Honors PreCalc Ch 5

Sinø = y Cosø = x Tanø = y/x Cscø = 1/y Secø = 1/x Cotø = x/y

Fundamental Identity of all 6 trig functions

Sin(x-y) = sin(x)cos(y) - sin(y)cos(x)

Sine difference identity

Sin(x+y) = sin(x)cos(y) + sin(y)cos(x)

Sine sum identity

Tan(x-y) = tan(x) - tan(y) / 1 + tan(x)tan(y)

Tangent difference identity

Quadrant in which x/2 lies

What determines the sign for the half angle formulas

Sin(x)sin(y) = 1/2[cos(x-y) - cos(x+y)] Cos(x)cos(y) = 1/2[cos(x-y) + cos(x+y)] Sin(x)cos(y) = 1/2[sin(x+y) + sin(x-y)] Cos(x)sin(y) = 1/2[sin(x+y) - sin(x-y)]

4 Product to Sum Formulas

Cos(-ø) = cosø Sec(-ø) = secø

2 Even Identities of trig functions

Sin(2x) = 2sin(x)cos(x) Cos(2x) = cos^2(x) - sin^2(x) = 1 - 2sin^2(x) = 2cos^2(x) - 1 Tan(2x) = 2tan(x) / 1 - tan^2(x)

3 Double Angle Formulas

Sin(x/2) = +/- <1 - cos(x) / 2> Cos(x/2) = +/- <1 + cos(x) / 2> Tan(x/2) = 1 - cos(x) / sin(x) = sin(x) / 1 + cos(x)

3 Half Angle Formulas

Sin^2(x) = 1 - cos(2x) / 2 Cos^2(x) = 1 + cos(2x) / 2 Tan^2(x) = 1 - cos(2x) / 1 + cos(2x)

3 Power Reducing Formulas

Sin(-ø) = -sinø Csc(-ø) = -cscø Tan(-ø) = -tanø Cot(-ø) = -cotø

4 Odd Identities of trig functions

Sin(x) + sin(y) = 2sin(x+y/2)cos(x-y/2) Sin(x) - sin(y) = 2sin(x-y/2)cos(x+y/2) Cos(x) + cos(y) = 2cos(x+y/2)cos(x-y/2) Cos(x) - cos(y) = -2sin(x+y/2)sin(x-y/2)

4 Sum to Product Formulas

1) Use basic identities to simplify trigonometric functions 2) Factor if necessary 3) Get trigonometric functions on one side and numerical values on the other 4) Solve for x using the Unit Circle

4 steps for solving Trigonometric Equations

Tanø = cot(pi/2 - ø) Cotø = tan(pi/2 - ø) Secø = csc(pi/2 - ø) Cscø = sec(pi/2 - ø) Sinø = cos(pi/2 - ø) Cosø = sin(pi/2 - ø)

Cofunction Identity of all 6 trig functions

Cos(x-y) = cos(x)cos(y) + sin(y)sin(x)

Cosine difference identity

Cos(x+y) = cos(x)cos(y) - sin(y)sin(x)

Cosine sum identity

Sin^2ø + cos^2ø = 1 Tan^2ø + 1 = sec^2ø 1 + cot^2ø = csc^2ø

Pythagorean Identity of all 6 trig functions (3)

Tanø = sinø/cosø Cotø = cosø/sinø

Quotient Identity of all 6 trig functions (2)

Sinø = 1/cscø Cosø = 1/secø Tanø = 1/cotø Cscø = 1/sinø Secø = 1/cosø Cotø = 1/tanø

Reciprocal Identity of all 6 trig functions

Tan(x+y) = tan(x) + tan(y) / 1 - tan(x)tan(y)

Tangent sum identity

0 (1, 0), pi/6 (<3>/2, 1/2), pi/4 (<2>/2, <2>/2), pi/3 (1/2, <3>/2), pi/2 (0, 1), 2pi/3 (-1/2, <3>/2), 3pi/4 (-<2>/2, <2>/2), 5pi/6 (-<3>/2, 1/2), pi (-1, 0), 7pi/6 (-<3>/2, -1/2), 5pi/4 (-<2>/2, -<2>/2), 4pi/3 (-1/2, -<3>/2), 3pi/2 (0, -1), 5pi/3 (1/2, -<3>/2), 7pi/4 (<2>/2, -<2>/2), 11pi/6 (<3>/2, -1/2)

Unit Circle from 0 to 11pi/6 w ordered pairs

pin

What to add to the solution of a trig equation when it could refer to more than one angle

2pin

What to add to the solution of a trig equation when it is only one specific angle and there are no restrictions