Trigonometric Functions

Calculus is full of things that repeat: waves, orbits, alternating current, the swing of a pendulum. The functions that describe them are the trigonometric functions, and calculus leans on them constantly, from the first limits to the integrals of Calculus 2. This review rebuilds them from scratch: first as ratios in a right triangle, then as coordinates on the unit circle, which is the version calculus actually uses. Angles here are in radians; if that’s rusty, start with radians and degrees.

Trig Functions in a Right Triangle

Pick one of the acute angles of a right triangle and call it . Seen from , the three sides have names: the hypotenuse opposite the right angle, the opposite side across from , and the adjacent side next to it.

The six trigonometric functions are the six possible ratios of those sides:

Example 1: A 3-4-5 Triangle

A right triangle has legs 3 and 4 and hypotenuse 5. For the angle opposite the side of length 3:

and the reciprocals are , , . The size of the triangle doesn’t matter: a 6-8-10 triangle gives the same ratios, which is why these depend only on the angle.

Trig Functions on the Unit Circle

A right triangle only allows angles between 0 and . Calculus needs for any angle, including negative ones and ones bigger than a full turn. The unit circle, a circle of radius 1 centred at the origin, provides that.

Turn an angle counterclockwise from the positive -axis. Where it meets the circle, the coordinates are defined to be

For angles between 0 and this agrees exactly with SOH CAH TOA, because the hypotenuse is the radius, 1. But it keeps working all the way around the circle, which is what lets sine and cosine become functions on the whole real line. It also explains the most important identity in the subject: the point is on a circle of radius 1, so

Exact Values at the Special Angles

These come up so often in calculus that they’re worth knowing cold:

A pattern helps: the sine row is , and the cosine row is the same list backwards.

Signs in Each Quadrant

Outside the first quadrant, the coordinates can be negative, so the trig functions can be too.

To evaluate a trig function at any angle, find the reference angle (the acute angle to the -axis), take the value from the table, and attach the sign for the quadrant.

Example 2: Angles Outside the First Quadrant

: the angle is in quadrant II with reference angle . Sine is positive there, so .

: quadrant III, reference angle , cosine negative, so .

: quadrant II, reference angle , tangent negative, so .

Graphs of the Trig Functions

Tangent looks different: it has vertical asymptotes wherever , and between them it rises from to . Sine and cosine aren’t one-to-one, which matters for their inverses, the next lesson in this chapter.

Amplitude and Period

Most trig functions in calculus problems are stretched or squeezed versions of these, written as

The amplitude is how far the wave rises above and falls below its centre line, and the period is how long one full cycle takes. For , the wave swings between and 3 and repeats every . These are the same function transformations used for any graph: stretches vertically and compresses horizontally.

Why Trig Functions Matter in Calculus

Their derivatives are each other: and . That makes them the natural functions for anything that oscillates, and the reason they show up in every chapter of calculus, from limits like to integrals in physics and engineering.

Common Mistakes to Avoid

• Mixing up opposite and adjacent. They depend on which angle you’re looking from.
• Forgetting the sign. is , not .
• Calculator in degree mode. means one radian here.
• Reading as . The reciprocal is ; is the inverse function.
• Writing for . means .

Practice Problems

Work each one before opening the answer.

Problem 1. A right triangle has opposite side 5, adjacent side 12 and hypotenuse 13. Find , and .

Problem 2. Find .

Problem 3. Find .

Problem 4. Find .

Problem 5. Find .

Problem 6. If and is in quadrant II, find .

Trig functions take an angle and give a number. The next lesson turns that around: given the number, which angle produced it?