Question

Given sin(−θ)=1/5 and tanθ=√6/12 .

What is the value of cosθ ?
A). √6/60
B). −2√6/5
C). −√6/60
D). 2√6/5

Answer 1

Given sin(−θ)=1/5, we know sin(θ)=−1/5, and tan(θ)=√6/12 is positive, indicating that cos(θ) must be positive in the fourth quadrant. Using the Pythagorean identity, we find that cos(θ)=2√6/5, which corresponds to option (D).

Step-by-step explanation:

The question asks for the value of cos(θ) given sin(−θ) = 1/5 and tan(θ) = √6/12.

First, recognize that sin(−θ) = −sin(θ), which means that sin(θ) = −1/5. Then, use the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 to find cos(θ).

sin^2(θ) = (1/5)^2 = 1/25
cos^2(θ) = 1 − (1/25) = 24/25
cos(θ) = ±√(24/25)

Since tan(θ) = √6/12 is positive, and tangent is the ratio of sine to cosine, and we know sine is negative (sin(θ) = −1/5), then cosine must be positive for the tangent to be positive. So, cos(θ) is the positive root.

cos(θ) = √(24/25) = 2√6/5. This matches with option D).

Answer 2

B). −2√6/5

Step-by-step explanation:

tan theta = sin theta/ cos theta

Multiply each side by cos theta

tan theta * cos theta = sin theta

Divide each side by tan theta

cos theta = sin theta/ tan theta

We know that the sin (- theta) = - sin theta since sin is and odd function

sin theta = - ( sin (-theta))

Putting this into the above equation,

cos theta = - ( sin (-theta)) / tan theta

cos theta = - 1/5 / (sqrt(6)/12)

Remember when dividing fractions, we use copy dot flip

cos theta = -1/5 * 12/ sqrt(6)

cos theta = -12/ (5 sqrt(6))

We cannot leave a sqrt in the denominator, so multiply the top and bottom by sqrt(6)/sqrt(6)

cos theta = -12/ (5 sqrt(6)) * sqrt(6)/sqrt(6)

cos theta = -12 sqrt(6) / 5*6

Simplify the fraction.

cos theta = -2 sqrt(60/5