01 / 21

Angle Addition &
Subtraction Formulas

Prove and apply the addition and subtraction identities for \(\sin\), \(\cos\), and \(\tan\) to evaluate exact values and solve trigonometric problems.

Precalculus

Why Do Angle Formulas Matter?

We know exact values for special angles like \(30^\circ\), \(45^\circ\), and \(60^\circ\). But what about \(75^\circ\) or \(15^\circ\)? The angle addition and subtraction formulas let us compute exact trigonometric values for any angle that can be written as a sum or difference of known angles.

The Unit Circle Foundation

P \(\alpha\) Q \(\beta\) y x \((\cos\alpha,\;\sin\alpha)\) \((\cos(\alpha+\beta),\;\sin(\alpha+\beta))\)

The Geometric Proof Strategy

The classic proof uses the distance formula on two pairs of points on the unit circle. By equating the distance between points separated by angle \((\alpha - \beta)\) measured in two different ways, we derive the cosine subtraction formula first, then obtain all others from it.

1
Place two points on the unit circle at angles \(\alpha\) and \(\beta\).
2
Compute the distance between them using coordinates: \((\cos\alpha - \cos\beta)^2 + (\sin\alpha - \sin\beta)^2\).
3
Rotate both points so one lies on the x-axis. The distance is now \((\cos(\alpha-\beta)-1)^2 + \sin^2(\alpha-\beta)\).
4
Set the distances equal and simplify to get \(\cos(\alpha-\beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta\).

The Cosine Subtraction Formula

This is the master identity. Notice the plus sign on the right even though we subtract on the left. Every other addition/subtraction formula can be derived from this single result.

The Cosine Addition Formula

Replace \(\beta\) with \(-\beta\) in the subtraction formula. Since \(\cos(-\beta) = \cos\beta\) and \(\sin(-\beta) = -\sin\beta\):

The sign flips: addition of angles gives a minus sign between the cosine-cosine and sine-sine products. This "opposite sign" pattern is a hallmark of the cosine formulas.

Sine Addition & Subtraction

Use the co-function identity \(\sin\theta = \cos\!\left(\frac{\pi}{2} - \theta\right)\) to convert cosine formulas into sine formulas:

Sine Addition
$$\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$$
Sine Subtraction
$$\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta$$

Tangent Addition & Subtraction

Since \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), divide the sine formula by the cosine formula, then divide numerator and denominator by \(\cos\alpha\cos\beta\):

Tangent Addition
$$\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}$$
Tangent Subtraction
$$\tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta}$$

Key Distinction: Sign Patterns

Function Addition (\(\alpha + \beta\)) Subtraction (\(\alpha - \beta\))
\(\cos\) \(\cos\alpha\cos\beta\) − \(\sin\alpha\sin\beta\) \(\cos\alpha\cos\beta\) + \(\sin\alpha\sin\beta\)
\(\sin\) \(\sin\alpha\cos\beta\) + \(\cos\alpha\sin\beta\) \(\sin\alpha\cos\beta\) − \(\cos\alpha\sin\beta\)
\(\tan\) Numerator: +, Denom: − Numerator: −, Denom: +

How to Apply the Formulas

1
Decompose the angle into a sum or difference of special angles (\(30^\circ\), \(45^\circ\), \(60^\circ\), \(90^\circ\), etc.).
2
Choose the right formula based on which trig function you need (\(\sin\), \(\cos\), or \(\tan\)) and whether you are adding or subtracting.
3
Substitute the known values of \(\sin\) and \(\cos\) for each special angle.
4
Simplify the arithmetic: combine fractions and radicals carefully.
5
Verify reasonableness: check the quadrant and approximate value match your expectation.

Example 1: Exact Value of \(\cos 75^\circ\)

Find the exact value of \(\cos 75^\circ\).

1
Decompose: \(75^\circ = 45^\circ + 30^\circ\)
2
Apply cosine addition: \(\cos(45^\circ + 30^\circ) = \cos 45^\circ\cos 30^\circ - \sin 45^\circ\sin 30^\circ\)
3
Substitute: \(= \dfrac{\sqrt{2}}{2}\cdot\dfrac{\sqrt{3}}{2} - \dfrac{\sqrt{2}}{2}\cdot\dfrac{1}{2}\)
4
Simplify: \(= \dfrac{\sqrt{6}}{4} - \dfrac{\sqrt{2}}{4} = \dfrac{\sqrt{6} - \sqrt{2}}{4}\)

Example 2: Exact Value of \(\sin 15^\circ\)

Find the exact value of \(\sin 15^\circ\).

