Secant, Cosecant, Cotangent
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 6 of 52.
Three more functions are the reciprocals of the first three:
sec(x) = 1/cos(x)
csc(x) = 1/sin(x)
cot(x) = 1/tan(x)
The last one needs tan x to have a value that is not 0. At 90° tan x has no value, but cot x, which is cos x over sin x, is 0 there.
Each pairs with the function whose name does not start like its own: secant goes with cosine, cosecant with sine. Those two are easy to swap by mistake. At 60°, cos is 1/2, so sec(60°) = 2.
A product of these turns into sines and cosines one chip at a time. Take cot(x)sec(x): it becomes (cos(x)/sin(x)) · (1/cos(x)). Multiply the tops and the bottoms to get cos(x)/(sin(x)cos(x)), cancel cos x to get 1/sin(x), and that is csc(x).
Challenge
Show that tan(x)csc(x) equals sec(x): rewrite it until the board reads sec(x). Then press Check.
Try it yourself
Simplify the expression
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Trig Identities
1Identities on the Circle
sin²x + cos²x = 1Squares Erase the SignsIdentity or Equation?Recap: the Circle Identity4Even, Odd and Cofunctions
Negative AnglesEven and Odd in a ProofEven and Odd GraphsCofunctionsRecap: Signs of Angles2Reciprocal and Quotient
The Quotient IdentitySecant, Cosecant, CotangentDividing by a FractionWhere They BreakRecap: Reciprocals5Proving Identities
Work on One SideRewrite in sin and cosFactor FirstMake One FractionProofs That Go WrongRecap: Prove It8Half Angles, Power Reduction
Power Reductionsin²x Is a WaveHalf-Angle Valuestan of a Half AngleRecap: Half Angles