Key takeaway: The sum and difference formulas give sin, cos and tan of α ± β from the values at α and β, such as cos(α + β) = cos α cos β − sin α sin β.
Grade 12 – PrecalculusF-TF.C.9Trigonometry
Definition of Sum and Difference Formulas
The sum and difference formulas let you find exact trig values for angles that are not on the standard list but can be written as a sum or difference of angles that are. Angles like 75° or 15° fall into this group, since 75° = 45° + 30° and 15° = 45° − 30°.
Trig functions do not distribute over addition, so cos(45° + 30°) is not cos 45° + cos 30°. Instead each formula mixes products of sines and cosines. The sign pattern is the part to memorize: sine keeps the sign of the angle sum, while cosine flips it.
Formula and notation
For angles \(\alpha\) and \(\beta\):
\[\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta,\quad \cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta,\]\[\tan(\alpha\pm\beta)=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}.\]
Key facts
- In the cosine formula the middle sign is opposite to the sign inside the parentheses.
- Setting both angles equal produces the double-angle formulas.
- With the sum and difference formulas, pairs such as 45° and 30° give exact answers with square roots.
- They also turn products like sin α cos β into sums, which is useful in calculus.
Sum and Difference Formulas example
Find the exact value of \(\cos 75^{\circ}\) by writing it as \(\cos(45^{\circ}+30^{\circ})\).
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Use the cosine of a sum: \(\cos 45^{\circ}\cos 30^{\circ}-\sin 45^{\circ}\sin 30^{\circ}\).
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Insert the exact values: the first product is root six over four, and the second product is root two over four.
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Subtract the two fractions, which share the denominator 4.
\(\cos 75^{\circ}=\)\(\frac{\sqrt{6}-\sqrt{2}}{4}\approx 0.2588\).
Common mistake
Watch out
Imani writes \(\cos(45^{\circ}+30^{\circ})=\cos 45^{\circ}+\cos 30^{\circ}\approx 0.707+0.866=1.573\).
Why it is wrong
None of the sum and difference formulas simply adds. Cosine does not distribute over a sum. A cosine can never exceed 1, so 1.573 is impossible. The correct product-and-subtract formula gives about 0.259.
Related terms and lessons
Browse the glossary: A–Z list · Precalculus vocabulary · Trigonometry