Download Midterm 2 Review Sheet: Trigonometry Identities and Equations and more Exams Mathematics in PDF only on Docsity! Midterm 2 Review Sheet Addition/Subtraction identities for sin and cos: You are supposed to know the addition/subtraction identities of sin(x±y) and cos(x±y). How to express sin 2x and cos 2x in terms of sin x and cos x? How to express sin 4x in terms of sinx and cos x? If sin x = 1 3 and x is in quadrant II, how to find the exact value of sin 4x. The addition/subtraction identities for tan: tan(x + y) = tanx + tan y 1− tanx tan y How to express tan(x + π4 ) in terms of tanx? Double Angle Identities: If sinx = .5 and x is in quadrant I, find the exact value of sin 2x (You need to find the value of cos x). If sinx = .5 (and we don’t know which quadrant x is in), find the exact value of cos 2x. (As cos 2x = 1− 2 sin2 x, we can find the exact value of cos 2x even if we cannot decide the value of cos x! cos x might be negative or positive) Half Angle Identities Given the value of cos x, we may find the value of sin x 2 and cos x 2 (don’t forget the plus/minus!), thus we may find the value of tan x 2 , which is tan x 2 = ± √ 1− cos x 1 + cos x If we know the value of both sinx and cos x, we can find the exact value of tan x 2 (without plus/minus!), that is tan x 2 = 1− cos x sinx 1 If we know sin x = 1 3 and x is in quadrant II, find the exact value of tan x 2 (without plus/minus!) Product and Factoring Identities Nothing tricky, but you need to memorize the boxed identities in textbook (page 515). How to prove sin t + sin 5t cos t + cos 5t = tan 3t? Inverse Trigonometric Functions: The domain/range of sin−1, cos−1 and tan−1. sin(cos−1 x) = cos(sin−1 x) = √ 1− x2. sin sin−1 x = x and cos cos−1 x = x, but it is NOT always true that sin−1(sinx) = x and cos−1(cos x) = x, why? How to find the exact value of cos(tan−1 3)? (let x = tan−1 3, then tanx = 3, as 1 + tan2 x = sec2 x, we know sec x = ± √ 10, as −π 2 < x < π 2 , and tanx is positive, we may claim that x is in quadrant I, thus sec x = √ 10, and cos x = 1 sec x = 1√ 10 ). Trigonometric Equations. How to solve sin x + cos 2x = 1? How to solve sec x = 2? How to solve sin x + sin2 x = 0? Complex Numbers, Polar Forms Arithmetic of complex numbers. Express complex numbers in polar form. For example, what is the polar form of z = −1− i? Polar form can simplify multiplication/division of complex numbers. For example, what is the value of (−1− i)40? Except for the topics/problems listed above, you are also supposed to be able to do the following problems. 1) Solve cos x + cos 2x = 1. 2) If z4 = i (z is complex number), find all the possible values of z. 3) If sinx = 1 4 and x is in quadrant I, find the exact value of sin 4x. 4) Find the exact value of tan(sin−1 1 3 ). 2
