Question

The angle

θ
1
θ
1
​
theta, start subscript, 1, end subscript is located in Quadrant
III
IIIstart text, I, I, I, end text, and
cos
⁡
(
θ
1
)
=
−
13
30
cos(θ
1
​
)=−
30
13
​
cosine, left parenthesis, theta, start subscript, 1, end subscript, right parenthesis, equals, minus, start fraction, 13, divided by, 30, end fraction .
What is the value of
sin
⁡
(
θ
1
)
sin(θ
1
​
)sine, left parenthesis, theta, start subscript, 1, end subscript, right parenthesis?

Answer 1

`sin\theta_1 = -(√(731))/(30)` is the correct answer.

Explanation:

It is given that
`\theta_(1)` is in third quadrant.

`cos\theta_(1)` is always negative in 3rd quadrant and also

`sin\theta_(1)` is always negative in 3rd quadrant.

Also, we know the following identity about and :

`sin^2\theta + cos^2\theta = 1`

Put
`\theta` as
`\theta_(1)`:

`sin^2\theta_1 + cos^2\theta_1 = 1`

We are given that
`cos\theta_1 = -(13)/(30)`

`sin^2\theta_1 + ((-13)/(30))^2 = 1\\`

`\Rightarrow sin^2\theta_1 = 1 - (169)/(900)\\\Rightarrow sin^2\theta_1 = (900-169)/(900)\\\Rightarrow sin^2\theta_1 = (731)/(900)\\\Rightarrow sin\theta_1 = +\sqrt{(731)/(900)}, -\sqrt{(731)/(900)}\\\Rightarrow sin\theta_1 = +(√(731))/(30), -(√(731))/(30)`

`\theta_(1)` is in 3rd quadrant so
`sin\theta_(1)` is negative.

So
`sin\theta_1 = -(√(731))/(30)`