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.
Sum and Difference of Trigonometric Functions Formulas: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
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
- Sine is opposite over hypotenuse.
- Substitute 5 and 13.
- Leave the ratio simplified.
Unit-circle cosine
- At angle 0, the point is (1, 0).
- Cosine is the x-coordinate.
- Read the x-value.
Try one before moving on
Sum and Difference of Trigonometric Functions Formulas: pop-up practice
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
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.
- \(\color{blue}{cos\:\frac{5\pi }{12}}\)
- \(\color{blue}{tan\:15^{\circ }}\)
- \(\color{blue}{sin\:75^{\circ }}\)
- \(\color{blue}{ \frac{\sqrt{6}-\sqrt{2}}{4}}\)
- \(\color{blue}{2-\sqrt{3}}\)
- \(\color{blue}{\frac{\sqrt{2+\sqrt{3}}}{2}}\)
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