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

The answer is in the pic, enjoy.

Answer 2

See explanation

Explanation:

Assuming

`\cos( \theta) = - (5)/( 6 )`

Using the Pythagorean identity:

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

`{ (- (5)/( 6 )) }^(2) + {sin}^(2) ( \theta) = 1`

`\sin \theta = \pm ( √(11) )/(6)`

since we are in the 3rd quadrant:

`\sin \theta = - ( √(11) )/(6)`

Also, assuming

`\tan( \beta) = (4)/(3)`

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

`{sec}^(2) ( \beta) = 1 + (16)/(9)`

`{sec}^(2) ( \beta) = (25)/(9)`

`{sec} ( \beta) = \pm (5)/(3)`

`{cos} ( \beta) = \pm (3)/(5)`

In the first quadrant;.

`{cos} ( \beta) = (3)/(5)`

and

`{sin} ( \beta) = (4)/(5)`

Now :

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

`\sin( \theta + \beta) = - ( √(11) )/(6) * (3)/(5) + - (5)/(6) * (4)/(5)`

`\sin( \theta + \beta) = - (2)/(3) - ( √(11) )/(10)`