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Inverse Trig Functions
Lesson 3.5
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Start with Sine Function
x y = sin(x) 0.0000 0.5236 0.5000 1.0472 0.8660 1.5708 1.0000 2.0944 2.6180 3.1416 3.6652 4.1888 4.7124 5.2360 5.7596 6.2832 Given y = sin (x) Table of values Graph What if we reversed the ordered pairs … y for x ?
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Reversed Ordered Pairs
x y 0.0000 0.5000 0.5236 0.8660 1.0472 1.0000 1.5708 2.0944 2.6180 3.1416 3.6652 4.1888 4.7124 5.2360 5.7596 6.2832 Problem This is not a function Fails the vertical line test There are multiple (x,y)'s where x = .5 Solution Limit the range
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The Inverse Trig Function
We say Similarly for inverse cosine The range of cos-1x is limited differently Note pg 258 for domain, range of other functions
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Evaluating Inverse Functions
Consider cos-1(-0.5) We are asking what angle has a cosine value of -0.5 Cosine negative in quadrants 2 and 3 But for cos-1(x) we look only in 1 & 2 Calculator also capable of evaluating inverse trig functions 2 -1
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Note: newer calculators will have these functions – find in Catalog
Try It Out Consider these Note: newer calculators will have these functions – find in Catalog
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Composition of Trig Functions and Inverses
Recall that in general f-1(f(x)) = f(f-1(x)) = x For trig functions this is the same sin(arcsin(x)) = arcsin(sin(x)) The restriction on the domain and range of the inverse functions must apply Thus sin-1(sin(3)) could not be 3 Note calculator results
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Composition of Trig Functions and Inverses
Try these … with and without calculator
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Solving Inverse Trig Equations
Given Strategy Isolate the sin-1x Take the sine of both sides of the equation
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Try it Out Try this one
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Assignment Lesson 3.5 Page 265 Exercises 1 – 65 EOO
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