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Sagot :
Answer:
63/65
Explanation:
Step 1
Given that a and b are first-quadrant angles. In addition:
[tex]\begin{gathered} \sin a=\frac{5}{13} \\ \cos b=\frac{3}{5} \end{gathered}[/tex]Using the double-angle formula:
[tex]\sin (a+b)=\sin a\cos b+\cos a\sin b[/tex]Step 2
We need to find the values of cos a and sin b.
(i)cos a
From trigonometric ratios:
[tex]\begin{gathered} \sin \theta=\frac{\text{Opposite}}{\text{Hypotenuse}} \\ \sin a=\frac{5}{13}\implies\text{Opp}=5,\text{Hyp}=13 \end{gathered}[/tex]We find the length of the adjacent side using the Pythagorean Theorem.
[tex]\begin{gathered} \text{Hyp}^2=\text{Opp}^2+\text{Adj}^2 \\ 13^2=5^2+\text{Adj}^2 \\ \text{Adj}^2=13^2-5^2=144=12^2 \\ \text{Adj}=12 \end{gathered}[/tex]Therefore:
[tex]\begin{gathered} \cos \theta=\frac{\text{Adjacent}}{\text{Hypotenuse}} \\ \cos a=\frac{12}{13} \end{gathered}[/tex](b) sin b
From trigonometric ratios:
[tex]\begin{gathered} \cos \theta=\frac{\text{Adjacent}}{\text{Hypotenuse}} \\ \cos b=\frac{3}{5}\implies\text{Adj}=3,\text{Hyp}=5 \end{gathered}[/tex]We find the length of the opposite side using the Pythagorean Theorem.
[tex]\begin{gathered} \text{Hyp}^2=\text{Opp}^2+\text{Adj}^2 \\ 5^2=\text{Opp}^2+\text{3}^2 \\ \text{Opp}^2=5^2-3^2=25-9=16 \\ \text{Opp}=\sqrt[]{16}=4 \end{gathered}[/tex]Therefore:
[tex]\begin{gathered} \sin \theta=\frac{\text{Opposite}}{\text{Hypotenuse}} \\ \sin b=\frac{4}{5} \end{gathered}[/tex]Step 3
Substitute the values of cos a and sin b into the double angle formula.
[tex]\begin{gathered} \sin (a+b)=\sin a\cos b+\cos a\sin b \\ =\frac{5}{13}\times\frac{3}{5}+\frac{12}{13}\times\frac{4}{5} \\ =\frac{15}{65}+\frac{48}{65} \\ =\frac{63}{65} \end{gathered}[/tex]The value of sin(a+b) is 63/65.
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