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# MAT110 Chapter 8 Study Guide MAT 110

Barry University

GPA 3.7

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This 8 page Study Guide was uploaded by Sterling Notetaker on Friday September 9, 2016. The Study Guide belongs to MAT 110 at Barry University taught by Dr. Singh in Fall 2016. Since its upload, it has received 11 views. For similar materials see Precalculus Mathematics 2 in Mathmatics at Barry University.

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Date Created: 09/09/16

MAT 110 Precalculus Mathematics 2 Chapter 8 Study Guide Notes L. Sterling September 15th, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 Double-Angle Formulas 1.1 sin(2 ▯) sin(2 ▯) = 2 sin ▯ cos ▯ 1.2 cos(2 ▯) 2 2 cos(2 ▯) = cos ▯ ▯ sin ▯ 2 cos(2 ▯) = 2 cos ▯ ▯ 1 2 cos(2 ▯) = 1 ▯ 2 sin ▯ 1 1.3 tan(2 ▯) tan(2 ▯) = 2 tan ▯ 1 ▯ tan ▯ 2 Half Angle Formulas 2 2.1 sin ▯ sin ▯ = 1(1 ▯ cos(2▯)) = ▯ cos(2▯) 2 2 2.2 cos ▯ 1 1 + cos(2▯) cos ▯ = (1 + cos(2▯)) = 2 2 2.3 tan ▯ sin ▯ 1▯cos(2▯) 1 ▯ cos(2▯) tan ▯ = 2 = 1+cos(2▯) cos ▯ 2 1 + cos(2▯) ▯ ▯ ▯ 3 Half Angle Formulas [for tan 2 ] ▯ ▯ ▯ 1 ▯ cos ▯ sin ▯ tan = = 2 sin ▯ 1 + cos ▯ 4 Inverse Trigonometric Functions and Proper- ties 4.1 Inverse Sine Function y = sin▯1(x) ! x = sin(y) ▯1 ▯ x ▯ 1 ▯ ▯ ▯ 2 ▯ y ▯ 2 2 ▯1 ▯ ▯ y = sin (x);▯1 ▯ x ▯ 1;▯ ▯ y ▯ 2 2 4.2 Properties of Inverse Sine Functions f ▯1(f (x)) = sin1(sin(x)) = x ▯ ▯ ▯ 2 ▯ x ▯ 2 ▯ ▯ ▯ ▯ f f ▯1 (x) = sin sin▯1(x) = x ▯1 ▯ x ▯ 1 4.3 Inverse Cosine Function y = cos▯1(x) ! x = cos(y) ▯1 ▯ x ▯ 1 0 ▯ y ▯ ▯ ▯1 y = cos (x);▯1 ▯ x ▯ 1;0 ▯ y ▯ ▯ 4.4 Properties of Inverse Cosine Functions ▯1 ▯1 f (f (x)) = cos (cos(x)) = x 0 ▯ x ▯ ▯ ▯ ▯ ▯ ▯ f f ▯1 (x) = cos cos▯1(x) = x ▯1 ▯ x ▯ 1 3 4.5 Inverse Tangent Function y = cos▯1 (x) ! x = cos(y) ▯1 < x < 1 ▯ ▯ ▯ ▯ x ▯ 2 2 ▯1 ▯ ▯ y = cos (x);▯1 < x < 1;▯ 2 ▯ x ▯ 2 4.6 Properties of Inverse Tangent Functions ▯1 ▯1 f (f (x)) = tan(tan(x)) = x ▯ ▯ ▯ ▯ x ▯ 2 2 ▯ ▯1 ▯ ▯ ▯1 ▯ f f (x) = tan tan (x) = x ▯1 < x < 1 4.7 Inverse Secant Function y = sec▯1 (x) ! x = sec(y) jxj ▯ 1 0 ▯ y ▯ ▯ y 6=▯ 2 ▯ y = sec▯1(x);jxj ▯ 1;0 ▯ y ▯ ▯;y 6= 2 4 4.8 Properties of Inverse Secant Functions f ▯1(f (x)) = sec1(sec(x)) = x 0 ▯ y ▯ ▯ ▯ y 6= 2 ▯ ▯1 ▯ ▯ ▯1 ▯ f f (x) = sec sec (x) = x jxj ▯ 1 4.9 Inverse Cosecant Function y = csc▯1(x) ! x = csc(y) jxj ▯ 1 ▯ ▯ ▯ ▯ y ▯ 2 2 y 6= 0 ▯1 ▯ ▯ y = csc (x);jxj ▯ 1;▯ ▯ y ▯ ;y 6= 0 2 2 4.10 Properties of Inverse Cosecant Functions f ▯1 (f (x)) = csc(csc(x)) = x ▯ ▯ ▯ y ▯ ▯ 2 2 y 6= 0 5 ▯ ▯ ▯ ▯ f f ▯1(x) = csc csc▯1 (x) = x jxj ▯ 1 4.11 Inverse Cotangent Function y = cot1 (x) ! x = cot(y) ▯1 < x < 1 ▯ ▯ ▯ ▯ x ▯ 2 2 ▯1 ▯ ▯ y = cot (x);▯1 < x < 1;▯ 2 ▯ x ▯ 2 4.12 Properties of Inverse Cotangent Functions ▯1 ▯1 f (f (x)) = cot (cot(x)) = x 0 ▯ y ▯ ▯ ▯ ▯ ▯ ▯ f f ▯1(x) = cot cot▯1(x) = x ▯1 < x < 1 5 Product-to-Sum Formulas 5.1 sin ▯ sin ▯ 1 cos(▯ ▯ ▯ ) ▯ cos(▯ + ▯ ) sin ▯ sin ▯ = [cos(▯ ▯ ▯) ▯ cos(▯ + ▯)] = 2 2 6 5.2 cos ▯ cos ▯ 1 cos(▯ ▯ ▯) + cos(▯ + ▯) cos ▯ cos ▯ = [cos(▯ ▯ ▯) + cos(▯ + ▯)] = 2 2 5.3 sin ▯ cos ▯ 1 sin(▯ + ▯ ) + sin(▯ ▯ ▯ ) sin ▯ cos ▯ = [sin(▯ + ▯) + sin(▯ ▯ ▯)] = 2 2 5.4 cos ▯ sin ▯ 1 sin(▯ + ▯ ) ▯ sin(▯ ▯ ▯ ) cos ▯ sin ▯ = [sin(▯ + ▯) ▯ sin(▯ ▯ ▯)] = 2 2 6 Squared Half Angle Formulas 2 ▯▯▯ 6.1 sin 2 2 ▯▯▯ 1 ▯ cos ▯ sin = 2 2 ▯▯▯ 6.2 cos2 2 2 ▯▯▯ 1 + cos ▯ cos = 2 2 ▯ ▯ 6.3 tan 2 ▯ 2 ▯ ▯ 2▯▯ ▯ 1▯cos ▯ tan2 ▯ = sin ▯ 2▯ = 2 = 1 ▯ cos ▯ 2 cos2 ▯ 1+cos ▯ 1 + cos ▯ 2 2 7 Sum and Di▯erence Formulas 7.1 Sum and Di▯erence Formula for Sines sin(▯ + ▯) = sin ▯ cos ▯ + cos ▯ sin ▯ sin(▯ ▯ ▯) = sin ▯ cos ▯ ▯ cos ▯ sin ▯ 7 7.2 Sum and Di▯erence Formula for Cosines cos(▯ + ▯) = cos ▯ cos ▯ ▯ sin ▯ sin ▯ cos(▯ ▯ ▯) = cos ▯ cos ▯ + sin ▯ sin ▯ 7.3 Sum and Di▯erence Formula for Tangents tan ▯ + tan ▯ tan(▯ + ▯) =1 ▯ tan ▯ ▯ tan ▯ tan ▯ ▯ tan ▯ tan(▯ ▯ ▯) = 1 + tan ▯ ▯ tan ▯ 8 Sum-to-Product Formulas 8.1 sin ▯ + sin ▯ ▯ ▯ + ▯▯ ▯ ▯ ▯ ▯ ▯ sin ▯ + sin ▯ = 2 sin cos 2 2 8.2 sin ▯ ▯ sin ▯ ▯ ▯ ▯ ▯ ▯ + ▯ ▯ ▯ ▯ sin ▯ ▯ sin ▯ = 2 cos2 sin 2 8.3 cos ▯ + cos ▯ ▯ ▯ ▯ ▯ cos ▯ + cos ▯ = 2 cos+ ▯ cos ▯ ▯ ▯ 2 2 8.4 cos ▯ ▯ cos ▯ ▯ ▯ ▯ ▯ ▯ + ▯ ▯ ▯ ▯ cos ▯ ▯ cos ▯ = ▯2 sin sin 2 2 8

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