The terminal side of an angle theta in standard position pas

The terminal side of an angle, theta, in standard position passes through the point (-10/3, -4/3). Calculate the values of the six trigonometric functions for angle theta.

Enter the exact answers and, if necessary, rationalize the denominator.

Solution

we know that Standard Position: An angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis. The ray on the x-axis is called the initial side and the other ray is called the terminal side.

the terminal side of an angle is theta , then we have to consider 180 -theta to calculate the values

given the standard position passes through the point (-10/3, -4/3).

that means x coordiate is -10/3 , and y coordinate is -4/3

tan(180 -theta) = y coordinate / x coordinate

     -tan(theta)                    = -4/3 * -3/10

   tan(theta) = -2/5

tan(theta) x cot(theta) =1

-2/5 x cot(theta) =1

cot(theta) = -5/2

sec^2(theta) - tan^2(theta) =1

sec^2(theta) - (4/25) =1

sec^2(theta) = 1+4/25

sec (theta) = sqrt(29)/5

sec(theta) = - sqrt(29)/5

so cos(theta) = -5/sqrt(29)

sin^2(theta) +cos^2(theta) =1

sin^2(theta) + 25/29 =1

sin^2 (theta) = 1-25/29

sin(theta) = 2/sqrt(29)

csc(theta) = sqrt(29 ) / 2