
1. Finding Values of tan,cot,sec,csc
2. You should be familiar with the process to find the cosine value, given the sine value and vice
versa.
In this lesson, we will define four other trig functions, the tangent, cotangent, secant and
cosecant, and find their values.
3. First, using the functions sine and cosine, we will define the other four trig functions. The
tangent is defined as the sine divided by the cosine. The other three functions are recipro-
cals. The cotangent is the reciprocal of the tangent. The cosecant, abbreviated c-s-c, is the
reciprocal of the sine. And the secant is the reciprocal of the cosine. Note that each pair has
one co-function. The reciprocal of sine is CO-secant, and the reciprocal of CO-sine is secant.
We can find the values of these functions, based on the values of sine and cosine.
4. For example, if sin θ=−
3
5, and cos θ=4
5, find the values of the other four trig functions.
This is done simply by using the definitions.
5. The tangent value is the sine value divided by the cosine value, the cotangent value is the
reciprocal of the tangent value, the secant value is the reciprocal of the cosine value, and the
cosecant value is the reciprocal of the sine value.
6. We actually gave you more information than you needed in order to find these trig values.
Once we know the sine value, the cosine value is also determined, up to a plus or a minus.
Here, we use the trig version of the Pythagorean Theorem to find the cosine. Since we are
given that the cosine is positive,
7. we know that cos θ= +4
5.
8. We could also have specified in which quadrant θlies,
9. or we could have specified the size of θ, either as a positive angle
10. or a negative angle.
11. To recap: Given either the sine value or cosine value, the standard procedure is to use the
trig version of the Pythagorean Theorem, that sin2θ+ cos2θ= 1, to find the other, and then
use the definitions.
12. If you are given either the secant or cosecant, first take the reciprocal to find the sine or cosine,
and then follow the standard procedure.
13. Here is an example. We are given the cosecant value. We can first find the sine value by
taking the reciprocal.
14. We next find the cosine value by using the the trig version of the Pythagorean Theorem.
15. Note that since the angle is between πand 3π
2, we are in the third quadrant, and therefore we
choose the negative value.
16. Now that we have the values for sine and cosine, we can find the value of tangent,
17. and then take reciprocals to find the value for cotangent