1
Decompose: \(15^\circ = 45^\circ - 30^\circ\)
2
Apply sine subtraction: \(\sin(45^\circ - 30^\circ) = \sin 45^\circ\cos 30^\circ - \cos 45^\circ\sin 30^\circ\)
3
Substitute: \(= \dfrac{\sqrt{2}}{2}\cdot\dfrac{\sqrt{3}}{2} - \dfrac{\sqrt{2}}{2}\cdot\dfrac{1}{2}\)
4
Simplify: \(= \dfrac{\sqrt{6} - \sqrt{2}}{4}\)

Example 3: Tangent of \(\frac{7\pi}{12}\)

Find the exact value of \(\tan\!\left(\dfrac{7\pi}{12}\right)\).

1
Decompose: \(\dfrac{7\pi}{12} = \dfrac{\pi}{3} + \dfrac{\pi}{4}\)
2
Apply tangent addition: \(\tan\!\left(\dfrac{\pi}{3}+\dfrac{\pi}{4}\right) = \dfrac{\tan\frac{\pi}{3} + \tan\frac{\pi}{4}}{1 - \tan\frac{\pi}{3}\cdot\tan\frac{\pi}{4}}\)
3
Substitute: \(= \dfrac{\sqrt{3} + 1}{1 - \sqrt{3}\cdot 1} = \dfrac{\sqrt{3}+1}{1-\sqrt{3}}\)
4
Rationalize: Multiply by \(\dfrac{1+\sqrt{3}}{1+\sqrt{3}}\): \(= \dfrac{(\sqrt{3}+1)(1+\sqrt{3})}{(1-\sqrt{3})(1+\sqrt{3})} = \dfrac{4+2\sqrt{3}}{-2} = -(2+\sqrt{3})\)

Deriving \(\sin(\alpha + \beta)\)

Starting from the cosine subtraction formula, we use the co-function identity \(\sin\theta = \cos\!\left(\frac{\pi}{2}-\theta\right)\):

1
Write: \(\sin(\alpha+\beta) = \cos\!\left(\frac{\pi}{2} - (\alpha+\beta)\right) = \cos\!\left(\left(\frac{\pi}{2}-\alpha\right) - \beta\right)\)
2
Apply cosine subtraction: \(= \cos\!\left(\frac{\pi}{2}-\alpha\right)\cos\beta + \sin\!\left(\frac{\pi}{2}-\alpha\right)\sin\beta\)
3
Use co-function identities: \(\cos\!\left(\frac{\pi}{2}-\alpha\right) = \sin\alpha\) and \(\sin\!\left(\frac{\pi}{2}-\alpha\right) = \cos\alpha\)
4
Result: \(\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta\) ✓

Common Mistakes to Avoid

✕ Distributing Trig Functions
Writing \(\sin(\alpha+\beta) = \sin\alpha + \sin\beta\). Trig functions are not linear: you must use the full formula.
✕ Wrong Sign for Cosine
Forgetting that \(\cos(\alpha+\beta)\) uses a minus sign. The cosine sign is always opposite to the operation.
✕ Mixing Up Terms
Confusing \(\sin\alpha\cos\beta\) with \(\cos\alpha\cos\beta\). For sine, terms are mixed (sin-cos); for cosine, terms are matched (cos-cos, sin-sin).

Complete Formula Summary

Identity Formula
\(\cos(\alpha-\beta)\) \(\cos\alpha\cos\beta + \sin\alpha\sin\beta\)
\(\cos(\alpha+\beta)\) \(\cos\alpha\cos\beta - \sin\alpha\sin\beta\)
\(\sin(\alpha+\beta)\) \(\sin\alpha\cos\beta + \cos\alpha\sin\beta\)
\(\sin(\alpha-\beta)\) \(\sin\alpha\cos\beta - \cos\alpha\sin\beta\)
\(\tan(\alpha+\beta)\) \(\dfrac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\)
\(\tan(\alpha-\beta)\) \(\dfrac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}\)

Question 1 of 5

What is the exact value of \(\sin 75^\circ\)?

Question 2 of 5

Which formula correctly expands \(\cos(\alpha + \beta)\)?

Question 3 of 5

What is the exact value of \(\tan 15^\circ\)?

Question 4 of 5

If \(\sin\alpha = \frac{3}{5}\) with \(\alpha\) in Q1, and \(\cos\beta = \frac{5}{13}\) with \(\beta\) in Q1, what is \(\sin(\alpha+\beta)\)?

Question 5 of 5

The cosine subtraction formula is the "master identity" because all other angle addition/subtraction formulas can be derived from it. Which identity is directly used to obtain the sine addition formula from the cosine subtraction formula?