Inverse Trigonometric Functions - Solved Problems

  • Review of Trigonometric Functions
  • Definition of Inverse Trigonometric Functions
  • Finding Inverse Trigonometric Functions using Special Triangles
  • Solving Equations with Inverse Trigonometric Functions
  • Ranges of Inverse Trigonometric Functions

Review of Trigonometric Functions

  • Trigonometric functions are mathematical functions that relate angles to the ratios of side lengths in right triangles.
  • The primary trigonometric functions are sin, cos, and tan.
  • These functions have specific properties and can be defined for specific ranges of angles.

Definition of Inverse Trigonometric Functions

  • The inverse trigonometric functions are the inverse functions of the primary trigonometric functions.
  • They allow us to find the angle whose trigonometric function value is known.
  • The inverse trigonometric functions are denoted as arcsin, arccos, and arctan.

Finding Inverse Trigonometric Functions using Special Triangles

  • We can use special triangles, such as the 30-60-90 and 45-45-90 triangles, to find exact values of inverse trigonometric functions.
  • These triangles have known ratios of side lengths, which can be used to determine the values of angles and their inverse trigonometric functions.

Solving Equations with Inverse Trigonometric Functions

  • Inverse trigonometric functions can be used to solve equations involving trigonometric functions.
  • By applying the inverse trigonometric functions to both sides of an equation, we can isolate the angle and find its value.
  • It is important to consider the domain and range of the inverse trigonometric functions when solving equations.

Ranges of Inverse Trigonometric Functions

  • The range of arcsin is [-π/2, π/2], which means it can only take on values between -90° and 90°.
  • The range of arccos is [0, π], which means it can only take on values between 0° and 180°.
  • The range of arctan is (-π/2, π/2), which means it can only take on values between -90° and 90°, excluding -90° and 90°.

Example: Finding the Value of arcsin

  • Given sin(x) = 1/2, find the value of x.
  • Using the inverse trigonometric function arcsin, we have arcsin(1/2) = x.
  • The value of x is 30° because sin(30°) = 1/2.

Example: Solving an Equation with arccos

  • Given cos(x) = 1/2, solve for x.
  • Applying arccos to both sides, we have arccos(1/2) = x.
  • The value of x is 60° because cos(60°) = 1/2.

