Difficulty: Foundational to Intermediate | Prerequisites: Algebra Foundations notes (real numbers, exponents, factoring).

Big Picture

Trigonometry is woven into almost every major topic in Calculus II. Trig substitution is a key integration technique, many series involve trig functions, and parametric curves often use sine and cosine. You need the identities, special values, and unit circle coordinates at your fingertips. If you have to look them up during a problem, you will run out of time on exams. This is material to memorise, not just understand.

TL;DR

There are six trig functions built from ratios of a right triangle's sides. A handful of identities (Pythagorean, addition/subtraction, double-angle) let you rewrite trig expressions into forms you can integrate or simplify. The unit circle gives you exact values for standard angles, and the laws of sines and cosines extend triangle solving beyond right triangles.

Key Terms

Radian

The angle subtended at the centre of a circle by an arc equal in length to the radius. pi radians = 180 degrees. In simple terms, radians measure angles using the circle's own radius as the ruler.

Unit circle

A circle of radius 1 centred at the origin. Every point on it has coordinates (cos theta, sin theta). Think of it as the master lookup table for trig values.

Fundamental identities (Pythagorean)

sin^2(theta) + cos^2(theta) = 1, and the two variants: 1 + tan^2(theta) = sec^2(theta), and 1 + cot^2(theta) = csc^2(theta). These let you convert between trig functions at will.

Addition and subtraction formulas

Formulas that expand sin(x +/- y) and cos(x +/- y) into products of sines and cosines of x and y separately. Essential for simplifying composite-angle expressions.

Double-angle formulas

Special cases of the addition formulas where y = x. They give sin(2x), cos(2x), and tan(2x) in terms of functions of x alone. The cos(2x) formula has three equivalent forms, which is a common exam point.

Law of sines

sin(A)/a = sin(B)/b = sin(C)/c, relating the sides and angles of any triangle. Used when you know an angle and its opposite side.

Law of cosines

a^2 = b^2 + c^2 - 2bc cos(A). A generalisation of the Pythagorean theorem to non-right triangles. In simple terms, it corrects the Pythagorean theorem with a cosine adjustment.

Core Content

Angle Measurement

• pi radians = 180 degrees

• 1 degree = pi/180 radians; 1 radian = 180/pi degrees

• Arc length: s = r * theta (theta in radians)

Right Triangle Definitions

• sin(theta) = opposite / hypotenuse

• cos(theta) = adjacent / hypotenuse

• tan(theta) = opposite / adjacent = sin(theta) / cos(theta)

• csc(theta) = 1 / sin(theta) = hypotenuse / opposite

• sec(theta) = 1 / cos(theta) = hypotenuse / adjacent

• cot(theta) = cos(theta) / sin(theta) = adjacent / opposite

Trigonometric Functions (Coordinate Definition)

For a point P = (x, y) on a circle of radius r:

• sin(theta) = y/r, cos(theta) = x/r, tan(theta) = y/x

• csc(theta) = r/y, sec(theta) = r/x, cot(theta) = x/y

Fundamental Identities

• sin^2(theta) + cos^2(theta) = 1

• 1 + tan^2(theta) = sec^2(theta)

• 1 + cot^2(theta) = csc^2(theta)

• sin(-theta) = -sin(theta) (odd function)

• cos(-theta) = cos(theta) (even function)

• tan(-theta) = -tan(theta) (odd function)

Co-function Identities

• sin(pi/2 - theta) = cos(theta)

• cos(pi/2 - theta) = sin(theta)

• tan(pi/2 - theta) = cot(theta)

Periodicity

• sin(theta + 2pi) = sin(theta)

• cos(theta + 2pi) = cos(theta)

• tan(theta + pi) = tan(theta)

Important Limits

• lim (theta to 0) sin(theta)/theta = 1

• lim (theta to 0) (1 - cos(theta))/theta = 0

Addition and Subtraction Formulas

• sin(x + y) = sin(x)cos(y) + cos(x)sin(y)

• sin(x - y) = sin(x)cos(y) - cos(x)sin(y)

• cos(x + y) = cos(x)cos(y) - sin(x)sin(y)

