The value of [tex]cos(a+b)[/tex] for the angles [tex]a[/tex] and [tex]b[/tex] in standard position in the first quadrant is [tex]-\frac{36}{85}[/tex]
We need to find the value of [tex]cos(a+b)[/tex]. To proceed, we need to use the compound angle formula
The cosine of the sum of two angles [tex]a[/tex] and [tex]b[/tex] is given below
[tex]cos(a+b)=cos(a)cos(b)-sin(a)sin(b)[/tex]
We are given
[tex]sin(a)=\dfrac{15}{17}\\\\cos(b)=\dfrac{3}{5}[/tex]
We need to find [tex]sin(b)[/tex] and [tex]cos(a)[/tex], using the identity
[tex]sin^2(\theta)+cos^2(\theta)=1[/tex]
To find [tex]sin(b)[/tex], note that
[tex]sin^2(b)+cos^2(b)=1\\\\\implies sin(b)=\sqrt{1-cos^2(b)}[/tex]
substituting [tex]\frac{3}{5}[/tex] for [tex]cos(b)[/tex] in the identity, we get
[tex]sin(b)=\sqrt{1-cos^2(b)}\\\\=\sqrt{1-\left(\dfrac{3}{5}\right)^2}=\dfrac{4}{5}[/tex]
To find [tex]cos(a)[/tex], note that
[tex]sin^2(a)+cos^2(a)=1\\\\\implies cos(a)=\sqrt{1-sin^2(a)}[/tex]
substituting [tex]\frac{15}{17}[/tex] for [tex]sin(a)[/tex] in the identity, we get
[tex]cos(a)=\sqrt{1-sin^2(a)}\\\\=\sqrt{1-\left(\dfrac{15}{17}\right)^2}=\dfrac{8}{17}[/tex]
We can now make use of the formula
[tex]cos(a+b)=cos(a)cos(b)-sin(a)sin(b)[/tex]
to find [tex]cos(a+b)[/tex].
[tex]cos(a+b)=cos(a)cos(b)-sin(a)sin(b)\\\\=\dfrac{8}{17}\cdot\dfrac{3}{5}-\dfrac{15}{17}\cdot\dfrac{4}{5}=-\dfrac{36}{85}[/tex]
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