Question

A point on the terminal side of an angle in standard position is given. Find the exact value of each of the six trigonometric functions of θ.
Point given: (2,-3)

Answer 1

Refer to Figure 1 for the diagram of the point and reference triangle needed to answer the problem.

Plot the point (2,-3) which is in quadrant 4

Draw a right triangle such that one leg is resting on the x axis and the other leg is perpendicular to the x axis. The hypotenuse goes from (0,0) to (2,-3). 

The horizontal leg is 2 units long. So a = 2
The vertical leg is 3 units long. So b = 3
The hypotenuse is unknown but we can use the pythagorean theorem to find it. We'll call it c for now

Use Pythagorean theorem to find c
a^2 + b^2 = c^2
2^2 + 3^2 = c^2
4 + 9 = c^2
13 = c^2
c^2 = 13
c = sqrt(13)
The hypotenuse is sqrt(13) which is shown in the attached image (Figure 1)

Using figure 1, now refer to figure 2 to compute the ratios for sine, cosine and tangent
Figure 3 shows how to compute the ratios for cosecant, sectant, and cotangent

Note: With figure 1, the leg furthest from angle theta is the opposite side (in this case -3). The leg closest to the angle theta is 2 units long.