Example: Solving an Equation with arctan

  • Given tan(x) = 1, solve for x.
  • Applying arctan to both sides, we have arctan(1) = x.
  • The value of x is 45° because tan(45°) = 1.
  1. Finding the Value of arccos
  • Given cos(x) = -1/2, find the value of x.
  • Using the inverse trigonometric function arccos, we have arccos(-1/2) = x.
  • The value of x is 120° because cos(120°) = -1/2.
  1. Solving an Equation with arcsin
  • Given arcsin(x) = π/6, solve for x.
  • Applying the trigonometric function sin to both sides, we have sin(arcsin(x)) = sin(π/6).
  • The value of x is 1/2 because sin(π/6) = 1/2.
  1. Solving an Equation with arcsec
  • Given arcsec(x) = π/4, solve for x.
  • Applying the trigonometric function sec to both sides, we have sec(arcsec(x)) = sec(π/4).
  • The value of x is √2 because sec(π/4) = √2.
  1. Graphical Representation of Inverse Trig Functions
  • The inverse trigonometric functions can also be represented graphically.
  • The graph of y = arcsin(x) is a restricted portion of the sine function.
  • The graph of y = arccos(x) is a restricted portion of the cosine function.
  • The graph of y = arctan(x) is a restricted portion of the tangent function.
  1. Properties of Inverse Trig Functions
  • The inverse trigonometric functions have certain properties that can be used to simplify expressions.
  • For example, arccos(x) + arcsin(x) = π/2.
  • Another property is arctan(1/x) = π/2 - arctan(x).
  1. Trig Substitutions in Integration
  • Inverse trigonometric functions are commonly used in integration techniques.
  • By making an appropriate trigonometric substitution, we can simplify the integral and solve it using the inverse trigonometric functions.
  • Trigonometric substitutions involve expressing a complicated expression as a function of a trigonometric function, allowing us to integrate it more easily.
  1. Applications of Inverse Trig Functions
  • Inverse trigonometric functions have many real-life applications.
  • For example, they are used in navigation and surveying to determine angles and distances.
  • They are also used in physics and engineering to calculate forces, angles of projections, and more.
  1. Example: Solving an Equation with arccos
  • Given arccos(x) = π/3, solve for x.
  • Applying the trigonometric function cos to both sides, we have cos(arccos(x)) = cos(π/3).
  • The value of x is 1/2 because cos(π/3) = 1/2.
  1. Example: Solving an Equation with arctan
  • Given arctan(x) = 3π/4, solve for x.
  • Applying the trigonometric function tan to both sides, we have tan(arctan(x)) = tan(3π/4).
  • The value of x is -1 because tan(3π/4) = -1.
  1. Summary
  • Inverse trigonometric functions are the inverse of the primary trigonometric functions.
  • They can be used to find the angle whose trigonometric function value is known.
  • We can find the values of inverse trigonometric functions using special triangles and simplifying trigonometric equations.
  • Inverse trigonometric functions have applications in various fields, including navigation, physics, and engineering.
  1. Example: Solving an Equation with arccos
  • Given arccos(x) = π/3, solve for x.
  • Applying the trigonometric function cos to both sides, we have cos(arccos(x)) = cos(π/3).
  • The value of x is 1/2 because cos(π/3) = 1/2.
  1. Example: Solving an Equation with arctan
  • Given arctan(x) = 3π/4, solve for x.
  • Applying the trigonometric function tan to both sides, we have tan(arctan(x)) = tan(3π/4).
  • The value of x is -1 because tan(3π/4) = -1.
  1. Example: Solving an Equation with arcsin
  • Given arcsin(x) = 1/2, solve for x.
  • Applying the trigonometric function sin to both sides, we have sin(arcsin(x)) = sin(1/2).
  • The value of x is 0.5236 (or approximately 0.5) because sin(1/2) = 0.5236.
  1. Example: Solving Trigonometric Equation using Inverse Trig Functions
  • Given sin(x) + tan(x) = 1, solve for x.
  • By applying the inverse trigonometric function arctan to both sides, we have arctan(sin(x) + tan(x)) = arctan(1).
  • Simplifying further, arctan(sin(x) + sin(x)/cos(x)) = arctan(1).
  • By using the identity tan(x) = sin(x)/cos(x), we have arctan(2sin(x)/cos(x)) = arctan(1).
  • The value of x is π/4 because arctan(2sin(π/4)/cos(π/4)) = arctan(1).
  1. Example: Solving Trigonometric Equation using Inverse Trig Functions
  • Given cos(x) = sin(x), solve for x.
  • Applying the inverse trigonometric function arcsin to both sides, we have arcsin(cos(x)) = arcsin(sin(x)).
  • Using the identity sin(x) = cos(π/2 - x), we have arcsin(cos(x)) = arcsin(cos(π/2 - x)).
  • The values of x are π/4 and 3π/4 because arcsin(cos(π/4)) = π/4 and arcsin(cos(3π/4)) = 3π/4.
  1. Example: Solving Trigonometric Equation using Inverse Trig Functions
  • Given sin(2x) = √3/2, solve for x.
  • By applying the inverse trigonometric function arcsin to both sides, we have arcsin(sin(2x)) = arcsin(√3/2).
  • Simplifying further, 2x = π/3 + 2kπ or 2x = 2π/3 + 2kπ, where k is an integer.
  • The values of x are π/6 + kπ or π/3 + kπ, where k is an integer.
  1. Summary
  • Inverse trigonometric functions can be used to solve equations involving trigonometric functions.
  • The solutions can be found by applying the appropriate inverse trigonometric function to both sides of the equation.
  • It is important to consider the domain and range of the inverse trigonometric functions when solving equations.
  • Examples have been provided to illustrate the process of solving equations using inverse trigonometric functions.
  1. Applications of Inverse Trig Functions
  • Inverse trigonometric functions have various real-life applications.
  • They are used in navigation to determine angles and distances.
  • In surveying, they help calculate angles and measurements.
  • In physics and engineering, they are used to analyze forces, angles of projection, and more.
  1. Practice Problems
  • Solve the equation tan(x) = 2 for x.
  • Find the value of arcsin(1) + arccos(-1/2).
  • Solve the equation arctan(x) = π/4 for x.
  • Find the value of sin(arcsin(3/5)).
  • Solve the equation arccos(sin(x)) = x for x.
  1. Conclusion
  • Inverse trigonometric functions play a crucial role in solving trigonometric equations and finding specific angle values.
  • They have applications in various fields, and understanding their properties and usage is essential.
  • By practicing solving problems involving inverse trigonometric functions, students can strengthen their understanding and skills in this topic.
  • It is important to keep in mind the ranges and properties of these functions while solving equations or working with real-life problems.