4}}{2}$. The cosine values follow the same pattern but in reverse order.

For the reciprocal functions at these angles:

Angle	$\csc\theta$	$\sec\theta$	$\cot\theta$
0°	undefined	1	undefined
30°	2	$\frac{2\sqrt{3}}{3}$	$\sqrt{3}$
45°	$\sqrt{2}$	$\sqrt{2}$	1
60°	$\frac{2\sqrt{3}}{3}$	2	$\frac{\sqrt{3}}{3}$
90°	1	undefined	0

Signs in Each Quadrant (ASTC)

As you move around the unit circle into different quadrants, the signs of your trigonometric functions change. We saw in the previous chapter that sine and cosine take their signs from the y and x coordinates, respectively. The other four functions inherit their signs from these.

Here is the complete picture:

Quadrant	Sine	Cosine	Tangent	Cosecant	Secant	Cotangent
I (0° to 90°)	+	+	+	+	+	+
II (90° to 180°)	+	-	-	+	-	-
III (180° to 270°)	-	-	+	-	-	+
IV (270° to 360°)	-	+	-	-	+	-

The classic mnemonic is “All Students Take Calculus” (ASTC), which tells you what is positive in each quadrant:

• All functions positive in Quadrant I
• Sine (and its reciprocal, cosecant) positive in Quadrant II
• Tangent (and its reciprocal, cotangent) positive in Quadrant III
• Cosine (and its reciprocal, secant) positive in Quadrant IV

The Pythagorean Identities

From the definition of the unit circle, every point $(x, y) = (\cos\theta, \sin\theta)$ satisfies $x^2 + y^2 = 1$. This gives us the most fundamental trigonometric identity:

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

This is called the Pythagorean identity because it comes directly from the Pythagorean theorem applied to the unit circle.

By dividing this identity by different quantities, we get two more Pythagorean identities:

Dividing by $\cos^2\theta$: $$\frac{\sin^2\theta}{\cos^2\theta} + \frac{\cos^2\theta}{\cos^2\theta} = \frac{1}{\cos^2\theta}$$ $$\tan^2\theta + 1 = \sec^2\theta$$

Dividing by $\sin^2\theta$: $$\frac{\sin^2\theta}{\sin^2\theta} + \frac{\cos^2\theta}{\sin^2\theta} = \frac{1}{\sin^2\theta}$$ $$1 + \cot^2\theta = \csc^2\theta$$

These three Pythagorean identities are workhorses in trigonometry. Memorize the first one; the other two follow naturally.

Reciprocal and Quotient Identities

The relationships between the six trigonometric functions can be summarized in two categories:

Reciprocal Identities: $$\csc\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}$$

Quotient Identities: $$\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta}$$

These are not really new information; they are the definitions we started with. But organizing them this way helps you see how the six functions form a coherent family.

Even and Odd Properties

Functions can be classified as even, odd, or neither, based on their symmetry. This classification turns out to be very useful for simplifying expressions.

A function $f$ is even if $f(-x) = f(x)$ (symmetric about the y-axis). A function $f$ is odd if $f(-x) = -f(x)$ (symmetric about the origin).

For the trigonometric functions:

Even functions: $\cos(-\theta) = \cos\theta$ and $\sec(-\theta) = \sec\theta$

Odd functions: $\sin(-\theta) = -\sin\theta$, $\tan(-\theta) = -\tan\theta$, $\csc(-\theta) = -\csc\theta$, $\cot(-\theta) = -\cot\theta$

Why does this make sense geometrically? On the unit circle, if you go to angle $\theta$, you reach the point $(\cos\theta, \sin\theta)$. If you go to angle $-\theta$ (the same angle but clockwise), you reach the point $(\cos\theta, -\sin\theta)$. The x-coordinate stays the same (cosine is even), but the y-coordinate flips sign (sine is odd).

Evaluating Trig Functions for Any Angle

To find the exact value of a trigonometric function at any angle, follow these steps:

1. Find a coterminal angle between 0° and 360° (or between 0 and $2\pi$) by adding or subtracting multiples of 360° (or $2\pi$).

2. Identify the quadrant where the terminal side lands.

3. Find the reference angle (the acute angle to the x-axis).

4. Look up the trig values for the reference angle (one of the special angles 0°, 30°, 45°, 60°, or 90°).

5. Adjust the signs based on the quadrant (using ASTC).

This method works because the trigonometric functions are periodic and the reference angle captures the “core” angle that determines the magnitude of each function value.

