Question

find the exact value of the expression.

\\( \sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } + \tan ^ { - 1 } \frac { 12 } { 5 } \
ight) \\)

\\( \sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } + \tan ^ { - 1 } \frac { 12 } { 5 } \
ight) = \square \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the

Explanation:

Step1: Let \(\alpha=\sin^{-1}\frac{3}{5}\) and \(\beta = \tan^{-1}\frac{12}{5}\)

By the definition of inverse - sine function, if \(\alpha=\sin^{-1}\frac{3}{5}\), then \(\sin\alpha=\frac{3}{5}\) and \(\cos\alpha=\sqrt{1 - (\frac{3}{5})^2}=\sqrt{\frac{25 - 9}{25}}=\frac{4}{5}\) (using the identity \(\sin^{2}\theta+\cos^{2}\theta = 1\) and since \(\alpha\in[-\frac{\pi}{2},\frac{\pi}{2}]\) and \(\sin\alpha>0\), \(\cos\alpha>0\)).

By the definition of inverse - tangent function, if \(\beta=\tan^{-1}\frac{12}{5}\), then \(\tan\beta=\frac{12}{5}\). Using the identity \(\tan\beta=\frac{\sin\beta}{\cos\beta}\) and \(\sin^{2}\beta+\cos^{2}\beta = 1\), we have \(\sin\beta=\frac{12}{13}\) and \(\cos\beta=\frac{5}{13}\) (since if \(\tan\beta=\frac{y}{x}=\frac{12}{5}\), then \(r=\sqrt{x^{2}+y^{2}}=\sqrt{25 + 144}=13\), \(\sin\beta=\frac{y}{r}\), \(\cos\beta=\frac{x}{r}\)).

Step2: Use the sum formula for sine \(\sin(A + B)=\sin A\cos B+\cos A\sin B\)

Here \(A=\alpha\) and \(B = \beta\).

Substitute \(\sin\alpha=\frac{3}{5}\), \(\cos\alpha=\frac{4}{5}\), \(\sin\beta=\frac{12}{13}\), and \(\cos\beta=\frac{5}{13}\) into the formula:

$$
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Answer:

\(\frac{63}{65}\)