If sin A = 4/5 and cos B = 5/13, Find sin(A+B), cos(A+B), sin(A−B), cos(A−B)
Question
If \[ \sin A = \frac{4}{5} \] and \[ \cos B = \frac{5}{13} \] where \[ 0 < A, B < \frac{\pi}{2} \] find the values of:
(i) \(\sin(A+B)\)
(ii) \(\cos(A+B)\)
(iii) \(\sin(A-B)\)
(iv) \(\cos(A-B)\)
Solution
Given: \[ \sin A = \frac{4}{5} \]
Using \[ \sin^2 A + \cos^2 A = 1 \]
\[ \cos A = \sqrt{1-\left(\frac{4}{5}\right)^2} \]
\[ = \sqrt{1-\frac{16}{25}} \]
\[ = \sqrt{\frac{9}{25}} \]
\[ \cos A = \frac{3}{5} \]
Also, \[ \cos B = \frac{5}{13} \]
Using \[ \sin^2 B + \cos^2 B = 1 \]
\[ \sin B = \sqrt{1-\left(\frac{5}{13}\right)^2} \]
\[ = \sqrt{1-\frac{25}{169}} \]
\[ = \sqrt{\frac{144}{169}} \]
\[ \sin B = \frac{12}{13} \]
(i) Find \(\sin(A+B)\)
Using formula: \[ \sin(A+B)=\sin A \cos B+\cos A \sin B \]
\[ =\frac{4}{5}\times\frac{5}{13}+\frac{3}{5}\times\frac{12}{13} \]
\[ =\frac{20}{65}+\frac{36}{65} \]
\[ =\frac{56}{65} \]
Therefore, \[ \boxed{\sin(A+B)=\frac{56}{65}} \]
(ii) Find \(\cos(A+B)\)
Using formula: \[ \cos(A+B)=\cos A \cos B-\sin A \sin B \]
\[ =\frac{3}{5}\times\frac{5}{13}-\frac{4}{5}\times\frac{12}{13} \]
\[ =\frac{15}{65}-\frac{48}{65} \]
\[ =-\frac{33}{65} \]
Therefore, \[ \boxed{\cos(A+B)=-\frac{33}{65}} \]
(iii) Find \(\sin(A-B)\)
Using formula: \[ \sin(A-B)=\sin A \cos B-\cos A \sin B \]
\[ =\frac{4}{5}\times\frac{5}{13}-\frac{3}{5}\times\frac{12}{13} \]
\[ =\frac{20}{65}-\frac{36}{65} \]
\[ =-\frac{16}{65} \]
Therefore, \[ \boxed{\sin(A-B)=-\frac{16}{65}} \]
(iv) Find \(\cos(A-B)\)
Using formula: \[ \cos(A-B)=\cos A \cos B+\sin A \sin B \]
\[ =\frac{3}{5}\times\frac{5}{13}+\frac{4}{5}\times\frac{12}{13} \]
\[ =\frac{15}{65}+\frac{48}{65} \]
\[ =\frac{63}{65} \]
Therefore, \[ \boxed{\cos(A-B)=\frac{63}{65}} \]