###### 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.

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# The Reciprocal Trigonometric Functions - Concept

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.

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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.