Why can't I just multiply the angles, like \cos(45° × 30°), or add the cosines separately?

The cosine function is not linear, so \cos(A + B) \neq \cos A + \cos B and angle multiplication doesn't work that way either. The addition formula tells you the correct way to expand \cos(A + B) into a specific combination of products of individual sines and cosines.

How do I remember whether to use plus or minus in the formula?

For angle addition (A + B), the formula uses a minus sign: \cos(A + B) = \cos A \cos B - \sin A \sin B. For angle subtraction (A - B), it flips to a plus: \cos(A - B) = \cos A \cos B + \sin A \sin B. The sign in the formula is opposite the angle operation.

What happens if I mix degrees and radians in the same problem?

You must convert both angles to the same unit first—either both degrees or both radians. Mixing units gives meaningless results because trig values depend entirely on the unit used. Convert one angle to match the other, then apply the formula.

My calculator gives a different answer than my formula result—what went wrong?

The most common issue is calculator mode: ensure it is set to degrees or radians to match your angles. Another frequent error is substituting decimal approximations like 0.707 instead of exact values like \frac{\sqrt{2}}{2}, which loses precision. Also verify you wrote the formula correctly with the right signs.

Cosine Addition Formula

Find the cosine of an angle sum or difference by using the cosine and sine values of the individual angles—essential for solving trigonometric equations.

\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B

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What each symbol means

What Cosine Addition Formula takes

Symbol	Meaning
$A$	An angle (in degrees or radians, but must match $B$ 's units) representing the first angle in the sum or difference; it can be any value, positive or negative.
$B$	An angle in the same units as $A$ representing the second angle; confusing the units (mixing degrees and radians) is a common mistake that ruins the entire calculation.

When to use it

Reach for this formula when you need to find the cosine of an angle that can be expressed as a sum or difference of angles whose trig values you already know.

Level

Usually taught in: Pre-Calculus

Worked examples

1. Find the exact cosine of a non-standard angle using addition

Problem

Find the exact value of

\cos(75°)

$75° = 45° + 30°$
We recognize that 75° is the sum of two standard angles whose trig values we know from the unit circle.
$\cos(75°) = \cos(45° + 30°)$
Rewrite the problem using angle addition.
$\cos(45° + 30°) = \cos(45°)\cos(30°) - \sin(45°)\sin(30°)$
Apply the cosine addition formula with a minus sign separating the two products.
$= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$
Substitute: $\cos(45°) = \frac{\sqrt{2}}{2}$ , $\cos(30°) = \frac{\sqrt{3}}{2}$ , $\sin(45°) = \frac{\sqrt{2}}{2}$ , $\sin(30°) = \frac{1}{2}$ .
$= \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$
Compute each product: $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{4}$ and $\frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{4}$ .
$= \frac{\sqrt{6} - \sqrt{2}}{4}$
Combine the fractions by subtracting the numerators.

Answer: $\frac{\sqrt{6} - \sqrt{2}}{4}$

This problem shows the core value of the formula: $\cos(75°)$ is not a standard value, but by breaking it into two familiar angles and using the formula, we find an exact expression.

2. Find the cosine of an obtuse angle resulting in a negative value

Problem

Find the exact value of

\cos(105°)

$105° = 60° + 45°$
Decompose 105° as a sum of two standard angles.
$\cos(105°) = \cos(60° + 45°)$
Rewrite the target angle as a sum.
$\cos(60° + 45°) = \cos(60°)\cos(45°) - \sin(60°)\sin(45°)$
Apply the cosine addition formula.
$= \frac{1}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$
Substitute: $\cos(60°) = \frac{1}{2}$ , $\cos(45°) = \frac{\sqrt{2}}{2}$ , $\sin(60°) = \frac{\sqrt{3}}{2}$ , $\sin(45°) = \frac{\sqrt{2}}{2}$ .
$= \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}$
Compute: $\frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4}$ and $\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4}$ .
$= \frac{\sqrt{2} - \sqrt{6}}{4}$
Combine into a single fraction; the result is negative because $\sqrt{6} > \sqrt{2}$ , which is correct for an obtuse angle.

Answer: $\frac{\sqrt{2} - \sqrt{6}}{4}$

This example demonstrates how the formula handles obtuse angles naturally—the negative result automatically emerges from the arithmetic without any special adjustment needed.

3. Combine two measured angles in a surveying context

Problem

A surveyor measures from a point: first 28° to landmark A, then an additional 37° to landmark B, totaling 65°. Using

\cos(28°) \approx 0.883

\sin(28°) \approx 0.469

\cos(37°) \approx 0.799

, and

\sin(37°) \approx 0.602

, find

\cos(65°)

to calculate the horizontal distance component.

