Question

The angle θ1\theta_1 θ 1 ​ theta, start subscript, 1, end subscript is located in Quadrant II\text{II} II start text, I, I, end text , and cos⁡(θ1)=−211\cos(\theta_1)=-\dfrac{2}{11} cos(θ 1 ​ )=− 11 2 ​ cosine, left parenthesis, theta, start subscript, 1, end subscript, right parenthesis, equals, minus, start fraction, 2, divided by, 11, end fraction . What is the value of sin⁡(θ1)\sin(\theta_1) sin(θ 1 ​ ) sine, left parenthesis, theta, start subscript, 1, end subscript, right parenthesis ?

Answer 1

Answer:

$\dfrac{3\sqrt{13}}{11}$

Step-by-step explanation:

Given that the angle $\theta_1$  is located in Quadrant II; and

$\cos(\theta_1)=-\dfrac{2}{11}$

In Quadrant II, x is negative and y is positive.

$\cos(\theta)=\dfrac{Adjacent}{Hypotenuse},\sin(\theta)=\dfrac{Opposite}{Hypotenuse}\\ Adjacent=-2\\Hypotenuse=11$

To find $\sin(\theta_1)$, we first determine the opposite angle of $\theta_1$.

This will be done using the Pythagoras theorem.

$Hypotenuse^2=Opposite^2+Adjacent^2\\11^2=Opposite^2+(-2)^2\\Opposite^2=121-4=117\\Opposite=\sqrt{117}=3\sqrt{13}$

Therefore:

$\sin(\theta_1)=\dfrac{Opposite}{Hypotenuse}=\dfrac{3\sqrt{13}}{11}$