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The inverse trigonometric functions The reciprocal trigonometric functions Trigonometric identities Examination-style question Contents © Boardworks Ltd of 35 The inverse trigonometric functions

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The inverse of the sine function Suppose we wish to find θ such that In other words, we want to find the angle whose sine is x. In this context, sin –1 x means the inverse of sin x. sin θ = x θ = sin –1 x or θ = arcsin x This is not the same as (sin x ) –1 which is the reciprocal of sin x,. Is y = sin –1 x a function? This is written as

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The inverse of the sine function The inverse of this graph is not a function because it is one-to-many: We can see from the graph of y = sin x between x = –2 π and x = 2 π that it is a many- to-one function: y x y = sin x y = sin –1 x x y

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The inverse of the sine function There is only one value of sin –1 x in this range, called the principal value. However, remember that if we use a calculator to find sin –1 x (or arcsin x ) the calculator will give a value between –90° and 90° (or between – x if working in radians). So, if we restrict the domain of f ( x ) = sin x to – x we have a one-to-one function: y 1 –1 x y = sin x

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x y 1–1 y = sin –1 x 1–1 y = sin –1 x The graph of y = sin –1 x The graph of y = sin –1 x is the reflection of y = sin x in the line y = x : The domain of sin –1 x is the same as the range of sin x : The range of sin –1 x is the same as the restricted domain of sin x : –1 x 1 1 –1 x y – sin –1 x (Remember the scale used on the x - and y -axes must be the same.) Therefore the inverse of f ( x ) = sin x, – x, is also a one-to-one function: f –1 ( x ) = sin –1 x

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The inverse of cosine and tangent We can restrict the domains of cos x and tan x in the same way as we did for sin x so that if f ( x ) = cos x for0 x π – < x

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y = cos x x 0 y 1–1 y = cos –1 x The graph of y = cos –1 x The domain of cos –1 x is the same as the range of cos x : The range of cos –1 x is the same as the restricted domain of cos x : 0 cos –1 x π –1 x 1 –1 x 0 1 y 1 y = cos –1 x

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y = tan x x y y = tan –1 x x The graph of y = tan –1 x The domain of tan –1 x is the same as the range of tan x : The range of tan –1 x is the same as the restricted domain of tan x : y x – < tan –1 x <

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Find the exact value of sin –1 in radians. Problems involving inverse trig functions To solve this, remember the angles whose trigonometric ratios can be written exactly: tan cos sin 90°60°45°30°0°0°degrees 0radians From this table sin –1 =

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Find the exact value of sin –1 in radians. Problems involving inverse trig functions This is equivalent to solving the trigonometric equation cos θ = – for 0 θ π this is the range of cos –1 x We know that cos = Sketching y = cos θ for 0 θ π : –1 θ 0 1 From the graph,cos =– = So, cos –1 =

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Find the exact value of cos (sin –1 ) in radians. Problems involving inverse trig functions Let sin –1 = θ so sin θ = Using the following right-angled triangle: θ 4 The length of the third side is 3 so a 2 = 16 cos θ = But sin –1 = θ socos (sin –1 ) = a = 3

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