1.3 Trigonometric Functions

Angle Measurements and Trigonometric Functions

Trigonometric functions connect angles to ratios, and they show up constantly in calculus. They're the main tool for describing anything periodic: circular motion, waves, oscillations. Before diving into derivatives and integrals of trig functions later in the course, you need a solid handle on how angles are measured, how the six trig functions work, and how their graphs behave.

Degree vs. Radian Measurements

A degree divides a full circle into 360 equal parts, so $360^\circ$ is one complete rotation. You've used degrees your whole life for compass directions, weather, etc.

A radian is defined by the unit circle: 1 radian is the angle created when the arc length equals the radius of the circle. A full circle has an arc length of $2\pi r$, so a full rotation is $2\pi$ radians.

Why do we care about radians in calculus? Because the derivative and integral formulas for trig functions only work cleanly in radians. For example, the derivative of $\sin(x)$ is $\cos(x)$ only when $x$ is in radians. If you use degrees, an ugly conversion factor appears. So from here on out, radians are the default.

Conversion formulas:

• Degrees to radians: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$
• Radians to degrees: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$

Common angles to memorize:

Trigonometric Functions in Triangles and Circles

Right triangle definitions (SOH-CAH-TOA):

For an acute angle $\theta$ in a right triangle:

• $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
• $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
• $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

These definitions work great for angles between $0$ and $\frac{\pi}{2}$, but they break down for larger or negative angles. That's where the unit circle comes in.

Unit circle definitions:

Place a circle of radius 1 centered at the origin. For any angle $\theta$ measured counterclockwise from the positive $x$-axis, the point where the terminal side of the angle meets the circle has coordinates $(\cos\theta,\, \sin\theta)$. This means:

• $\cos(\theta) = x\text{-coordinate}$
• $\sin(\theta) = y\text{-coordinate}$
• $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{y}{x}$

The unit circle definition handles any angle, positive or negative, well beyond $360^\circ$. It also makes clear why $\sin$ and $\cos$ are always between $-1$ and $1$: the point never leaves the unit circle.

Properties of Trigonometric Functions

Each trig function has a specific domain (valid inputs), range (possible outputs), and period (how often it repeats).

• Sine: Domain is all real numbers. Range is $[-1, 1]$. Period is $2\pi$.
• Cosine: Domain is all real numbers. Range is $[-1, 1]$. Period is $2\pi$.
• Tangent: Domain is all real numbers except $\theta = \frac{\pi}{2} + n\pi$ (where $\cos\theta = 0$). Range is $(-\infty, \infty)$. Period is $\pi$.

Trigonometric Identities and Graphs

Trigonometric Identities

Trig identities are equations that hold true for all valid angles. You'll use these constantly in calculus to simplify expressions before differentiating or integrating.

Reciprocal identities define three additional trig functions:

• $\csc(\theta) = \frac{1}{\sin(\theta)}$
• $\sec(\theta) = \frac{1}{\cos(\theta)}$
• $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}$

Pythagorean identities come directly from the equation of the unit circle, $x^2 + y^2 = 1$:

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

The second and third identities are just the first one divided by $\cos^2(\theta)$ and $\sin^2(\theta)$, respectively. You don't need to memorize all three if you remember how to derive them from the first.

Graphs of Basic Trigonometric Functions

Sine $y = \sin(\theta)$:

• Starts at 0, rises to 1 at $\frac{\pi}{2}$, returns to 0 at $\pi$, drops to $-1$ at $\frac{3\pi}{2}$, and completes the cycle at $2\pi$
• Range: $[-1, 1]$
• Odd function: $\sin(-\theta) = -\sin(\theta)$ (symmetric about the origin)

Cosine $y = \cos(\theta)$:

• Starts at 1, drops to 0 at $\frac{\pi}{2}$, reaches $-1$ at $\pi$, returns to 0 at $\frac{3\pi}{2}$, and completes the cycle at $2\pi$
• Range: $[-1, 1]$
• Even function: $\cos(-\theta) = \cos(\theta)$ (symmetric about the $y$-axis)

Notice that cosine is just sine shifted left by $\frac{\pi}{2}$: $\cos(\theta) = \sin\!\left(\theta + \frac{\pi}{2}\right)$.

Tangent $y = \tan(\theta)$:

• Period of $\pi$ (repeats twice as fast as sine and cosine)
• Range: $(-\infty, \infty)$
• Vertical asymptotes at $\theta = \frac{\pi}{2} + n\pi$ for any integer $n$, because $\cos(\theta) = 0$ at those points
• Odd function: $\tan(-\theta) = -\tan(\theta)$

Effects of Parameters on Trigonometric Graphs

The general form is $y = A\sin(B\theta + C) + D$ (same idea applies to cosine). Each parameter controls a different transformation:

• Amplitude $|A|$: Stretches the graph vertically. The function oscillates between $-|A| + D$ and $|A| + D$. If $A$ is negative, the graph also flips over the horizontal axis.
• Period $\frac{2\pi}{|B|}$: Controls horizontal compression or stretching. A larger $|B|$ means more cycles squeezed into the same interval. For example, $y = \sin(2\theta)$ completes a full cycle in $\pi$ instead of $2\pi$.
• Phase shift $-\frac{C}{B}$: Shifts the graph horizontally. A positive result shifts right; a negative result shifts left. Be careful with the sign here: in $y = \sin(2\theta - \pi)$, the phase shift is $\frac{\pi}{2}$ to the right, not $\pi$.
• Vertical shift $D$: Moves the entire graph up ($D > 0$) or down ($D < 0$). The midline of the wave becomes $y = D$ instead of $y = 0$.

Quick example: For $y = 3\sin(2\theta - \pi) + 1$, you have amplitude 3, period $\frac{2\pi}{2} = \pi$, phase shift $\frac{\pi}{2}$ to the right, and midline at $y = 1$.