Find the exact value of each of the following under the given conditions below.

(a)
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Question

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

UpStudy AI · Accepted Answer

To find $$ \sin(\alpha + \beta) $$, we'll use the sine addition formula:
$$
\sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta
$$
Let's break down each component step by step.

### Given:
- $$ \tan \alpha = -\frac{12}{5} $$ with $$ \frac{\pi}{2} < \alpha < \pi $$
- $$ \cos \beta = \frac{1}{2} $$ with $$ 0 < \beta < \frac{\pi}{2} $$

### Step 1: Determine $$ \sin\alpha $$ and $$ \cos\alpha $$

Since $$ \tan \alpha = \frac{\sin \alpha}{\cos \alpha} = -\frac{12}{5} $$, we can consider a right triangle where:
- Opposite side to angle $$ \alpha $$ is 12
- Adjacent side to angle $$ \alpha $$ is -5 (negative because $$ \alpha $$ is in the second quadrant)
- Hypotenuse $$ r = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = 13 $$

Thus:
$$
\sin \alpha = \frac{12}{13}, \quad \cos \alpha = -\frac{5}{13}
$$

### Step 2: Determine $$ \sin\beta $$

Given $$ \cos \beta = \frac{1}{2} $$ and $$ \beta $$ is in the first quadrant:
$$
\sin \beta = \sqrt{1 - \cos^2 \beta} = \sqrt{1 - \left(\frac{1}{2}\right)^2} = \frac{\sqrt{3}}{2}
$$

### Step 3: Apply the Sine Addition Formula

$$
\sin(\alpha + \beta) = \left(\frac{12}{13}\right)\left(\frac{1}{2}\right) + \left(-\frac{5}{13}\right)\left(\frac{\sqrt{3}}{2}\right)
$$
$$
\sin(\alpha + \beta) = \frac{12}{26} - \frac{5\sqrt{3}}{26} = \frac{12 - 5\sqrt{3}}{26}
$$

### Final Answer:

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

Answer:
\boxed{\dfrac{12\,-\,5\,\sqrt{3}}{26}}
$$ \sin(\alpha + \beta) = \frac{12 - 5\sqrt{3}}{26} $$

Find the exact value of each of the following under the given conditions below.

(a)
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

Step 1: Determine and

Step 2: Determine and

Step 3: Calculate the required values

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Latest Trigonometry Questions

Find the exact value of each of the following under the given conditions below. (a) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

Step 1: Determine and

Step 2: Determine and

Step 3: Calculate the required values

preguntas relacionadas

Latest Trigonometry Questions

Find the exact value of each of the following under the given conditions below.

(a)
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)