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 \sin(\theta_1)=\dfrac{1}{4}sin(θ 1 ​ )= 4 1 ​ sine, (, theta, start subscript, 1, end subscript, ), equals, start fraction, 1, divided by, 4, end fraction . What is the value of \cos(\theta_1)cos(θ 1 ​ )cosine, (, theta, start subscript, 1, end subscript, )?

Answer 1

`\cos(\theta_1)=-(√(15))/(4)`

Explanation:

The angle
`\theta_1` is located in Quadrant II.

`\sin(\theta_1)=(1)/(4)`

From trigonometry, we know that:

`\sin(\theta)=(Opposite)/(Hypotenuse)\\$Therefore:\\Opposite=1\\Hypotenuse=4\\Using Pythagorean theorem:\\Hypotenuse^2=Opposite^2+Adjacent^2\\4^2=1^2+Adjacent^2\\Adjacent^2=16-1\\Adjacent^2=15\\Adjacent=√(15)`

Now, in Quadrant II,

• The x-axis is negative
• The y-axis is positive

Therefore, the Adjacent angle to
`\theta_1 =-√(15)`

Therefore:

`\cos(\theta_1)=(Adjacent)/(Hypotenuse)=(-√(15))/(4)\\\\\cos(\theta_1)=-(√(15))/(4)`