Find the exact value of each of the following under the given conditions below. $$ \tan \alpha=-\frac{12}{5}, \frac{\pi}{2}

Question

Find the exact value of each of the following under the given conditions below. $$ \tan \alpha=-\frac{12}{5}, \frac{\pi}{2}<\alpha<\pi ; \cos \beta=\frac{1}{2}, 0<\beta<\frac{\pi}{2} $$ (a) $$ \sin (\alpha+\beta) $$ (b) $$ \cos (\alpha+\beta) $$ (c) $$ \sin (\alpha-\beta) $$ (d) $$ \boldsymbol{\operatorname { t a n }}(\alpha-\beta) $$ (a) $$ \sin (\alpha+\beta)=\frac{12-5 \sqrt{3}}{26} $$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) (b) $$ \cos (\alpha+\beta)= $$ $$ \square $$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

UpStudy AI · Accepted Answer

To find $$ \cos (\alpha + \beta) $$, we'll use the cosine addition formula:

$$
\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta
$$

**Given:**

- $$ \tan \alpha = -\frac{12}{5} $$ with $$ \frac{\pi}{2} < \alpha < \pi $$ (Quadrant II)
  
  In Quadrant II:
  $$
  \sin \alpha = \frac{12}{13}, \quad \cos \alpha = -\frac{5}{13}
  $$

- $$ \cos \beta = \frac{1}{2} $$ with $$ 0 < \beta < \frac{\pi}{2} $$ (Quadrant I)
  
  In Quadrant I:
  $$
  \sin \beta = \frac{\sqrt{3}}{2}, \quad \cos \beta = \frac{1}{2}
  $$

**Calculating $$ \cos (\alpha + \beta) $$:**

$$
\cos (\alpha + \beta) = \left( -\frac{5}{13} \right) \left( \frac{1}{2} \right) - \left( \frac{12}{13} \right) \left( \frac{\sqrt{3}}{2} \right)
$$

$$
= -\frac{5}{26} - \frac{12\sqrt{3}}{26}
$$

$$
= -\frac{5 + 12\sqrt{3}}{26}
$$

**Final Answer:**

$$
\cos (\alpha + \beta) = -\frac{5 + 12\sqrt{3}}{26}
$$

Answer:
Problem b Answer

$$
\cos (\alpha+\beta) = -\,\frac{5 + 12\,\sqrt{3}}{26}
$$
$$ \cos (\alpha + \beta) = -\frac{5 + 12\sqrt{3}}{26} $$

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