Sum and Difference of Trigonometric Functions Formulas

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 \(0^{\circ },\:30^{\circ },\:45^{\circ },\:60^{\circ },\:90^{\circ },\:180^{\circ }\).

Related Topics

• How to Solve Double Angle Identities
• Pythagorean Identities

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:

 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.

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

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