Inverse Trigonometric Functions - Relations between various inverse trig functions

• In this lecture, we will explore the relations between various inverse trigonometric functions.
• We will focus on the inverse sine function, inverse cosine function, and inverse tangent function.
• We will discover how these functions are related to each other and how they can be used to find angles.

Inverse Sine Function (sin^-1 x)

• The inverse sine function, denoted as sin^-1 x, is defined as the inverse of the sine function.
• It is also known as arcsin x or asin x.
• The domain of this function is -1 ≤ x ≤ 1, and the range is -π/2 ≤ y ≤ π/2.
• Its principal values lie in the first and fourth quadrants. Example:
• If sinθ = 0.5, then θ = sin^-1(0.5) = 30° or π/6.

Inverse Cosine Function (cos^-1 x)

• The inverse cosine function, denoted as cos^-1 x, is defined as the inverse of the cosine function.
• It is also known as arccos x or acos x.
• The domain of this function is -1 ≤ x ≤ 1, and the range is 0 ≤ y ≤ π.
• Its principal values lie in the first and second quadrants. Example:
• If cosθ = 0.5, then θ = cos^-1(0.5) = 60° or π/3.

Inverse Tangent Function (tan^-1 x)

• The inverse tangent function, denoted as tan^-1 x, is defined as the inverse of the tangent function.
• It is also known as arctan x or atan x.
• The domain of this function is -∞ < x < ∞, and the range is -π/2 < y < π/2.
• Its principal values lie in the first and fourth quadrants. Example:
• If tanθ = 1, then θ = tan^-1(1) = 45° or π/4.

Properties of Inverse Trig Functions

1. sin(sin^-1 x) = x for -1 ≤ x ≤ 1

2. cos(cos^-1 x) = x for -1 ≤ x ≤ 1

3. tan(tan^-1 x) = x for -∞ < x < ∞ Note: These properties establish the relationship between the trigonometric function and its inverse. Example:

• sin(sin^-1 0.5) = 0.5
• cos(cos^-1 0.5) = 0.5
• tan(tan^-1 1) = 1

Relations Between Inverse Trig Functions

1. sin^-1 x + cos^-1 x = π/2 for -1 ≤ x ≤ 1

2. tan^-1 x + cot^-1 x = π/2 for x > 0 Note: These relations can be used to find the angles between various trig functions. Example:

• sin^-1(0.6) + cos^-1(0.8) = π/2
• tan^-1(2) + cot^-1(0.5) = π/2

Evaluating Inverse Trig Functions

• The values of inverse trig functions can be evaluated using a calculator or trigonometric identities.
• It is important to understand the principal values and choose the appropriate quadrants. Example:
• Find the value of sin^-1(-0.4).
• Solution: Since the domain of sin^-1 x is -1 ≤ x ≤ 1 and the range is -π/2 ≤ y ≤ π/2, we have sin^-1(-0.4) = -sin^-1(0.4) = -0.4115 radians or -23.58°.

Slide 11: Composition of Inverse Trig Functions

• We can also compose inverse trig functions with each other to obtain new values.
• For example, sin(sin^-1 x) is equal to x, as mentioned earlier.
• Similarly, cos(cos^-1 x) and tan(tan^-1 x) are also equal to x.
• These compositions help in simplifying expressions and solving equations involving trig functions. Example:
• Simplify the expression sin(sin^-1 0.3).
• Solution: sin(sin^-1 0.3) = 0.3, since sin(sin^-1 x) = x.

Slide 12: Inverse Trig Function Identities

• Similar to the trigonometric identities, there are also identities related to inverse trig functions.
• Some of these identities include:
  • sin^-1(-x) = -sin^-1(x)
  • cos^-1(-x) = π - cos^-1(x)
  • tan^-1(-x) = -tan^-1(x) Example:
• Evaluate cos^-1(-0.7).
• Solution: cos^-1(-0.7) = π - cos^-1(0.7) = 2.4509 radians or 140.68°.

