Answer:
[tex]\displaystyle \cos(\alpha+\beta)=\frac{21}{221}[/tex]
Step-by-step explanation:
First, we can draw two right triangles to represent the given information.
Please refer to the attachment.
We will ignore the negatives for now.
The triangle on the left represents the ratio:
[tex]\displaystyle \tan(\alpha)=\frac{15}{8}[/tex]
And the triangle on the right represents the ratio:
[tex]\displaystyle \sin(\beta)=\frac{5}{13}[/tex]
And the unknown side, c and d, were determined using the Pythagorean Theorem.
We want to find:
[tex]\displaystyle \cos(\alpha+\beta), \; \pi/2<\alpha<\pi, \; \pi/2<\beta<\pi[/tex]
So, both α and β are in QII.
Using the Sum Identity, we can write our expression as:
[tex]\displaystyle\cos(\alpha+\beta) =\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)[/tex]
Now, we will use our triangles and what we know about our angles and quadrants.
Since α and β are in QII, cosine is always negative, sine is always positive, and tangent is always negative.
Now, we can use our trig ratios. Recall SohCahToa.
According to the first triangle, cos(α) is 8/17.
However, since α is in QII, cos(α) must be -8/17.
Likewise, according to the second triangle, cos(β) is 12/13.
Since β is in QII, cos(β) is -12/13.
Now, we can determine our sine ratios.
According to the first triangle, sin(α) is 15/17.
And since α is in QII, this stays positive.
And, sin(β), as given to us, is 5/13.
Therefore, we will substitute this into our equation. So:
[tex]\displaystyle \cos(\alpha+\beta)=(-\frac{8}{17})(-\frac{12}{13})-(\frac{15}{17})(\frac{5}{13})[/tex]
Evaluate:
[tex]\displaystyle \cos(\alpha+\beta)=\frac{96}{221}-\frac{75}{221}[/tex]
Hence:
[tex]\displaystyle \cos(\alpha+\beta)=\frac{21}{221}[/tex]
