Question

Cosθ=−5√6 , where π≤θ≤3π2 . tanβ=43 , where 0≤β≤π2 . What is the exact value of sin(θ+β) ? Enter your answer, as a fraction in simplified form, in the box.

Answer 1

Answer:

See explanation

Step-by-step explanation:

Assuming

$\cos( \theta) = - \frac{5}{ 6 }$

Using the Pythagorean identity:

${cos}^{2} ( \theta) + {sin}^{2} ( \theta) = 1$

${ (- \frac{5}{ 6 }) }^{2} + {sin}^{2} ( \theta) = 1$

$\sin \theta = \pm \frac{ \sqrt{11} }{6}$

since we are in the 3rd quadrant:

$\sin \theta = - \frac{ \sqrt{11} }{6}$

Also, assuming

$\tan( \beta) = \frac{4}{3}$

${sec}^{2} ( \beta) = 1 + \tan ^{2} ( \beta)$

${sec}^{2} ( \beta) = 1 + \frac{16}{9}$

${sec}^{2} ( \beta) = \frac{25}{9}$

${sec} ( \beta) = \pm \frac{5}{3}$

${cos} ( \beta) = \pm \frac{3}{5}$

In the first quadrant;.

${cos} ( \beta) = \frac{3}{5}$

and

${sin} ( \beta) = \frac{4}{5}$

Now :

$\sin( \theta + \beta) = \sin( \theta) \cos \beta + \cos( \theta) \sin \beta$

$\sin( \theta + \beta) = - \frac{ \sqrt{11} }{6} \times \frac{3}{5} + - \frac{5}{6} \times \frac{4}{5}$

$\sin( \theta + \beta) = - \frac{2}{3} - \frac{ \sqrt{11} }{10}$

Answer 2

Answer:

The answer is in the pic, enjoy.