3.5 Inverse Trigonometric Functions
When we studied inverse functions, we learned that in order to have an inverse, a function must be one­to­one, which means its graph should pass the horizontal line test. Clearly, the graphs of sine and cosine do not. (Nor do the graphs of tangent, cotangent, secant, or cosecant.) However, if we restrict their domains, we can get them to be. Consider the following:
y = tan x
y = sin x
y = cos x
We can get one­to­one functions if we restrict the domain of sine to [­π/2, π/2], the domain of cosine to [0, π], and the domain of tangent to (­π/2, π/2). The range for sine and cosine is [­1, 1], and for tangent the range is (­∞, ∞).
So when we are taking the inverse trig function, our input will be a real number (between ­1 and 1 for sine and cosine), and the output will be an angle in the restricted domain!
i.e. sin­1x = y sin (y) = x; cos­1x = y cos (y) = x;
tan­1x = y tan (y) = x
tan­1(1) = π/4 because tan (π/4) = 1
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Alternate notation for inverse trig functions:
sin­1(x) = arcsin (x) cos­1(x) = arccos (x) tan­1(x) = arctan (x) csc­1(x) = arccsc (x) sec­1(x) = arcsec (x) cot­1(x) = arccot (x) We can find the exact values of some expressions by creating a right triangle.
Ex: 1) tan(sin­1(1/3))
2) sin(arccos(7/10))
Note that we have the following cancellation equations for inverse functions:
sin­1(sin x) = x for x∈ [­π/2, π/2] sin(sin­1(x)) = x for x ∈ [­1, 1]
cos­1(cos x) = x for x∈ [0, π]
cos(cos­1(x)) = x for x ∈
[­1, 1]
tan­1(tan x) = x for x∈ (­π/2, π/2) tan(tan­1(x)) = x for all x
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Evaluate each of the following expressions.
1) arcsin(1/2)
2) arccos(0)
3) sin(sin­1(3/5))
4) tan­1(tan(π/3))
5) arcsin(sin(3π))
6) cos(arctan(7))
Simplify each of the following expressions. Hint: Draw a right triangle.
1) tan(sin­1(x))
2) cos(tan­1(x))
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The inverse trig functions are all differentiable, and their derivatives are as follows:
Naturally, any of these can be combined with the Chain Rule. For example, if u is a function of x, then Ex: Find y' for each of the following:
y = sin­1(8x5)
y = cos­1(ln x)
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y = tan­1(x2)
y = sin­1(ex) + cot­1(3x)
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Ex: Find y' for each of the following:
y = 9x2 sin­1(sin (x))
Ex: Find the derivative of the function below and state the domain of both the function and its derivative.
y = arcsin (5x + 3)
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Ex: Find the equation of the tangent line to the curve y = arccos (4x) ­ 3
at the point (0, π/2­3).
Find the equation of the tangent line to the curve y = 6sin­1(x/2)
at the point (1, π).
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