Question

The angle \theta_1θ 1 ​ theta, start subscript, 1, end subscript is located in Quadrant \text{I}Istart text, I, end text, and \cos(\theta_1)=\dfrac{3}{8}cos(θ 1 ​ )= 8 3 ​ cosine, (, theta, start subscript, 1, end subscript, ), equals, start fraction, 3, divided by, 8, end fraction . What is the value of \sin(\theta_1)sin(θ 1 ​ )sine, (, theta, start subscript, 1, end subscript, )? Express your answer exactly.

Answer 1

`\sin(\theta_1) =(√(55) )/(8)`

Explanation:

The angle
`\theta_1` is located in Quadrant I and
`\cos(\theta_1)=(3)/(8)`

From Trigonometric ratio, In the First Quadrant

`\cos \theta=(Adjacent)/(hypotenuse)`

Adjacent =3, Hypotenuse =8

Using Pythagoras Theorem

`Hypotenuse^2=Opposite^2+Adjacent^2\\8^2=Opposite^2+3^2\\Opposite^2=64-9=55\\Opposite=√(55)`

Therefore:

`\sin(\theta_1)=(Opposite)/(Hypotenuse)\\\sin(\theta_1) =(√(55) )/(8)`