Cofunctions
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 18 of 52.
In a right triangle the two acute angles add to 90°. Call one x; the other is 90° - x. The side opposite x is the side next to 90° - x, so the sine of one angle is the cosine of the other:
sin(90° - x) = cos(x)
cos(90° - x) = sin(x)
The name comes from the Latin complementi sinus, the sine of the complement: the angle that completes 90°. The pairs go on: tan(90° - x) = cot(x), sec(90° - x) = csc(x). In radians the co-angle of x is π/2 - x, so cos(π/2 - x) = sin(x).
Check it at 30°: sin(60°) = √(3)/2, and cos(30°) = √(3)/2 too. The identities hold at every angle where both sides are defined, not only for the acute angles of a triangle; the chapter on sum and difference formulas proves them.
Challenge
Draw the graph of y = sin(π/2 - x). Then press Check.
Try it yourself
Draw the graph
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