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Understanding Inverse Trigonometric Functions: Domains, Ranges, and Graphs

This homework assignment focuses on the exploration of inverse trigonometric functions, including sine, cosine, and tangent. Students will graph these functions and determine their domains and ranges. Key concepts covered include how to identify if a function has an inverse, the role of the horizontal line test (HLT), and memorization quizzes. We will solve specific trigonometric equations, analyze functions like y = 3.61sin(x + 0.59), and discuss properties critical for mastering inverses in trigonometry.

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Understanding Inverse Trigonometric Functions: Domains, Ranges, and Graphs

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  1. Pg. 385 Homework • Pg. 395 #13 – 41 odd, graph the three inverse trig functions and label the domain and range of each.Memorization quiz through inverse trig functions on Thursday!! • #43 y = 3.61 sin (x + 0.59) • #44 y = -5.83 sin (x + 2.6) • #46 y = 5 sin (2x – 0.64) • #51 D: (∞, ∞); R: [-5.39, 5.39]; P: 2π; max (0.38, 5.39); min (3.52, -5.39) • #9 Graph • #10 Graph • #35 • #36 No Solution!! • #37 x = ±2.66 + 4kπ, where k is any integer • #38 (-3.98, -3.75)U(-1.39, 0)U(1.39, 3.75)U(3.98, ∞)

  2. 7.2 Inverse Trigonometric Functions Inverse Functions Inverse sin x Consider y = sin x on the interval [-π/2, π/2]. Will it pass the HLT? Will it have an inverse? An inverse can be defined as long as the domain of the original function lends itself to an inverse. • What is an inverse? • How can you tell it is an inverse both algebraically and graphically? • Will trig functions have an inverse?

  3. 7.2 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Functions The inverse cosine function, denoted y = cos-1 x or y = arccosx is the function with a domain of [-1, 1] and a range of [0, π] that satisfies the relation cosy = x. If f(x) = cosx and f-1(x) = cos-1 x(f-1 ◦ f)(x) = x on [0, π] and(f ◦ f-1)(x) = x on [-1, 1] • The inverse sine function, denoted y = sin-1 x or y = arcsinx is the function with a domain of [-1, 1] and a range of [-π/2, π/2] that satisfies the relation sin y = x. • If f(x) = sin x and f-1(x) = sin-1 x(f-1 ◦ f)(x) = x on [-π/2, π/2] and(f ◦ f-1)(x) = x on [-1, 1]

  4. 7.2 Inverse Trigonometric Functions Inverse Tangent Function Finding the Domain and Range f(x) = sin-1 (2x) g(x) = sin-1 (⅓ x) • The inverse tangent function, denoted y = tan-1 x or y = arctanx is the function with a domain of (-∞, ∞) and a range of (-π/2, π/2) that satisfies the relation tan y = x. • If f(x) = tan x and f-1(x) = tan-1 x(f-1 ◦ f)(x) = x on (-π/2, π/2) and(f ◦ f-1)(x) = x on (-∞, ∞)

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