Question

The angle \theta_1θ

1
​
theta, start subscript, 1, end subscript is located in Quadrant \text{II}IIstart text, I, I, end text, and \cos(\theta_1)=-\dfrac{12}{19}cos(θ
1
​
)=−
19
12
​
cosine, left parenthesis, theta, start subscript, 1, end subscript, right parenthesis, equals, minus, start fraction, 12, divided by, 19, end fraction .
What is the value of \sin(\theta_1)sin(θ
1
​
)sine, left parenthesis, theta, start subscript, 1, end subscript, right parenthesis?

Answer 1

− square root 15/4

Explanation:

Answer 2

`sin\theta_1 = (√(217))/(19)`

Explanation:

It is given that:

`cos\theta_1 = -(12)/(19)`

And we have to find the value of
`sin\theta_1 = ?`

As per trigonometric identities, the relation between
`sin\theta\ and \ cos\theta` can represented as:

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

Putting
`\theta_1` in place of
`\theta` Because we are given

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

Putting value of cosine:

`cos\theta_1 = -(12)/(19)`

`sin^2\theta_1 + ((12)/(19))^2 = 1\\\Rightarrow sin^2\theta_1 + (144)/(361) = 1\\\Rightarrow sin^2\theta_1 = 1-(144)/(361)\\\Rightarrow sin^2\theta_1 = (361-144)/(361)\\\Rightarrow sin^2\theta_1 = (217)/(361)\\\Rightarrow sin\theta_1 = +\sqrt{(217)/(361)}, -\sqrt{(217)/(361)}\\\Rightarrow sin\theta_1 = +(√(217))/(19), -(√(217))/(19)`

It is given that
`\theta_1` is in 2nd quadrant and value of sine is always positive in 2nd quadrant. So, the answer is.

`\Rightarrow sin\theta_1 = (√(217))/(19)`