Inverse Trigonometric Functions - Other properties of inverse trig functions
Recap of Inverse Trigonometric Functions
Definition: Inverse function of a trigonometric function
Range of inverse trig functions
Domain and Range of Inverse Trigonometric Functions
Domain: Range of corresponding trig functions
Range: Restricted domains of inverse trig functions
Basic Properties of Inverse Trig Functions
Identity Property: sin^(-1)(sin(x)) = x
Identity Property: cos^(-1)(cos(x)) = x
Identity Property: tan^(-1)(tan(x)) = x
Odd and Even Properties of Inverse Trig Functions
Odd Property: sin^(-1)(-x) = -sin^(-1)(x)
Even Property: cos^(-1)(-x) = cos^(-1)(x)
Odd Property: tan^(-1)(-x) = -tan^(-1)(x)
Pythagorean Identities of Inverse Trig Functions
Pythagorean Identity: sin^(-1)(x) = cos^(-1)(√(1 - x^2))
Pythagorean Identity: cos^(-1)(x) = sin^(-1)(√(1 - x^2))
Examples: Identifying Domain and Range
Example 1: Find the domain and range of sin^(-1)(x)
Example 2: Find the domain and range of cos^(-1)(x)
Complementary Angles Property
Complementary Angles Property: sin^(-1)(x) + cos^(-1)(x) = π/2
Complementary Angles Property: tan^(-1)(x) + cot^(-1)(x) = π/2
Examples: Complementary Angles Property
Example 1: Find the value of sin^(-1)(3/5) + cos^(-1)(4/5)
Example 2: Find the value of tan^(-1)(2) + cot^(-1)(3)
Composition of Inverse Trig Functions
Composition Property: sin(sin^(-1)(x)) = x
Composition Property: cos(cos^(-1)(x)) = x
Composition Property: tan(tan^(-1)(x)) = x
Examples: Composition Property
Example 1: Prove that sin(sin^(-1)(x)) = x
Example 2: Prove that tan(tan^(-1)(x)) = x
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Examples: Identifying Domain and Range
Example 1: Find the domain and range of sin^(-1)(x)
Domain: -1 ≤ x ≤ 1 (range of sin function)
Range: -π/2 ≤ sin^(-1)(x) ≤ π/2 (restricted range)
Example 2: Find the domain and range of cos^(-1)(x)
Domain: -1 ≤ x ≤ 1 (range of cos function)
Range: 0 ≤ cos^(-1)(x) ≤ π (restricted range)
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Complementary Angles Property
Complementary Angles Property: sin^(-1)(x) + cos^(-1)(x) = π/2
The sine and cosine of complementary angles always add up to π/2
Useful in finding missing angles in right-angled triangles
Complementary Angles Property: tan^(-1)(x) + cot^(-1)(x) = π/2
The tangent and cotangent of complementary angles always add up to π/2
Used to find missing angles in right-angled triangles
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Examples: Complementary Angles Property
Example 1: Find the value of sin^(-1)(3/5) + cos^(-1)(4/5)
Using the Complementary Angles Property: sin^(-1)(3/5) + cos^(-1)(4/5) = π/2
Example 2: Find the value of tan^(-1)(2) + cot^(-1)(3)
Using the Complementary Angles Property: tan^(-1)(2) + cot^(-1)(3) = π/2
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Composition of Inverse Trig Functions
Composition Property: sin(sin^(-1)(x)) = x
The sine of the arcsine of x is x itself
Composition Property: cos(cos^(-1)(x)) = x
The cosine of the arccosine of x is x itself
Composition Property: tan(tan^(-1)(x)) = x
The tangent of the arctangent of x is x itself
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Examples: Composition Property
Example 1: Prove that sin(sin^(-1)(x)) = x
Let y = sin^(-1)(x)
Then, sin(y) = x (definition of arcsine)
Taking the sine of both sides, sin(sin^(-1)(x)) = x
Example 2: Prove that tan(tan^(-1)(x)) = x
Let y = tan^(-1)(x)
Then, tan(y) = x (definition of arctangent)
Taking the tangent of both sides, tan(tan^(-1)(x)) = x
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Derivatives of Inverse Trig Functions
Derivative of arcsin(x) = 1/√(1 - x^2)
Derivative of arccos(x) = -1/√(1 - x^2)
Derivative of arctan(x) = 1/(1 + x^2)
Integrals of Inverse Trig Functions
Integral of 1/√(1 - x^2) = arcsin(x) + C
Integral of -1/√(1 - x^2) = arccos(x) + C
Integral of 1/(1 + x^2) = arctan(x) + C
Logarithmic Properties of Inverse Trig Functions
Logarithmic Property: ln(1 + x) = arctan(x) + C
Logarithmic Property: ln(1 - x) = -arctan(x) + C
Properties of Inverse Hyperbolic Trig Functions
Inverse Hyperbolic Functions: arcsinh(x), arccosh(x), arctanh(x)
Similar properties as inverse trig functions
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Derivatives of Inverse Trig Functions
