The Complete Trigonometry Guide: From Basic Trig Functions to Advanced Identities

Trigonometry is one of the most important branches of mathematics, connecting angles and side lengths in ways that show up everywhere — from physics and engineering to architecture and computer graphics. Whether you are just starting with sine and cosine or working through complex trig equations, this guide covers everything you need to master trigonometry.

This guide is organized so you can read it straight through or jump to the section you need. We will start with the foundational trig functions, move through the unit circle and identities, and finish with graphing, inverse trig, and solving equations.

The Six Trigonometric Functions

Trigonometry begins with a right triangle. Given a right triangle with an acute angle theta, the six trig functions are defined as ratios of the triangle’s sides.

SOH-CAH-TOA: The Core Three

The three primary trig functions are:

• Sine (sin): opposite / hypotenuse
• Cosine (cos): adjacent / hypotenuse
• Tangent (tan): opposite / adjacent

The mnemonic SOH-CAH-TOA helps you remember which ratio goes with which function. “SOH” means Sine = Opposite over Hypotenuse, and so on.

The Reciprocal Functions

The other three trig functions are reciprocals of the first three:

• Cosecant (csc): hypotenuse / opposite (reciprocal of sine)
• Secant (sec): hypotenuse / adjacent (reciprocal of cosine)
• Cotangent (cot): adjacent / opposite (reciprocal of tangent)

Example Problem

Problem: In a right triangle, the side opposite angle A is 5 and the hypotenuse is 13. Find sin(A), cos(A), and tan(A).

Approach: First, find the adjacent side using the Pythagorean theorem: adjacent = sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12. Then apply the ratios:

• sin(A) = 5/13
• cos(A) = 12/13
• tan(A) = 5/12

Try it yourself: Screenshot any trigonometry problem and get step-by-step solutions with Math.Photos — free browser extension.

The Unit Circle

The unit circle extends trigonometry beyond right triangles. It is a circle with radius 1 centered at the origin of the coordinate plane. Any point on the unit circle can be written as (cos(theta), sin(theta)), where theta is the angle measured counterclockwise from the positive x-axis.

Key Angles to Memorize

The most important angles on the unit circle, along with their sine and cosine values, are:

Angle (degrees)	Angle (radians)	cos	sin
0	0	1	0
30	pi/6	sqrt(3)/2	1/2
45	pi/4	sqrt(2)/2	sqrt(2)/2
60	pi/3	1/2	sqrt(3)/2
90	pi/2	0	1

These five angles in the first quadrant generate all the other key values. In quadrant II, cosine becomes negative. In quadrant III, both sine and cosine are negative. In quadrant IV, sine is negative.

Reference Angles

A reference angle is the acute angle that a given angle makes with the x-axis. To evaluate trig functions for any angle, find its reference angle, compute the trig value for that reference angle, and then apply the correct sign based on the quadrant.

Example: Find sin(150 degrees).

The reference angle is 180 - 150 = 30 degrees. Since 150 degrees is in quadrant II, sine is positive. So sin(150) = sin(30) = 1/2.

Radians vs. Degrees

Radians are the standard angle measure in higher math. The conversion is straightforward: 180 degrees = pi radians. To convert degrees to radians, multiply by pi/180. To convert radians to degrees, multiply by 180/pi.

Try it yourself: Screenshot any trigonometry problem and get step-by-step solutions with Math.Photos — free browser extension.

Trigonometric Identities

Trig identities are equations that are true for all valid values of the variable. They are essential for simplifying expressions and solving equations. For a comprehensive reference, see our trig identities cheat sheet.

Pythagorean Identities

The most fundamental identity comes directly from the Pythagorean theorem applied to the unit circle:

• sin^2(x) + cos^2(x) = 1

Dividing through by cos^2(x) or sin^2(x) gives two more:

• tan^2(x) + 1 = sec^2(x)
• 1 + cot^2(x) = csc^2(x)

Sum and Difference Formulas

These let you find exact values for trig functions of sums or differences of angles:

• sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
• sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
• cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
• cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

Double Angle Formulas

Setting A = B in the sum formulas gives the double angle formulas:

• sin(2x) = 2sin(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))

The three forms of cos(2x) are all equivalent and useful in different situations.

Half Angle Formulas

Derived from the double angle formulas, these are especially useful in calculus for integration:

• sin(x/2) = plus or minus sqrt((1 - cos(x)) / 2)
• cos(x/2) = plus or minus sqrt((1 + cos(x)) / 2)

The sign depends on the quadrant of x/2.

Example Problem

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

Approach: Replace tan(x) with sin(x)/cos(x). The second term becomes sin(x) cos(x) sin(x)/cos(x) = sin^2(x). So the expression simplifies to sin^2(x) + sin^2(x) = 2sin^2(x). You could also write this as 1 - cos(2x) using the double angle identity.

Try it yourself: Screenshot any trigonometry problem and get step-by-step solutions with Math.Photos — free browser extension.

