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The Reciprocal Trigonometric Functions - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

There are three reciprocal trigonometric functions, making a total of six including cosine, sine, and tangent. The reciprocal cosine function is secant: sec(theta)=1/cos(theta). The reciprocal sine function is cosecant, csc(theta)=1/sin(theta). The reciprocal tangent function is cotangent, expressed two ways: cot(theta)=1/tan(theta) or cot(theta)=cos(theta)/sin(theta).

I want to talk about the reciprocal of trigonometric functions. Now just like the other 3 trigonometric functions, the reciprocal functions have unit circle definitions. Recall the definition of cosine, sine and tangent. Cosine theta equals x, sine theta equals y and tangent theta equals y over x, where x and y are the coordinates of the point on the terminal side of the angle. And the 3 new functions are secant theta, cosecant theta and cotangent theta.
Secant theta is defined as 1 over x. Cosecant theta is 1 over y and cotangent is x over y. Let's develop some identities with these new functions. So because secant theta is defined as 1 over x, we can use the fact that x equals cosine theta to write 1 over cosine theta. So secant theta equals 1 over cosine theta and that's a really useful identity when you're trying to understand what secant theta is.
Cosecant theta is 1 over y and y is sine theta. So cosecant theta is the reciprocal of sine.
Tangent, we've already done this one, tangent is sine theta over cosine theta. Cotangent because it's defined as x over y is cosine theta over sine theta. And you can actually see that these 2 are reciprocals of one other as well.
Now you remember the main Pythagorean identity cosine squared plus sine squared equals 1. There are actually 2 other Pythagorean identities. One of them is cotangent squared theta plus 1 equals cosecant squared theta, and the other is 1 plus tangent squared theta is secant squared theta. And we'll be using these Pythagorean identities later.

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