Use sin(A±B)=sinAcosB±cosAsinB and cos(A±B)=cosAcosB∓sinAsinB. Tangent of a sum or difference uses a quotient whose denominator can vanish.
Exact side ratios can be substituted directly without first finding decimal angles. Check that the chosen signs match the sum or difference in the question.
Check the conditions
The two angles are acute in these cases.
The tangent quotient must have a nonzero denominator.
Three worked examples
Example 1 · Compound-angle formulas
A and B are acute, with \(\displaystyle \sin A=\frac{12}{37},\cos A=\frac{35}{37},\sin B=\frac{16}{65},\cos B=\frac{63}{65}\).
Find sin(A+B).
Find cos(A+B).
Find tan(A−B).
Find cos(A−B).
Show the reasoning
(a)
Substitute into the compound-angle formula with the correct signs.
\(\displaystyle \frac{1316}{2405}\)
Answer: \(\displaystyle \frac{1316}{2405}\).
(b)
Substitute into the compound-angle formula with the correct signs.
\(\displaystyle \frac{2013}{2405}\)
Answer: \(\displaystyle \frac{2013}{2405}\).
(c)
Substitute into the compound-angle formula with the correct signs.
\(\displaystyle \frac{196}{2397}\)
Answer: \(\displaystyle \frac{196}{2397}\).
(d)
Substitute into the compound-angle formula with the correct signs.
\(\displaystyle \frac{2397}{2405}\)
Answer: \(\displaystyle \frac{2397}{2405}\).
Example 2 · Compound-angle formulas
A and B are acute, with \(\displaystyle \sin A=\frac{7}{25},\cos A=\frac{24}{25},\sin B=\frac{5}{13},\cos B=\frac{12}{13}\).
Find sin(A+B).
Find cos(A+B).
Find tan(A−B).
Find cos(A−B).
Show the reasoning
(a)
Substitute into the compound-angle formula with the correct signs.
\(\displaystyle \frac{204}{325}\)
Answer: \(\displaystyle \frac{204}{325}\).
(b)
Substitute into the compound-angle formula with the correct signs.
\(\displaystyle \frac{253}{325}\)
Answer: \(\displaystyle \frac{253}{325}\).
(c)
Substitute into the compound-angle formula with the correct signs.
\(\displaystyle - \frac{36}{323}\)
Answer: \(\displaystyle - \frac{36}{323}\).
(d)
Substitute into the compound-angle formula with the correct signs.
\(\displaystyle \frac{323}{325}\)
Answer: \(\displaystyle \frac{323}{325}\).
Example 3 · Compound-angle formulas
A and B are acute, with \(\displaystyle \sin A=\frac{3}{5},\cos A=\frac{4}{5},\sin B=\frac{12}{37},\cos B=\frac{35}{37}\).
Find sin(A+B).
Find cos(A+B).
Find tan(A−B).
Find cos(A−B).
Show the reasoning
(a)
Substitute into the compound-angle formula with the correct signs.
\(\displaystyle \frac{153}{185}\)
Answer: \(\displaystyle \frac{153}{185}\).
(b)
Substitute into the compound-angle formula with the correct signs.
\(\displaystyle \frac{104}{185}\)
Answer: \(\displaystyle \frac{104}{185}\).
(c)
Substitute into the compound-angle formula with the correct signs.
\(\displaystyle \frac{57}{176}\)
Answer: \(\displaystyle \frac{57}{176}\).
(d)
Substitute into the compound-angle formula with the correct signs.
\(\displaystyle \frac{176}{185}\)
Answer: \(\displaystyle \frac{176}{185}\).
Common mistakes
Cosine uses the opposite sign before the sine product.