• cos(x - y) = cos(x)cos(y) + sin(x)sin(y)

• tan(x + y) = (tan(x) + tan(y)) / (1 - tan(x)tan(y))

• tan(x - y) = (tan(x) - tan(y)) / (1 + tan(x)tan(y))

Double-Angle Formulas

• sin(2x) = 2 sin(x) cos(x)

• cos(2x) = cos^2(x) - sin^2(x) = 2cos^2(x) - 1 = 1 - 2sin^2(x)

• tan(2x) = 2tan(x) / (1 - tan^2(x))

Half-Angle (Power-Reducing) Formulas

• sin^2(x) = (1 - cos(2x)) / 2

• cos^2(x) = (1 + cos(2x)) / 2

These are critical for integrating even powers of sin and cos.

Special Values Table

Formulas

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

\sin(x+y) = \sin x \cos y + \cos x \sin y

\cos(x+y) = \cos x \cos y - \sin x \sin y

\sin 2x = 2\sin x \cos x

\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x

\sin^2 x = \frac{1 - \cos 2x}{2}, \quad \cos^2 x = \frac{1 + \cos 2x}{2}

\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}

a^2 = b^2 + c^2 - 2bc\cos A

Common Misconceptions

• Students often forget that the sin(theta)/theta limit only equals 1 when theta is in radians. In degrees, this limit is pi/180.

• Confusing the three forms of cos(2x). All three are equivalent: cos^2(x) - sin^2(x) = 2cos^2(x) - 1 = 1 - 2sin^2(x). Pick the form that eliminates the function you want gone.

• Mixing up signs in the addition formulas. In sin(x + y), the terms are added. In cos(x + y), the product sin(x)sin(y) is subtracted. A memory aid: "sine is friendly (same sign as the formula), cosine is contrary (opposite sign)."

• Thinking tan(90 degrees) is zero. It is undefined, because cos(90 degrees) = 0 and division by zero is not allowed.

Why It Matters / Exam Flags

• The power-reducing formulas (sin^2 and cos^2) are used constantly in Calc II to integrate even powers of trig functions. Know them cold.

• Trig substitution problems require instant recall of identities: if you have to derive them from scratch, you will run out of time.

• Unit circle values appear in definite integral evaluations. When you substitute bounds, you need exact values of sin and cos at standard angles.

• The double-angle identity for cos(2x) in the form 1 - 2sin^2(x) is especially common in integration problems.

Quick Self-Test

1. Convert 150 degrees to radians.

1. True or false: sin^2(theta) + cos^2(theta) = 2.

1. What is cos(pi/3)?

1. Expand sin(2x) using the double-angle formula.

1. What is lim (theta to 0) sin(theta)/theta?

Practice Q&A

Q: Simplify sin^2(x) + cos^2(x) + tan^2(x).

A: sin^2(x) + cos^2(x) = 1, so the expression becomes 1 + tan^2(x) = sec^2(x).

Q: Find the exact value of sin(75 degrees).

A: Write 75 = 45 + 30. Apply the addition formula: sin(45 + 30) = sin(45)cos(30) + cos(45)sin(30) = (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2) = (sqrt(6) + sqrt(2)) / 4.

Q: Rewrite cos^2(x) in a form suitable for integration.

A: Use the power-reducing formula: cos^2(x) = (1 + cos(2x)) / 2.

Q: If sin(theta) = 3/5 and theta is in the first quadrant, find cos(theta).

A: From sin^2(theta) + cos^2(theta) = 1: cos^2(theta) = 1 - 9/25 = 16/25. Since theta is in the first quadrant, cos(theta) = 4/5.

Q: What is cos(2x) if sin(x) = 1/3?

A: Use cos(2x) = 1 - 2sin^2(x) = 1 - 2(1/9) = 1 - 2/9 = 7/9.

Connections to Other Topics

Trig identities feed directly into integration techniques (trig integrals and trig substitution). The special values table is needed when evaluating definite integrals with trig bounds. Power-reducing formulas connect to the next set of notes on integration. The graphs of trig functions underpin understanding of Fourier series, which some Calc II courses introduce near the end.

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