sin(A + B) and sin(A - B)
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 26 of 52.
The sine of a sum is not the sum of the sines. At 30° and 30°, sin(60°) is about 0.87, while sin(30°) + sin(30°) is 1. The true formulas mix sine and cosine:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
They give exact values at angles the special triangles do not reach. 75° is 45° + 30°:
sin(45° + 30°) = (√(2)/2) · (√(3)/2) + (√(2)/2) · (1/2)
which is √(6)/4 + √(2)/4, so sin(75°) = (√(6) + √(2))/4, about 0.966. The sign in the middle follows the sign in the angle.
Challenge
Find the exact value of sin(15°).
On this board the formula chip writes the values unworked; type each next line yourself in the box, and a line that does not equal the one above is refused.
Type 15° as a difference of two special angles, use the formula chip, type the line with the exact values put in, then work it down to one fraction. 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