Slide 13: Solving Equations with Inverse Trig Functions

• Inverse trig functions are often used to solve equations involving trigonometric functions.
• For example, if sin^-1 x = 0.5, we can find x by taking the sine of both sides: sin(sin^-1 x) = sin(0.5).
• This simplifies to x = sin(0.5).
• Similarly, we can solve equations involving cos^-1 x and tan^-1 x. Example:
• Solve the equation sin^-1 x = 0.3.
• Solution: Taking the sine of both sides, we get x = sin(0.3) ≈ 0.2955.

Slide 14: Pythagorean Identity for Inverse Trig Functions

• The Pythagorean identity for inverse trig functions is similar to the Pythagorean identity for trig functions.
• It states that sin² θ + cos² θ = 1, where θ represents any angle in the domain of the inverse trig functions.
• This identity is useful in simplifying trigonometric expressions involving inverse trig functions. Example:
• Simplify the expression sin²(sin^-1 x) + cos²(sin^-1 x).
• Solution: Using the Pythagorean identity, we have x² + (1 - x²) = 1.

Slide 15: Graphs of Inverse Trig Functions

• The graphs of inverse trig functions have distinct characteristics.
• The graph of y = sin^-1 x is a quarter of a circle with center at the origin and a radius of 1.
• The graph of y = cos^-1 x is also a quarter of a circle but with center at (1, 0).
• The graph of y = tan^-1 x is a curve that approaches π/2 and -π/2 as x approaches ∞ and -∞, respectively. Example:
• Draw the graph of y = tan^-1 x.
• Solution: The graph of y = tan^-1 x has a horizontal asymptote at y = π/2 and -π/2, and it approaches these values as x approaches ∞ and -∞.

Slide 16: Trig Functions and Inverse Trig Functions

• Trigonometric functions and inverse trig functions are closely related and can be used interchangeably.
• For example, sin(sin^-1 x) = x and sin(theta) = x imply theta = sin^-1 x.
• These relationships can be used to find angles or unknown values in trigonometric equations. Example:
• If cos(theta) = 0.2, find theta.
• Solution: Since cos(theta) = x implies theta = cos^-1 x, we have theta = cos^-1 0.2 ≈ 78.46°.

Slide 17: Domain and Range of Inverse Trig Functions

• The domain of inverse trig functions depends on the original trigonometric function.
• For example, the domain of sin^-1 x is -1 ≤ x ≤ 1, while the domain of tan^-1 x is -∞ < x < ∞.
• The range of inverse trig functions also varies but is generally limited to certain intervals. Example:
• Determine the domain and range of tan^-1 x.
• Solution: The domain of tan^-1 x is -∞ < x < ∞, and the range is -π/2 < y < π/2.

Slide 18: Applications of Inverse Trig Functions

• Inverse trig functions have numerous applications in real-world scenarios.
• They are used in physics, engineering, computer graphics, and navigation, among other fields.
• For example, in physics, inverse trig functions are used to find angles in projectile motion or rotational motion problems. Example:
• A projectile is launched with an initial velocity of 30 m/s at an angle of 45°. Find the maximum height reached by the projectile.
• Solution: The maximum height can be found using the equation H = (v0² sin² θ) / (2g), where v0 is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity. Plugging in the values, we get H = (30² sin² 45°) / (2 * 9.8) ≈ 45.9 meters.

Slide 19: Summary

• Inverse trig functions, such as sin^-1 x, cos^-1 x, and tan^-1 x, are defined as the inverse of the corresponding trigonometric functions.
• They have specific domains and ranges that depend on the original trig function.
• Inverse trig functions can be composed with each other and used to solve trigonometric equations.
• They also have important identities and can be graphed to visualize their behavior.
• Inverse trig functions have diverse applications in various fields, including physics and engineering.

Slide 20: Questions

• What is the domain of sin^-1 x?
• How do inverse trig functions help in solving trigonometric equations?
• What is the identity for cos(cos^-1 x)?
• Describe the graph of y = tan^-1 x.
• What are some real-life applications of inverse trig functions?