Notation and Terminology

Term	Symbol	Meaning
Sine	$\sin\theta$	The y-coordinate on the unit circle at angle $\theta$
Cosine	$\cos\theta$	The x-coordinate on the unit circle at angle $\theta$
Tangent	$\tan\theta$	$\frac{\sin\theta}{\cos\theta}$; the slope of the terminal side
Cosecant	$\csc\theta$	$\frac{1}{\sin\theta}$; the reciprocal of sine
Secant	$\sec\theta$	$\frac{1}{\cos\theta}$; the reciprocal of cosine
Cotangent	$\cot\theta$	$\frac{\cos\theta}{\sin\theta}$; the reciprocal of tangent
Pythagorean identity	$\sin^2\theta + \cos^2\theta = 1$	Fundamental relationship from the unit circle
Even function	$f(-x) = f(x)$	Symmetric about the y-axis
Odd function	$f(-x) = -f(x)$	Symmetric about the origin
ASTC	“All Students Take Calculus”	Mnemonic for signs in quadrants

Examples

Key Properties and Rules

The Three Pythagorean Identities

$$\sin^2\theta + \cos^2\theta = 1$$ $$\tan^2\theta + 1 = \sec^2\theta$$ $$1 + \cot^2\theta = \csc^2\theta$$

Reciprocal Relationships

$$\sin\theta \cdot \csc\theta = 1$$ $$\cos\theta \cdot \sec\theta = 1$$ $$\tan\theta \cdot \cot\theta = 1$$

Quotient Relationships

$$\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta}$$

Even/Odd Properties

Even Functions	Odd Functions
$\cos(-\theta) = \cos\theta$	$\sin(-\theta) = -\sin\theta$
$\sec(-\theta) = \sec\theta$	$\tan(-\theta) = -\tan\theta$
	$\csc(-\theta) = -\csc\theta$
	$\cot(-\theta) = -\cot\theta$

Domain Restrictions

Not all trigonometric functions are defined for all angles:

Function	Undefined When	Because
$\tan\theta$	$\theta = 90° + 180°n$ (i.e., $\frac{\pi}{2} + n\pi$)	$\cos\theta = 0$
$\cot\theta$	$\theta = 180°n$ (i.e., $n\pi$)	$\sin\theta = 0$
$\sec\theta$	$\theta = 90° + 180°n$ (i.e., $\frac{\pi}{2} + n\pi$)	$\cos\theta = 0$
$\csc\theta$	$\theta = 180°n$ (i.e., $n\pi$)	$\sin\theta = 0$

Complete Special Angles Reference Table

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

Real-World Applications

Navigation and Surveying

Surveyors use trigonometric functions to calculate distances and heights that cannot be measured directly. By measuring angles and using tangent, they can determine the height of a building or the distance across a river without actually crossing it. GPS systems rely heavily on trigonometric calculations to determine your position from satellite signals.

Physics: Waves and Oscillations

The motion of a pendulum, the vibration of a guitar string, and the oscillation of electrical current in a circuit are all described using sine and cosine functions. When you see a sound wave displayed on a screen, that wave shape is the graph of a trigonometric function. The “frequency” of a note is directly related to how rapidly the sine function oscillates.

Engineering: Force Analysis

Engineers use trigonometric functions to break forces into components. When a cable supports a bridge at an angle, the tension in the cable can be decomposed into horizontal and vertical components using sine and cosine. This allows engineers to ensure structures can handle the loads they will experience.

Computer Graphics and Game Development

Every rotation in a video game or 3D animation involves trigonometric functions. When a character turns or a camera pans, the computer calculates new positions using sine and cosine. The smooth circular motion of objects, from planets in a space simulator to spinning wheels in a racing game, relies on the unit circle and its associated functions.

Medicine: Imaging and Diagnostics

CT scans and MRI machines use mathematical techniques that rely on trigonometric functions to reconstruct images of the inside of your body. The Fourier transform, which is built on sine and cosine functions, allows doctors to see detailed internal structures from raw scan data.

Music and Audio Processing

Digital audio relies on representing sound waves as combinations of sine waves (a technique called Fourier analysis). When you use an equalizer to boost the bass or reduce treble, you are adjusting the amplitudes of different sine-wave components. Noise-canceling headphones use trigonometric principles to generate waves that cancel out unwanted sound.

Self-Test Problems

Summary

• The six trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. Sine and cosine are defined as coordinates on the unit circle; the others are built from these two.

• Tangent equals $\frac{\sin\theta}{\cos\theta}$ and represents the slope of the terminal side. Cotangent is its reciprocal.

• Secant and cosecant are the reciprocals of cosine and sine, respectively.

• Special angles (0°, 30°, 45°, 60°, 90°) have exact trigonometric values that should be memorized. The values involve 0, 1, $\frac{1}{2}$, $\frac{\sqrt{2}}{2}$, and $\frac{\sqrt{3}}{2}$.

• ASTC (“All Students Take Calculus”) tells you which functions are positive in each quadrant: All in I, Sine in II, Tangent in III, Cosine in IV.

• The Pythagorean identities ($\sin^2\theta + \cos^2\theta = 1$ and its variations) are essential for simplifying expressions and solving equations.

• Cosine and secant are even functions (symmetric about the y-axis); the other four are odd functions (symmetric about the origin).

• To evaluate trig functions at any angle: (1) find a coterminal angle, (2) identify the quadrant, (3) find the reference angle, (4) use special angle values, (5) adjust signs.

• Trigonometric functions appear throughout science and engineering, from analyzing waves and forces to powering computer graphics and medical imaging.