Inverse Trigonometric Functions

Inverse trig functions let you find an angle when you know a trig ratio. If sin(theta) = 0.5, then theta = arcsin(0.5) = 30 degrees (or pi/6 radians).

Domains and Ranges

Because trig functions are periodic, their inverses must be restricted to give a single output:

• arcsin(x): domain [-1, 1], range [-pi/2, pi/2]
• arccos(x): domain [-1, 1], range [0, pi]
• arctan(x): domain (all real numbers), range (-pi/2, pi/2)

These restricted ranges are critical. When your calculator gives you arcsin(0.5) = 30 degrees, that is the principal value. There may be other angles with the same sine value that you need to find separately.

Example Problem

Problem: Find all solutions to sin(x) = -sqrt(2)/2 in [0, 2pi).

Approach: The reference angle is arcsin(sqrt(2)/2) = pi/4 (45 degrees). Since sine is negative, x must be in quadrants III and IV. The solutions are x = pi + pi/4 = 5pi/4 and x = 2pi - pi/4 = 7pi/4.

Connection to Calculus

Inverse trig functions appear frequently in calculus, particularly in derivatives and integrals. The derivative of arcsin(x) is 1/sqrt(1 - x^2), and the derivative of arctan(x) is 1/(1 + x^2). For more on this topic, see our guide on how to find derivatives of trig functions.

Try it yourself: Screenshot any trigonometry problem and get step-by-step solutions with Math.Photos — free browser extension.

Law of Sines and Law of Cosines

Not every triangle is a right triangle. The Law of Sines and Law of Cosines extend trigonometry to all triangles, making them essential tools for solving oblique triangles.

Law of Sines

For any triangle with sides a, b, c opposite angles A, B, C:

a/sin(A) = b/sin(B) = c/sin(C)

Use the Law of Sines when you know:

• Two angles and one side (AAS or ASA)
• Two sides and an angle opposite one of them (SSA) — but watch for the ambiguous case

The Ambiguous Case (SSA)

When you have two sides and a non-included angle, there may be zero, one, or two valid triangles. This is the trickiest part of the Law of Sines. You must check whether the sine value you compute leads to a valid angle, and if so, whether the supplement of that angle also produces a valid triangle.

Law of Cosines

For any triangle:

c^2 = a^2 + b^2 - 2ab*cos(C)

This is a generalization of the Pythagorean theorem (when C = 90 degrees, cos(C) = 0 and you get c^2 = a^2 + b^2). Use the Law of Cosines when you know:

• Two sides and the included angle (SAS)
• All three sides (SSS)

Example Problem

Problem: A triangle has sides a = 8, b = 11, and included angle C = 37 degrees. Find side c.

Approach: Apply the Law of Cosines: c^2 = 8^2 + 11^2 - 2(8)(11)cos(37). Computing: c^2 = 64 + 121 - 176(0.7986) = 185 - 140.55 = 44.45. So c = sqrt(44.45) which is approximately 6.67.

Try it yourself: Screenshot any trigonometry problem and get step-by-step solutions with Math.Photos — free browser extension.

Graphing Trigonometric Functions

Understanding the graphs of trig functions is essential for visualizing periodic behavior and solving equations.

The Sine and Cosine Graphs

Both sin(x) and cos(x) produce smooth wave patterns. Their key properties:

• Period: 2pi (the length of one complete cycle)
• Amplitude: 1 (the distance from the midline to the peak)
• Range: [-1, 1]
• Midline: y = 0

The cosine graph is identical to the sine graph shifted left by pi/2. This is because cos(x) = sin(x + pi/2).

Transformations: y = A*sin(Bx - C) + D

Each parameter transforms the graph:

• A (amplitude): stretches or compresses vertically. |A| is the amplitude. If A is negative, the graph flips vertically.
• B (frequency): changes the period to 2pi/|B|. Larger B means more cycles in the same interval.
• C (phase shift): shifts the graph horizontally by C/B units to the right.
• D (vertical shift): moves the midline to y = D.

Example Problem

Problem: Find the amplitude, period, and phase shift of y = 3sin(2x - pi/3) + 1.

Approach: Comparing to y = A*sin(Bx - C) + D: A = 3 (amplitude = 3), B = 2 (period = 2pi/2 = pi), C = pi/3 (phase shift = (pi/3)/2 = pi/6 to the right), D = 1 (midline at y = 1).

The Tangent Graph

The tangent function has a different shape from sine and cosine:

• Period: pi (not 2pi)
• Vertical asymptotes: at x = pi/2 + n*pi for any integer n
• No amplitude: tangent has no maximum or minimum value
• Range: all real numbers

The graph increases from negative infinity to positive infinity between each pair of asymptotes.

Try it yourself: Screenshot any trigonometry problem and get step-by-step solutions with Math.Photos — free browser extension.

Solving Trigonometric Equations

Solving trig equations means finding all angles that satisfy the equation. This combines your knowledge of identities, the unit circle, and algebraic techniques.