Derivative of arcsin(x) = 1/√(1 - x^2)
Example: Find the derivative of y = arcsin(2x)
Derivative of arccos(x) = -1/√(1 - x^2)
Example: Find the derivative of y = arccos(3x)
Derivative of arctan(x) = 1/(1 + x^2)
Example: Find the derivative of y = arctan(4x)
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Integrals of Inverse Trig Functions
Integral of 1/√(1 - x^2) = arcsin(x) + C
Example: Evaluate the integral ∫(1/√(1 - x^2)) dx
Integral of -1/√(1 - x^2) = arccos(x) + C
Example: Evaluate the integral ∫(-1/√(1 - x^2)) dx
Integral of 1/(1 + x^2) = arctan(x) + C
Example: Evaluate the integral ∫(1/(1 + x^2)) dx
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Logarithmic Properties of Inverse Trig Functions
Logarithmic Property: ln(1 + x) = arctan(x) + C
Example: Prove that ln(1 + x) = arctan(x) + C
Logarithmic Property: ln(1 - x) = -arctan(x) + C
Example: Prove that ln(1 - x) = -arctan(x) + C
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Properties of Inverse Hyperbolic Trig Functions
Inverse Hyperbolic Functions: arcsinh(x), arccosh(x), arctanh(x)
Similar to inverse trig functions, but for hyperbolic functions
Inverse Hyperbolic Identity: sinh(arcsinh(x)) = x
Inverse Hyperbolic Identity: cosh(arccosh(x)) = x
Inverse Hyperbolic Identity: tanh(arctanh(x)) = x
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Properties of Inverse Hyperbolic Trig Functions (contd.)
Inverse Hyperbolic Identity: arcsinh(sinh(x)) = x
Inverse Hyperbolic Identity: arccosh(cosh(x)) = x, x ≥ 1
Inverse Hyperbolic Identity: arccosh(cosh(x)) = -x, x ≤ -1
Inverse Hyperbolic Identity: arctanh(tanh(x)) = x
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Graphs of Inverse Trig Functions
Graph of y = sin^(-1)(x)
Graph of y = cos^(-1)(x)
Graph of y = tan^(-1)(x)
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Graphs of Inverse Hyperbolic Trig Functions
Graph of y = sinh^(-1)(x)
Graph of y = cosh^(-1)(x)
Graph of y = tanh^(-1)(x)
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Solving Equations Involving Inverse Trig Functions
Method: Convert inverse trig equation to a regular trig equation
Example: Solve the equation sin^(-1)(x) + cos^(-1)(x) = π/2
Example: Solve the equation tan^(-1)(x) = π/4
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Summary
Inverse Trig Functions: arcsin(x), arccos(x), arctan(x)
Properties of Inverse Trig Functions
Derivatives and Integrals of Inverse Trig Functions
Logarithmic Properties of Inverse Trig Functions
Properties of Inverse Hyperbolic Trig Functions
Solving Equations Involving Inverse Trig Functions
Graphs of Inverse Trig Functions and Inverse Hyperbolic Trig Functions
Resume presentation
Inverse Trigonometric Functions - Other properties of inverse trig functions Recap of Inverse Trigonometric Functions Definition: Inverse function of a trigonometric function Range of inverse trig functions Domain and Range of Inverse Trigonometric Functions Domain: Range of corresponding trig functions Range: Restricted domains of inverse trig functions Basic Properties of Inverse Trig Functions Identity Property: sin^(-1)(sin(x)) = x Identity Property: cos^(-1)(cos(x)) = x Identity Property: tan^(-1)(tan(x)) = x Odd and Even Properties of Inverse Trig Functions Odd Property: sin^(-1)(-x) = -sin^(-1)(x) Even Property: cos^(-1)(-x) = cos^(-1)(x) Odd Property: tan^(-1)(-x) = -tan^(-1)(x) Pythagorean Identities of Inverse Trig Functions Pythagorean Identity: sin^(-1)(x) = cos^(-1)(√(1 - x^2)) Pythagorean Identity: cos^(-1)(x) = sin^(-1)(√(1 - x^2)) Examples: Identifying Domain and Range Example 1: Find the domain and range of sin^(-1)(x) Example 2: Find the domain and range of cos^(-1)(x) Complementary Angles Property Complementary Angles Property: sin^(-1)(x) + cos^(-1)(x) = π/2 Complementary Angles Property: tan^(-1)(x) + cot^(-1)(x) = π/2 Examples: Complementary Angles Property Example 1: Find the value of sin^(-1)(3/5) + cos^(-1)(4/5) Example 2: Find the value of tan^(-1)(2) + cot^(-1)(3) Composition of Inverse Trig Functions Composition Property: sin(sin^(-1)(x)) = x Composition Property: cos(cos^(-1)(x)) = x Composition Property: tan(tan^(-1)(x)) = x Examples: Composition Property Example 1: Prove that sin(sin^(-1)(x)) = x Example 2: Prove that tan(tan^(-1)(x)) = x