Write the first expression in terms of the second if the terminal point determined by t is in the given quadrant.

Question

Write the first expression in terms of the second if the terminal point determined by t is in the given quadrant.

tan(t), cos(t);

 Quadrant III

tan(t) =

Answer 1

For an angle t in Quadrant III, the tangent function tan(t) in terms of the cosine function cos(t) is expressed as follows:

Since both sine and cosine are negative in Quadrant III, the tangent, being the ratio of sine to cosine, will be positive. The Pythagorean identity sin^2(t) + cos^2(t) = 1 is used. Solving for sin(t):

sin(t) = -√(1 - cos^2(t)), because sin(t) is negative in Quadrant III.

Therefore, tan(t) = -√(1 - cos^2(t)) / cos(t). This expression reflects the fact that sin(t) is negative in this quadrant.