M259 (7) - Analytic Trigonometry

Objective 7.7

- Express products as sums - Express sums as products

Objective 7.4

- Use algebra to simplify trig expressions - Establish identities

y = sin^-1 x (arcsin x)

x = sin y given D: [-1, 1] R: [-π/2, π/2]

Sum and Difference formulas (6)

- Cos (α + β) = cos α cos β - sin α sin β - Cos (α - β) = cos α cos β + sin α sin β - Sin (α + β) = sin α cos β + cos α sin β - Sin (α - β) = sin α cos β - sin α cos β - Tan (α + β) = (tan α + tan β)/(1 - tan α tan β) - Tan (α - β) = (tan α - tan β)/(1 + tan α tan β)

Objective 7.1

- Find the exact value of an inverse sine, cosine, or tangent function - Find the approximate value of an inverse sine function - Use properties of an inverse function to find the exact values of composite functions - Find the inverse function of a trigonometric function - Solve equations involving inverse trigonometric functions

Objective 7.2

- Find the exact value of the expressions involving the inverse, sine, cosine and tangent functions - Define the inverse sec, csc, cot functions - Use a calculator to evaluate arcsec x, arccsc x, arccot x - Write a trigonometric expression as an algebraic function

Double-Angle Identities

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

Sum-to-Product Formulas

- Sin α + sin β = 2 sin (α + β)/2 cos (α - β)/2 - Cos α + cos β = 2 cos (α + β)/2 cos (α - β)/2 - Sin α - sin β = 2 sin (α - β)/2 cos (α + β)/2 - Cos α - cos β = -2 sin (α + β)/2 sin (α - β)/2

Product-to-Sum Formulas

- Sin α sin β = 1/2 [cos (α - β) - cos (α + β)] - Cos α - cos β = 1/2 [cos (α - β) + cos (α + β)] - Sin α - cos β = 1/2 [cos (α + β) + cos (α + β)]

Half- Angle Formulas

- Sin^2 α/2 = (1 - cos α)/2 - Sin α/2 = ± ((1 - cos α)/2)^1/2 - Cos^2 α/2 = (1- cos α)/2 - Cos α/2 = ± ((1- cos α)/2)^1/2 - Tan^2 α/2 = (1 - cos α)/(1 + cos α) - Tan α/2 = ± ((1 - cos α)/(1 + cos α))^1/2 = (1 - cos α)/sin α = sin α/(1 + cos α) * + or - is determined by quadrant of α/2

Objective 7.3

- Solve equations involving a single trig function - Solve trig equations using a calculator - Solve trig equations in quadratic form - Solve trig equations using fundamental identities - Solve trig equations using a graphing utility

Objective 7.6

- Use double-angle formulas to find exact values - Use double-angle formulas to establish identities - Use half-angle formulas to find exact values

Objective 7.5

- Use the sum and difference formulas to find each values - Use sum and difference formulas to establish identities - Use sum and difference formulas involving inverse trig functions - Solve linear trig equations in sine and cosine

Inverse trigonometric functions

- law postulates inverse functions must be of 1-1 ratio; since trigonometric functions are periodic their inverse functions can only be defined for a monotone increase/ decrease (fundamental period) period/range

y = cos^-1 x (arccos x)

x = cos y given D: [-1, 1] R: [0, π]

y = cot^-1 x (arccot x)

x = cot y given D: (-∞, ∞) R: (-1/2π, 0] or [0, 1/2π)

y = csc^-1 x (arccsc x)

x = csc y given D: (-∞, ∞) R: [-1/2π, 0)U(0,1/2π]

y = sec^-1 x (arcsec x)

x = sec y given D: (-∞, ∞) R: [0, 1/2π) or (1/2π, π]

y = tan^-1 x (arctan x)

x = tan y given D: (-∞, ∞) R: (-1/2π, 1/2π)