Slide 21: Evaluating Inverse Trig Functions

• Inverse trig functions can be evaluated using calculators or tables.
• The output of inverse trig functions is typically given in radians.
• To convert from radians to degrees, multiply by (180/π).
• To convert from degrees to radians, multiply by (π/180). Example:
• Evaluate sin^-1(0.4) in radians and degrees.
  • Solution: Using a calculator, we find sin^-1(0.4) ≈ 0.4115 radians or ≈ 23.58°.

Slide 22: Domain and Range Examples

• Find the domain and range of the following inverse trig functions:
  • a) cos^-1(x)
  • b) tan^-1(x) Example:
• a) The domain of cos^-1(x) is -1 ≤ x ≤ 1, and the range is 0 ≤ y ≤ π.
• b) The domain of tan^-1(x) is -∞ < x < ∞, and the range is -π/2 < y < π/2.

Slide 23: Trig Equations with Inverse Functions

• Inverse trig functions can be used to solve trigonometric equations.
• To solve equations involving inverse trig functions, use the definition of the function and apply appropriate properties.
• Remember to check the solutions for extraneous values. Example:
• Solve the equation sin(sinx) = 0.
  • Solution: The equation implies sinx = 0, which means x = 0, π.

Slide 24: Simplifying Expressions

• Inverse trig functions can be used to simplify expressions involving trigonometric functions.
• Use the properties and identities of inverse trig functions to simplify expressions.
• Simplify expressions by expressing the functions in terms of inverse trig functions. Example:
• Simplify the expression tan(sin^-1(x)).
  • Solution: Using the property tan(tan^-1(x)) = x, we have tan(sin^-1(x)) = x.

Slide 25: Trig and Inverse Trig Compositions

• Compositions of trig and inverse trig functions can be used to solve equations or simplify expressions.
• Apply the relevant inverse trig functions to the trig functions involved in the composition. Example:
• Solve the equation sin(sin^-1(x)) = 1/2.
  • Solution: The equation simplifies to x = 1/2.

Slide 26: Pythagorean Identity for Inverse Trig Functions

• A Pythagorean identity exists for inverse trig functions as well.
• sin²(sin^-1(x)) + cos²(sin^-1(x)) = 1. Example:
• Simplify the expression sin²(sin^-1(0.3)) + cos²(sin^-1(0.3)).
  • Solution: Using the Pythagorean identity, the expression simplifies to 1.

Slide 27: Inverse Trig Functions in Right Triangles

• Inverse trig functions can be used in right triangles to find missing angles or sides.
• Use the appropriate inverse trig functions based on the given information. Example:
• Given a right triangle with a side length of 3 and a hypotenuse of 5, find the measure of angle θ.
  • Solution: Using sinθ = opposite/hypotenuse, we have sinθ = 3/5. Therefore, θ = sin^-1(3/5).

Slide 28: Inverse Trig Functions in Calculus

• Inverse trig functions are used extensively in calculus.
• They are used in finding derivatives and integrals involving trigonometric functions. Example:
• Find the derivative of y = sin^-1(x).
  • Solution: Using the chain rule and the derivative of sin(x), the derivative of y = sin^-1(x) is 1/√(1-x^2).

Slide 29: Graphing Inverse Trig Functions

• Graphs of inverse trig functions show the relationship between input and output.
• The vertical asymptotes and horizontal asymptotes vary for different inverse trig functions.
• Use the properties of inverse trig functions to sketch their graphs. Example:
• Sketch the graph of y = sin^-1(x).
  • Solution: The graph is a quarter of a circle centered at the origin with a radius of 1.

Slide 30: Summary and Wrap-Up

• Inverse trig functions are the inverses of trigonometric functions.
• They have specific domains, ranges, identities, and properties.
• Inverse trig functions can be used to evaluate expressions, solve equations, simplify trigonometric expressions, and find missing angles or sides in right triangles.
• They are also applicable in calculus and have diverse real-world applications.
• Understanding inverse trig functions helps in understanding the relationship between trigonometric functions and their inverses.