Question

If cos(x)=−1/4 and x is in quadrant III, find the exact values of the following without solving for x:

Answer 1

When given cos(x) = −1/4 in quadrant III, we find sin(x) = −√15/4 and tan(x) = √15 using Pythagorean identity and the tangent definition.

Step-by-step explanation:

The problem states that cos(x) = −1/4 and that x is in the quadrant III. In this quadrant, both sine and cosine functions are negative, as the coordinate values (x,y) are both negative.

To find the exact values of sine and tangent, we use the Pythagorean identity: sin2(x) + cos2(x) = 1. Given that cos(x) = −1/4, we can solve for sin(x):

1. sin2(x) = 1 - cos2(x)
2. sin2(x) = 1 - (−1/4)2 = 1 - 1/16
3. sin2(x) = 15/16
4. sin(x) will be the negative square root because x is in quadrant III, so sin(x) = -√(15/16) = −√15/4.

Next, for the tangent function, we use the identity tan(x) = sin(x)/cos(x):

1. tan(x) = (−√15/4) / (−1/4)
2. tan(x) = √15