Find the values of the trigonometric functions of theta from

Find the values of the trigonometric functions of theta from the information given cos(theta) = -7/9, cos(theta) > 0 sin(theta) = cos(theta) = tan(theta) = csc(theta) = sec(theta) =

Solution

Solution:

Quadrant 1: All angles are positive (sin/cos/tan)
Quadrant 2: Sine is positive (cos/tan are negative)
Quadrant 3: Tan is positive (sin/cos are negative)
Quadrant 4: Cos is positive (tan/sin are negative)

If cot < 0, the angle lies in either quadrants 2 or 4.
If cos > 0, the angle lies in either quadrants 1 or 4.

Since 4 is a common quadrant in both of the above statements, the triangle being described must lie in quadrant four. Draw a triangle in which the central angle is theta, and opposite/adjacent,

cot theta = 1/tan theta = 1/opposite/adjacent = adjacent/opposite = -7/9

To find the hypotenuse, use the Pythagorean theorem

hypotenuse = Sqrt((7)^2 + (9)^2) = Sqrt(49+81) = Sqrt(130)

sin theta = opposite/hypotenuse = -9/Sqrt(130)
cos theta = adjacent/hypotenuse = 7/Sqrt(130)
tan theta = 1/cot theta = -9/7
csc theta = 1/sin theta = -Sqrt(130)/9
sec theta = 1 / cos theta = Sqrt(130)/7