Double and Half Angle Formulas Dr. Barry Shildneck

Double Angle Formulas Given that you know the sum and difference formulas, derive a formula for the Sine, Cosine, and Tangent of twice an angle.

sin(1st)cos(2nd) [same op] sin(2nd)cos(1st) cos(1st)cos(2nd) [opposite op] sin(1st)sin(2nd) Note: You do not have to remember the tangent formula if you just remember the quotient identities.

Example: Derive the formula for Question: What does “2x” mean? 2x = (2)(x)= x + x So what does the sin(2x) equal?: sin(2x)= sin(x + x)

Deriving sin(2x) sin(2x)= sin(x + x) = sin(x)cos(x) + sin(x)cos(x)

Deriving tan(2x)

Deriving Cos(2x) OR OR

Double Angle Formulas Note: The sign of the value of the double angle is taken care of by the formula.

Example 1 sin(2θ) cos(2θ) tan(2θ) csc(2θ) cot(2θ) Use the triangle to determine the Values of each of the following. sin(2θ) cos(2θ) tan(2θ) csc(2θ) cot(2θ)

Example 2 Find the values of the cos2x given

Example 3 Use identities to rewrite each expression in terms of a single trigonometric function. 1) 4sinxcosx 2) 5–10cos2x

Sinus Minus Cosine Snot Half Angle Formulas Sinus Minus Cosine Snot

Example 4 Use the half-angle formula to find the exact value of the sin112.5o.

Example 5 Given , find the value of .

Assignment Assignment 10 – Multiple and Half Angle Identities Worksheet