T.3.3 – Trigonometric Identities – Double Angle Formulas

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Presentation transcript:

T.3.3 – Trigonometric Identities – Double Angle Formulas IB Math SL - Santowski

Fast Five Graph the following functions and develop an alternative equation for the graphed function (i.e. Develop an identity for the given functions) f(x) = 2sin(x)cos(x) g(x) = cos2(x) – sin2(x)

Fast Five

(A) Intro To continue working with trig identities, we will introduce the DOUBLE ANGLE formulas WITHOUT any attempt at a proof. To see a proof based upon the SUM & DIFFERENCE formulas, see: (a) The Math Page To see videos on DOUBLE ANGLE FORMULAS: (a) http://www.youtube.com/watch?v=VzKUIr-s5O0&feature=related (b) http://www.youtube.com/watch?v=FAtQrZegxfI&feature=related (c) http://www.youtube.com/watch?v=B8psoIRChkM&feature=related

(B) Double Angle Formulas sin(2x) = 2 sin(x) cos(x) cos(2x) = cos2(x) – sin2(x)

(B) Double Angle Formulas Working with cos(2x) = cos2(x) – sin2(x) But recall the Pythagorean Identity where sin2(x) + cos2(x) = 1 So sin2(x) = 1 – cos2(x) And cos2(x) = 1 – sin2(x) So cos(2x) = cos2(x) – (1 – cos2(x)) = 2cos2(x) - 1 And cos(2x) = (1 – sin2(x)) – sin2(x) = 1 – 2sin2(x)

(B) Double Angle Formulas Working with cos(2x) cos(2x) = cos2(x) – sin2(x) cos(2x) = 2cos2(x) - 1 cos(2x) = 1 – 2sin2(x)

(B) Double Angle Formulas cos(2x) = 2cos2(x) – 1 So we can get into “power reducing” formulas for cos2x  ½(1 + cos(2x)) cos(2x) = 1 – 2sin2(x) So we can get into “power reducing” formulas for sin2x  ½(1 - cos(2x))

(C) Double Angle Identities - Applications (A) Simplifying applications (B) Power reducing applications (C) Trig equations applications

(C) Double Angle Identities - Applications (a) Determine the value of sin(2x) and cos(2x) if (b) Determine the values of sin(2x) and cos(2x) if (c) Determine the values of sin(2x) and cos(2x) if

(C) Double Angle Identities - Applications Simplify the following expressions:

(C) Double Angle Identities - Applications Write as a single function:

(C) Double Angle Identities - Applications Use the double angle identities to prove that the following equations are identities:

(C) Double Angle Identities - Applications Solve the following equations, making use of the double angle formulas for key substitutions:

(C) Double Angle Identities - Applications Use the idea of “power reduction” to rewrite an equivalent expression for:

Homework HH Text EX 13J, Q1-8, page 293