Sum and Difference of Trigonometric Functions Formulas

The sum and difference formulas give sin(A ± B), cos(A ± B) and tan(A ± B) in terms of the individual angles, which is how exact values for angles like 15° and 75° are found. Below: all six formulas, the sign pattern that trips students on cosine, and worked examples.

Tutor-style math help

Sum and Difference of Trigonometric Functions Formulas: what to notice and how to work it

Trigonometry skill

Trigonometry connects an angle to a triangle ratio, a unit-circle coordinate, or a repeating graph. Choosing the right picture makes the problem much easier.

What to notice first

Decide whether the problem is triangle-based, circle-based, or graph-based. Then use the matching definition.

Common student mistake

Do not mix degrees and radians. The angle unit must match the formula, graph scale, or calculator setting.

Key formulas and cues

\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)

\(\tan\theta=\frac{\sin\theta}{\cos\theta}\)

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

A reliable path

Choose the modelUse a right triangle, the unit circle, or a transformed graph.
Track unitsConvert degrees and radians when needed.
Use identitiesReplace complicated trig expressions with equivalent simpler ones.

Worked examples

Right-triangle sine

Example: opposite = 5, hypotenuse = 13

Sine is opposite over hypotenuse.
Substitute 5 and 13.
Leave the ratio simplified.

Answer: \(\sin\theta=\frac5{13}\)

Unit-circle cosine

Example: \(\cos(0)\)

At angle 0, the point is (1, 0).
Cosine is the x-coordinate.
Read the x-value.

Answer: \(1\)

Open pop-up practice

Try one before moving on

Try: In a right triangle, tangent equals which ratio?

Answer: Opposite over adjacent.

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

The formulas for sum and difference in trigonometry are used to find the value of trigonometric functions at specific angles where it is easier to express the angle as a sum or the difference of unique angles \(0^{\circ },\:30^{\circ },\:45^{\circ },\:60^{\circ },\:90^{\circ },\:180^{\circ }\).

A step-by-step guide to sum and difference formulas

We have six sum and difference formulas for the trigonometric functions including the sine function, cosine function, and tangent function.

These formulas help us to estimate the value of trigonometric functions at angles that can be expressed as the sum or difference of specific angles \(0^{\circ },\:30^{\circ },\:45^{\circ },\:60^{\circ },\:90^{\circ },\:180^{\circ }\)

The list of sum and difference formulas is as follows:

\(\color{blue}{sin\:\left(A\:+\:B\right)=\:sin\:A\:cos\:B\:+\:cos\:A\:sin\:B}\)
\(\color{blue}{sin\:\left(A\:-\:B\right)=\:sin\:A\:cos\:B\:-\:cos\:A\:sin\:B}\)
\(\color{blue}{cos\:\left(A\:+\:B\right)=\:cos\:A\:cos\:B\:-\:sin\:A\:sin\:B}\)
\(\color{blue}{cos\:\left(A\:-\:B\right)\:=\:cos\:A\:cos\:B\:+\:sin\:A\:sin\:B}\)
\(\color{blue}{tan\:\left(A\:+\:B\right)=\:\frac{\:\left(tan\:A+\:tan\:B\right)}{\:\left(1-\:tan\:A\:tan\:B\right)}}\)
\(\color{blue}{tan\:\left(A\:-\:B\right)=\:\frac{\:\left(tan\:A\:-\:tan\:B\right)}{\:\left(1+\:tan\:A\:tan\:B\right)}}\)

Sum and Difference Formulas , Example 1:

Trigonometry for Beginners The Ultimate Step by Step Guide to Acing Trigonometry

Find the value of \(cos 105°\).

Solution:

We can write \(105°\) as \(105°= 60° + 45°\). So, using the sum formula of \(cos\), \(\color{blue}{cos\:\left(A\:+\:B\right)=\:cos\:A\:cos\:B\:-\:sin\:A\:sin\:B}\)

\(cos 105° = cos\:\left(60°\:+\:45°\right)= cos\:60°\:cos\:45°\:-\:sin\:60°\:sin\:45°\)

\(= (\frac{1}{2}) (\frac{\sqrt{2}}{2}) – (\frac{\sqrt{3}}{2})(\frac{\sqrt{2}}{2})\)

\(=\frac{1}{2\sqrt{2}} – \frac{\sqrt{3}}{2\sqrt{2}}\)

\(=\frac{1-\sqrt{3}}{2\sqrt{2}}\)

Exercises for Sum and Difference Formulas

Find the value of each trigonometric function.

\(\color{blue}{cos\:\frac{5\pi }{12}}\)
\(\color{blue}{tan\:15^{\circ }}\)
\(\color{blue}{sin\:75^{\circ }}\)

\(\color{blue}{ \frac{\sqrt{6}-\sqrt{2}}{4}}\)
\(\color{blue}{2-\sqrt{3}}\)
\(\color{blue}{\frac{\sqrt{2+\sqrt{3}}}{2}}\)

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Effortless Math Team

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Sum and Difference of Trigonometric Functions Formulas