Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition Assume that: 0 < θ < or 0 < θ < 90 Unit Circle Definition Assume θ can be any angle. y x, y hypotenuse opposite θ x y x θ adjacent sin θ = opp hyp csc θ = hyp opp sin θ = y csc θ = y cos θ = adj hyp sec θ = hyp adj cos θ = x sec θ = x tan θ = opp adj cot θ = adj opp tan θ = y x cot θ = x y Domains of the Trig Functions sin θ, θ, cos θ, θ, tan θ, θ n +, where n Z csc θ, sec θ, cot θ, θ n, where n Z θ n +, where n Z θ n, where n Z Ranges of the Trig Functions sin θ cos θ tan θ csc θ and csc θ sec θ and sec θ cot θ Periods of the Trig Functions The period of a function is the number, T, such that f θ +T = f θ. So, if ω is a fixed number and θ is any angle we have the following periods. sinωθ T = ω cscωθ T = ω cosωθ T = ω secωθ T = ω tanωθ T = ω cotωθ T = ω
Identities and Formulas Tangent and Cotangent Identities tan θ = sin θ cos θ Reciprocal Identities sin θ = csc θ cos θ = sec θ tan θ = cot θ cot θ = cos θ sin θ csc θ = sin θ sec θ = cos θ cot θ = tan θ Pythagorean Identities sin θ + cos θ = tan θ + = sec θ + cot θ = csc θ Even and Odd Formulas sin θ = sin θ cos θ = cos θ tan θ = tan θ Periodic Formulas If n is an integer sinθ + n = sin θ cosθ + n = cos θ tanθ + n = tan θ Double Angle Formulas sinθ = sin θ cos θ cosθ = cos θ sin θ = cos θ = sin θ tanθ = tan θ tan θ csc θ = csc θ sec θ = sec θ cot θ = cot θ cscθ + n = csc θ secθ + n = sec θ cotθ + n = cot θ Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then: 80 = t x t = x and x = 80 t 80 Half Angle Formulas cosθ sin θ = ± + cosθ cos θ = ± cosθ tan θ = ± + cosθ Sum and Difference Formulas sinα ± β = sin α cos β ± cos α sin β cosα ± β = cos α cos β sin α sin β tanα ± β = tan α ± tan β tan α tan β Product to Sum Formulas sin α sin β = [cosα β cosα + β] cos α cos β = [cosα β + cosα + β] sin α cos β = [sinα + β + sinα β] cos α sin β = [sinα + β sinα β] Sum to Product Formulas α + β α β sin α + sin β = sin cos α + β α β sin α sin β = cos sin α + β α β cos α + cos β = cos cos α + β α β cos α cos β = sin sin Cofunction Formulas sin θ = cos θ csc θ = sec θ tan θ = cot θ cos θ = sin θ sec θ = csc θ cot θ = tan θ
Unit Circle 0,, 90,,,, 0, 0,, 5, 5,, 50, 5 0,, 0 80, 0,, 0 0, 7, 5, 5 5, 7 0,,, 0, 00, 5,, 70,, 0, F or any ordered pair on the unit circle x, y : cos θ = x and sin θ = y Example cos 7 = sin 7 =
Inverse Trig Functions Definition θ = sin x is equivalent to x = sin θ Inverse Properties These properties hold for x in the domain and θ in the range θ = cos x is equivalent to x = cos θ θ = tan x is equivalent to x = tan θ Domain and Range sinsin x = x coscos x = x tantan x = x sin sinθ = θ cos cosθ = θ tan tanθ = θ Function θ = sin x θ = cos x θ = tan x Domain x x x Range θ 0 θ < θ < Other Notations sin x = arcsinx cos x = arccosx tan x = arctanx Law of Sines, Cosines, and Tangents β a c γ α b sin α a Law of Sines = sin β b = sin γ c Law of Cosines a = b + c bc cos α b = a + c ac cos β c = a + b ab cos γ Law of Tangents a b a + b = tan α β tan α + β b c b + c = tan β γ tan β + γ a c a + c = tan α γ tan α + γ
Complex Numbers i = i = i = i i = a = i a, a 0 a + bi + c + di = a + c + b + di a + bi c + di = a c + b di a + bic + di = ac bd + ad + bci a + bia bi = a + b a + bi = a + b Complex Modulus a + bi = a bi Complex Conjugate a + bia + bi = a + bi Example: Let z = i, find z. DeMoivre s Theorem Let z = rcos θ θ, and let n be a positive integer. Then: z n = r n cos nθ nθ. Solution: First write z in polar form. r = + = θ = argz = tan = Polar Form: z = cos Applying DeMoivre s Theorem gives : z = cos = cos = 80 + i = 8i 5
Finding the nth roots of a number using DeMoivre s Theorem Example: Find all the complex fourth roots of. x =. That is, find all the complex solutions of r n We are asked to find all complex fourth roots of. These are all the solutions including the complex values of the equation x =. For any positive integer n, a nonzero complex number z has exactly n distinct nth roots. More specifically, if z is written in the trigonometric form rcos θ θ, the nth roots of z are given by the following formula. cos θ n + 0 k n θ n + 0 k n, for k = 0,,,..., n. Remember from the previous example we need to write in trigonometric form by using: r = b a + b and θ = argz = tan. a So we have the complex number a + ib = + i0. Therefore a = and b = 0 So r = + 0 = and 0 θ = argz = tan = 0 Finally our trigonometric form is = cos 0 0 Using the formula above with n =, we can find the fourth roots of cos 0 0 For k = 0, 0 cos + 0 0 0 + 0 0 = cos0 0 = For k =, 0 cos + 0 0 + 0 = cos90 90 = i For k =, 0 cos + 0 0 + 0 = cos80 80 = For k =, 0 cos + 0 0 + 0 = cos70 70 = i Thus all of the complex roots of x = are:, i,, i.
Formulas for the Conic Sections Circle StandardF orm : x h + y k = r W here h, k = center and r = radius Ellipse Standard F orm for Horizontal Major Axis : x h y k + = a b Standard F orm for V ertical Major Axis : x h y k + = b a Where h, k= center a=length of major axis b=length of minor axis 0 < b < a Foci can be found by using c = a b Where c= foci length 7
More Conic Sections Hyperbola Standard F orm for Horizontal T ransverse Axis : x h y k = a b Standard F orm for V ertical T ransverse Axis : y k x h = a b Where h, k= center a=distance between center and either vertex Foci can be found by using b = c a Where c is the distance between center and either focus. b > 0 Parabola Vertical axis: y = ax h + k Horizontal axis: x = ay k + h Where h, k= vertex a=scaling factor 8
fx fx = sinx x 0 5 7 5 5 7 - Example : sin 5 = fx fx = cosx x 0 5 7 5 5 7 - Example : cos 7 = 9
fx fx = tan x 5 0 5 x 0