Trig Functions and Equations

Main idea

Walk a point around a circle of radius 1: its left-right position is the cosine of the angle and its height is the sine. The other four trig functions are built from those two, and in calculus angles are always measured in radians.

{\htmlData{sym=0}{\cos^2\theta}} + {\htmlData{sym=1}{\sin^2\theta}} = 1

What the symbols mean

$\cos^2\theta$
x-coordinate squared
cos θ is the x-coordinate of the point at angle θ on the unit circle.
$\sin^2\theta$
y-coordinate squared
sin θ is the y-coordinate. The identity is the Pythagorean theorem on the unit circle.

\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \csc\theta = \frac{1}{\sin\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}

Explanation

The trig functions are simply a way to describe where you are on a circle once you know how far around you've gone. All we're really saying is: stand on a circle of radius 1 (the unit circle), turn through an angle $\theta$ , and look at where you are. The main identity has two pieces: $\cos\theta$ (how far right or left of center you are) and $\sin\theta$ (how far up or down). And $\cos^2\theta + \sin^2\theta = 1$ is just the Pythagorean theorem: those two distances are the legs of a right triangle whose long side is the radius, 1. The other four functions are just sine and cosine divided into each other or into 1.

That picture explains a lot of facts you might have memorized. Sine and cosine always stay between $-1$ and $1$ , because you never leave a circle of radius 1. They repeat every $2\pi$ , because after one full turn you're back where you started. And sine is negative in the bottom half of the circle, because that's where you're below the center.

Radians. An angle in radians is the arc length it cuts on a unit circle, so a full turn is $2\pi$ and $\pi = 180^\circ$ . Calculus formulas like $\frac{d}{dx}\sin x = \cos x$ are only true in radians.

A common mistake. It's tempting to leave your calculator in degree mode, because degrees are the angles you grew up with. In calculus that gives wrong answers: $\sin 1$ means the sine of 1 radian (about $0.841$ ), not 1 degree (about $0.017$ ). Use radians everywhere unless a problem says otherwise.

Values to know:

$\theta$	$0$	$\frac\pi6$	$\frac\pi4$	$\frac\pi3$	$\frac\pi2$
$\sin\theta$	$0$	$\frac12$	$\frac{\sqrt2}{2}$	$\frac{\sqrt3}{2}$	$1$
$\cos\theta$	$1$	$\frac{\sqrt3}{2}$	$\frac{\sqrt2}{2}$	$\frac12$	$0$

Every other angle reduces to these by symmetry: in the second quadrant sine stays positive and cosine turns negative, and so on.

Solving trig equations.

Isolate the trig function.
Find every solution in one period, usually $[0, 2\pi)$ . There are typically two.
Add multiples of the period ( $2\pi n$ for sine and cosine, $\pi n$ for tangent) for all solutions.

With $\sin(2x)$ or $\cos(3x)$ , solve for the inside first, then divide. Doing that gives more solutions per period, not fewer.

See it

This is a point going around the unit circle, with its height carried across to the right, where the angle is laid out along the axis in radians. As you can see, the heights trace out the sine wave: up to 1 at a quarter turn (π/2 ≈ 1.57), back to 0 at a half turn (π), down to −1 at three quarters, and home at 2π ≈ 6.28. Cosine is the same point's left-right position.

Worked examples

Example 1

Solve $2\cos x = 1$ .

1
Isolate the cosine: $\cos x = \frac12$ .

Example 2

Solve $\sin(2x) = 0$ for $0 \le x < 2\pi$ .

1
Solve for the inside first. As $x$ runs over $[0, 2\pi)$ , $2x$ runs over $[0, 4\pi)$ , two full turns, and sine is zero at $2x = 0, \pi, 2\pi, 3\pi$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Convert $150^\circ$ to radians.

Problem 2+5 XP

Find $\cos\frac{2\pi}{3}$ .

Problem 3+5 XP

Solve $2\sin x - 1 = 0$ for $0 \le x < 2\pi$ .

Problem 4+5 XP

Find $\sec\frac\pi3$ .

Problem 5+5 XP

How many solutions does $\sin x = 0.3$ have in $[0, 2\pi)$ ?

Problem 6+5 XP

A Ferris wheel of radius $20$ m has its center $25$ m above the ground and turns once every $4$ minutes. Your height after $t$ minutes is $h(t) = 25 - 20\cos\frac{\pi t}{2}$ . When do you first reach $35$ m?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Solve $2\cos^2x + 3\sin x = 3$ for $0 \le x < 2\pi$ .

Where it's used

Anything that repeats is built from sine and cosine: sound, alternating current, tides, seasons, the position of a point on a wheel.

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