Co-Functions
Introduction:
In this lesson, the six fundamental trigonometric functions will be compared using the idea of a “co-function.”
Background Information:
In a right triangle, one angle is and the side across from this angle is called the hypotenuse. The two sides which form the angle are called the legs of the right triangle. We show a right triangle below. The legs are defined as either “opposite” or “adjacent” (next to) the angle A.

We shall call the opposite side “opp,” the adjacent side “adj” and the hypotenuse “hyp.”
The Lesson:
The six trigonometric functions for angle A in this triangle are defined as follows:
 sin(A) = cos(A) = tan(A) = csc(A) = sec(A) = cot(A) =

1. Observe that since the sum of the angles in a triangle is 180º, the sum of the measures of angles A and B must be 90º. Angles with a sum of 90º are called “complementary.” In the triangle shown above, we have B = 90º - A.
2. Observe that the sin(A) = is the same as the cos(B) = . This is because cos(B) is the ratio of the side adjacent to B and the hypotenuse and the side adjacent to B is the side that is also opposite to A.
3. We state this as an equation: sin(A) = cos(90º - B). The word “cosine” is a combination of “co” for “complementary” and “sine.”
To summarize, the sine of an angle is the same as the cosine of the complement of the angle.
For example, the sin(23º) = cos(90º - 23º) = cos(67º).
We can generalize these observations to the other cofunctions.
tan(A) = cot(90º - A) and sec(A) = csc(90º - A).
Let's Practice:
1. If the tan, what is cot(27º)?
cot(27º) which is the same as tan(27°) since 27º and 63º are complementary angles.
2. If the sin(X) = 0.2, what is the cosine of 90º - X?
The cosine will be the same as the sine since these angles are complementary. Therefore, cos(X) = 0.2.
3. If A = 137º which is a second quadrant angle, show that the sin(137º) = cos(90º - 137º).
The cofunction relationship works for any angle A. Using a calculator, we see that sin. The sine is positive for this quadrant two angle.

The cosine of (90º - 137º) = -43º is also positive because the cosine of a quadrant four angle is positive. We have also.

Examples
 If the tan(A) = 5, what is the cot(90º - A)? What is your answer?
 If the and A is a quadrant three angle, what is the cos(90º - A) and in what quadrant is the angle (90º - A)? What is your answer?

M Ransom

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