Strategy for Solving Trig Equations

1. Isolate the trig function if possible.
2. Use identities to rewrite the equation in terms of one trig function.
3. Find the reference angle using inverse trig functions.
4. Determine all solutions in the given interval, considering which quadrants give the correct sign.
5. Write the general solution if no interval is specified, using + 2npi for sine/cosine or + npi for tangent.

Example: Linear Trig Equation

Problem: Solve 2cos(x) - 1 = 0 for x in [0, 2pi).

Approach: Isolate: cos(x) = 1/2. The reference angle is pi/3. Cosine is positive in quadrants I and IV, so x = pi/3 and x = 2pi - pi/3 = 5pi/3.

Example: Quadratic Trig Equation

Problem: Solve 2sin^2(x) - sin(x) - 1 = 0 for x in [0, 2pi).

Approach: This factors as (2sin(x) + 1)(sin(x) - 1) = 0. So sin(x) = -1/2 or sin(x) = 1.

For sin(x) = -1/2: reference angle is pi/6, and sine is negative in quadrants III and IV. Solutions: x = 7pi/6 and x = 11pi/6.

For sin(x) = 1: x = pi/2.

The solution set is {pi/2, 7pi/6, 11pi/6}.

Example: Using Identities to Solve

Problem: Solve sin(2x) = cos(x) for x in [0, 2pi).

Approach: Replace sin(2x) with 2sin(x)cos(x): the equation becomes 2sin(x)cos(x) = cos(x). Rearrange: 2sin(x)cos(x) - cos(x) = 0. Factor: cos(x)(2sin(x) - 1) = 0. So cos(x) = 0 or sin(x) = 1/2.

For cos(x) = 0: x = pi/2 and x = 3pi/2. For sin(x) = 1/2: x = pi/6 and x = 5pi/6.

The solution set is {pi/6, pi/2, 5pi/6, 3pi/2}.

Common Mistakes to Avoid

• Dividing by a trig function instead of factoring. If you divide both sides of sin(2x) = cos(x) by cos(x), you lose the solutions where cos(x) = 0.
• Forgetting solutions in other quadrants. Always check all quadrants where the trig function has the required sign.
• Mixing up degrees and radians. Stay consistent throughout the problem.

Try it yourself: Screenshot any trigonometry problem and get step-by-step solutions with Math.Photos — free browser extension.

Trigonometry in Calculus and Beyond

Trigonometry does not end with precalculus. It becomes even more important in calculus, physics, and engineering.

In calculus, you will need trig identities for integration techniques like trigonometric substitution. Derivatives and integrals of trig functions form a core part of the curriculum. Check out our calculus guide for how trig connects to derivatives and integrals.

In physics, trig is used to decompose forces into components, model oscillations and waves, and analyze circular motion.

In engineering, Fourier analysis uses sine and cosine functions to break down complex signals into simple components.

If you want to verify your work on any trig or calculus problem, our free math checker can help. You can also learn more about how screenshot math solving works with AI to give you step-by-step explanations.

Frequently Asked Questions

What is trigonometry used for?

Trigonometry is used in a wide range of fields. In physics, it helps analyze forces, waves, and projectile motion. In engineering, it is essential for structural analysis and signal processing. In computer graphics, trig functions drive rotations and animations. Even in everyday life, trigonometry helps with navigation, surveying land, and understanding periodic patterns like tides and sound waves.

What is the easiest way to memorize the unit circle?

Focus on the first quadrant. Memorize the sine and cosine values for 0, 30, 45, 60, and 90 degrees. Notice the pattern: sine values go 0, 1/2, sqrt(2)/2, sqrt(3)/2, 1 while cosine values are the same sequence in reverse. For other quadrants, use reference angles and remember the sign rules (All Students Take Calculus: All positive in QI, Sine in QII, Tangent in QIII, Cosine in QIV).

What is the difference between sin, cos, and tan?

All three are ratios of sides in a right triangle relative to a given angle. Sine is the ratio of the opposite side to the hypotenuse, cosine is adjacent to hypotenuse, and tangent is opposite to adjacent. Tangent can also be expressed as sin/cos. Each function has different values for the same angle and different graph shapes.

How do I know when to use the Law of Sines vs. the Law of Cosines?

Use the Law of Sines when you have a known angle-side pair plus one additional piece (AAS, ASA, or SSA configurations). Use the Law of Cosines when you know two sides and the included angle (SAS) or all three sides (SSS). If you have SSA, be careful of the ambiguous case where two triangles might be possible.

Why do trig identities matter?

Trig identities are not just abstract formulas to memorize. They are practical tools that let you simplify complex expressions, solve equations that would otherwise be unsolvable, and compute exact values without a calculator. In calculus, identities like the double angle and half angle formulas are essential for evaluating integrals. Without identities, many real-world problems in physics and engineering would be far more difficult to solve.

Can AI help me solve trigonometry problems?

Yes. Tools like Math.Photos let you screenshot any trig problem — whether it is from a textbook, worksheet, or online assignment — and get a detailed, step-by-step solution. The AI works through the problem the same way a tutor would, showing you each step and explaining the reasoning. This is especially helpful when you are stuck on a specific problem or want to check your work before submitting an assignment.