$65° = 28° + 37°$
The total viewing angle is the sum of the two individual angle measurements.
$\cos(65°) = \cos(28° + 37°)$
Express the unknown angle as a sum of the known angles.
$\cos(28° + 37°) = \cos(28°)\cos(37°) - \sin(28°)\sin(37°)$
Apply the cosine addition formula.
$= (0.883)(0.799) - (0.469)(0.602)$
Substitute the given decimal approximations for each trig value.
$= 0.706 - 0.282$
Compute the products: $0.883 \times 0.799 = 0.706$ and $0.469 \times 0.602 = 0.282$ .
$= 0.424$
Subtract to find the final result.

Answer: $\cos(65°) \approx 0.424$

In surveying and navigation, angles are measured sequentially, and the cosine addition formula combines these separate measurements into the result for the total angle—essential for distance and position calculations.

Common mistakes

Where Cosine Addition Formula usually goes wrong

The mistake	Why it is wrong	The fix
$\cos(A + B) = \cos A \cos B + \sin A \sin B$	Students often forget the minus sign in the formula when adding angles, which changes the sign of the entire calculation.	The correct formula is $\cos(A + B) = \cos A \cos B - \sin A \sin B$ ; use a minus sign between the products when angles are added.
$\cos(A + B) = \sin A \cos B + \cos A \sin B$	This is the Sine Addition Formula, not the cosine formula—the patterns look similar structurally, causing confusion.	For cosine, you multiply each function by itself: $\cos A$ times $\cos B$ and $\sin A$ times $\sin B$ ; for sine, you cross-multiply the functions as shown in the mistake.
$\cos(A - B) = \cos A \cos B - \sin A \sin B$	When subtracting angles, the sign of the sine product term reverses—students incorrectly apply the addition formula pattern to subtraction.	For angle differences, use $\cos(A - B) = \cos A \cos B + \sin A \sin B$ with a plus sign between the products, which is the opposite of the addition formula.

Tips and when to use something else

The key pattern: cosines multiply with cosines, and sines multiply with sines in separate products—if you see anything else, check your work.
When facing an unfamiliar angle like $\cos(73°)$ , try decomposing it into a sum or difference of standard angles (30°, 45°, 60°, 90°) where all four trig values are known exactly.
The signs flip between addition and subtraction: addition uses minus between products, but subtraction ( $A - B$ ) uses plus instead—this reversal is where most sign errors happen.
If you're solving a trigonometric equation, this formula is powerful for expanding and simplifying; but if you only need one value, check whether the Sine Addition Formula or Half Angle Formulas might be faster for your specific problem.

Related formulas

Sine Addition FormulaThe Sine Addition Formula expands sine of a sum or difference into a product of sines and cosines, making it useful for exact angle calculations.Double Angle FormulasDouble angle formulas express trig functions of 2θ in terms of single angles, letting you simplify complex expressions and solve trigonometric equations.Unit CircleThe unit circle is a fundamental tool that shows how angles map to sine and cosine coordinates: the point (cos θ, sin θ) always lies on x² + y² = 1.Solving Trigonometric EquationsFind all angles that produce a specific sine value by using inverse sine and the periodic nature of trigonometric functions.

Frequently asked questions

Why can't I just multiply the angles, like $\cos(45° × 30°)$ , or add the cosines separately?: The cosine function is not linear, so $\cos(A + B) \neq \cos A + \cos B$ and angle multiplication doesn't work that way either. The addition formula tells you the correct way to expand $\cos(A + B)$ into a specific combination of products of individual sines and cosines.
How do I remember whether to use plus or minus in the formula?: For angle addition ( $A + B$ ), the formula uses a minus sign: $\cos(A + B) = \cos A \cos B - \sin A \sin B$ . For angle subtraction ( $A - B$ ), it flips to a plus: $\cos(A - B) = \cos A \cos B + \sin A \sin B$ . The sign in the formula is opposite the angle operation.
What happens if I mix degrees and radians in the same problem?: You must convert both angles to the same unit first—either both degrees or both radians. Mixing units gives meaningless results because trig values depend entirely on the unit used. Convert one angle to match the other, then apply the formula.
My calculator gives a different answer than my formula result—what went wrong?: The most common issue is calculator mode: ensure it is set to degrees or radians to match your angles. Another frequent error is substituting decimal approximations like 0.707 instead of exact values like $\frac{\sqrt{2}}{2}$ , which loses precision. Also verify you wrote the formula correctly with the right signs.

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Reviewed 2026-09-18