Final answer:
To express the tangent of an angle t in terms of the cosine of t for an angle in Quadrant IV, use the equation tan t = -√(1 - cos^2 t) / cos t, because in Quadrant IV, sine is negative and cosine is positive.
Step-by-step explanation:
To express the tangent of an angle t in terms of the cosine of t, we can use the Pythagorean identity for trigonometric functions: sin2t + cos2t = 1. In Quadrant IV, cosine is positive but sine is negative. As the tangent function is the ratio of the sine to the cosine, we can rearrange the Pythagorean identity to solve for sine:
- sin2t = 1 - cos2t
- sin t = -√(1 - cos2t) (negative because t is in Quadrant IV)
Now we can express the tangent function as:
tan t = sin t / cos t = -√(1 - cos2t) / cos t
So the tangent of angle t in Quadrant IV can be written in terms of the cosine of t as -√(1 - cos2t) / cos t.