Turning by 90° and 180°
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 28 of 52.
Adding a quarter or a half turn to an angle changes its sine and cosine in a fixed way, and the sum formulas show exactly how. The values of sine and cosine at 90°, 180° and 270° are 0, 1 or -1, so one product in each formula vanishes.
Take sin(x + 180°). The formula gives sin(x)cos(180°) + cos(x)sin(180°). Put in cos(180°) = -1 and sin(180°) = 0: sin(x)(-1) + cos(x)(0), which is -sin(x). Half a turn sends the point to the opposite side of the centre, so both coordinates change sign.
The cofunctions come out the same way: sin(90° - x) = sin(90°)cos(x) - cos(90°)sin(x), which is cos x.
Challenge
Show that cos(x + 270°) equals sin(x).
Type each next line yourself; a line that does not equal the one above is refused.
Expand it with the formula chip, type the line with the values at 270° put in, then work it down until the board reads sin(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 Angles3Pythagorean Identities
1 + tan²x = sec²x1 + cot²x = csc²xA Difference of SquaresSquaring a SumRecap: Three Forms6Sum and Difference
sin(A + B) and sin(A - B)cos(A + B) and cos(A - B)Turning by 90° and 180°Read the Formula Backwardstan(A + B) and tan(A - B)Recap: Sum Formulas