Inverse Trigonometric Functions - Domain range and graph of cosine inverse

  • The inverse of the cosine function is called the cosine inverse or arccosine function.
  • It is denoted as cos^(-1)(x) or arccos(x).
  • The domain of the cosine inverse function is [-1, 1].
  • The range of the cosine inverse function is [0, π].
  • The graph of the cosine inverse function is a reflection of the graph of cosine function about the line y = x.

Inverse Trigonometric Functions - Domain range and graph of cosine inverse

  • The cosine function is defined for all real numbers (-∞, ∞).
  • However, the inverse cosine function is only defined for values between -1 and 1, inclusive.
  • The domain of cosine inverse is [-1, 1].
  • The range of cosine inverse is [0, π].

Inverse Trigonometric Functions - Graph of cosine inverse

  • The graph of the function y = cos^(-1)(x) is a curve that is symmetric about the line y = x.
  • The graph starts at (1, 0) and approaches (0, π) as x approaches -1.
  • The graph is decreasing in the interval [-1, 1].
  • It is important to note that the graph of the cosine inverse function is not a function but a relation.

Inverse Trigonometric Functions - Examples

  1. Find the value of cos^(-1)(1/2).
    • Since the cosine function is positive in the first quadrant, we look for an angle in the first quadrant whose cosine is 1/2.
    • The angle θ in the first quadrant that satisfies cos(θ) = 1/2 is 60 degrees or π/3.
    • Therefore, cos^(-1)(1/2) = 60 degrees or π/3.
  1. Find the value of cos^(-1)(0)
    • The cosine function is 0 at π/2 and 3π/2.
    • Therefore, cos^(-1)(0) = π/2 and 3π/2.

Inverse Trigonometric Functions - Key Concepts

  • The inverse trigonometric functions are used to find angles given trigonometric ratios.
  • The cosine inverse function is the inverse of the cosine function.
  • The domain of the cosine inverse function is [-1, 1].
  • The range of the cosine inverse function is [0, π].
  • The graph of the cosine inverse function is a reflection of the graph of the cosine function about the line y = x.

Inverse Trigonometric Functions - Key Concepts

  • The cosine inverse function is not a true function but a relation.
  • The values of the cosine inverse function are angles in radians.
  • The values of the cosine inverse function are often expressed as exact values or approximations in terms of π.
  • The cosine inverse function is used in solving trigonometric equations and applications in various fields such as physics, engineering, and geometry.

Inverse Trigonometric Functions - Quiz

Question 1: Find the value of cos^(-1)(-1). a) 0 b) π/2 c) π d) 2π Question 2: Find the value of cos^(-1)(1/2). a) 30 degrees b) π/3 c) 60 degrees d) 90 degrees

Inverse Trigonometric Functions - Quiz (Answers)

Question 1: Find the value of cos^(-1)(-1).

  • Answer: b) π/2 Question 2: Find the value of cos^(-1)(1/2).
  • Answer: c) 60 degrees
  1. Inverse Trigonometric Functions - Examples
  • Example 1: Find the value of cos^(-1)(√2/2).
    • The cosine function is positive in the first and fourth quadrants.
    • In the first quadrant, the angle θ that satisfies cos(θ) = √2/2 is 45 degrees or π/4.
    • In the fourth quadrant, the angle θ that satisfies cos(θ) = √2/2 is 315 degrees or 7π/4.
    • Therefore, cos^(-1)(√2/2) = 45 degrees or π/4, 315 degrees or 7π/4.
  • Example 2: Find the value of cos^(-1)(-√3/2).
    • The cosine function is negative in the second and third quadrants.
    • In the second quadrant, the angle θ that satisfies cos(θ) = -√3/2 is 120 degrees or 2π/3.
    • In the third quadrant, the angle θ that satisfies cos(θ) = -√3/2 is 240 degrees or 4π/3.
    • Therefore, cos^(-1)(-√3/2) = 120 degrees or 2π/3, 240 degrees or 4π/3.
  • Example 3: Find the value of cos^(-1)(-1/2).
    • The cosine function is negative in the second and third quadrants.
    • In the second quadrant, the angle θ that satisfies cos(θ) = -1/2 is 120 degrees or 2π/3.
    • In the third quadrant, the angle θ that satisfies cos(θ) = -1/2 is 240 degrees or 4π/3.
    • Therefore, cos^(-1)(-1/2) = 120 degrees or 2π/3, 240 degrees or 4π/3.
  • Example 4: Find the value of cos^(-1)(0.5).
    • The cosine function is positive in the first and fourth quadrants.
    • In the first quadrant, the angle θ that satisfies cos(θ) = 0.5 is 60 degrees or π/3.
    • In the fourth quadrant, the angle θ that satisfies cos(θ) = 0.5 is 300 degrees or 5π/3.
    • Therefore, cos^(-1)(0.5) = 60 degrees or π/3, 300 degrees or 5π/3.
  • Example 5: Find the value of cos^(-1)(-0.5).
    • The cosine function is negative in the second and third quadrants.
    • In the second quadrant, the angle θ that satisfies cos(θ) = -0.5 is 120 degrees or 2π/3.
    • In the third quadrant, the angle θ that satisfies cos(θ) = -0.5 is 240 degrees or 4π/3.
    • Therefore, cos^(-1)(-0.5) = 120 degrees or 2π/3, 240 degrees or 4π/3.
  1. Inverse Trigonometric Functions - Domain range and graph of sine inverse
  • The inverse of the sine function is called the sine inverse or arcsine function.
  • It is denoted as sin^(-1)(x) or arcsin(x).
  • The domain of the sine inverse function is [-1, 1].
  • The range of the sine inverse function is [-π/2, π/2].
  • The graph of the sine inverse function is a reflection of the graph of the sine function about the line y = x.
  • The sine function is defined for all real numbers (-∞, ∞).
  • However, the inverse sine function is only defined for values between -1 and 1, inclusive.
  • The domain of sine inverse is [-1, 1].
  • The range of the sine inverse function is [-π/2, π/2].
  • The graph of the sine inverse function is a reflection of the graph of the sine function about the line y = x.
  • The graph of the function y = sin^(-1)(x) is a curve that is symmetric about the line y = x.
  • The graph starts at (-1, -π/2) and approaches (1, π/2) as x approaches -∞ and ∞.
  • The graph is increasing in the interval [-1, 1].
  • Example: Find the value of sin^(-1)(√2/2).
    • Since the sine function is positive in the first and second quadrants, we look for an angle in these quadrants whose sine is √2/2.
    • The angle θ in the first quadrant that satisfies sin(θ) = √2/2 is 45 degrees or π/4.
    • The angle θ in the second quadrant that satisfies sin(θ) = √2/2 is 135 degrees or 3π/4.
    • Therefore, sin^(-1)(√2/2) = 45 degrees or π/4, 135 degrees or 3π/4.
  1. Inverse Trigonometric Functions - Examples
  • Example 1: Find the value of sin^(-1)(1/2).
    • Since the sine function is positive in the first and second quadrants, we look for an angle in these quadrants whose sine is 1/2.
    • The angle θ in the first quadrant that satisfies sin(θ) = 1/2 is 30 degrees or π/6.
    • The angle θ in the second quadrant that satisfies sin(θ) = 1/2 is 150 degrees or 5π/6.
    • Therefore, sin^(-1)(1/2) = 30 degrees or π/6, 150 degrees or 5π/6.
  • Example 2: Find the value of sin^(-1)(0).
    • The sine function is 0 at 0 and π.
    • Therefore, sin^(-1)(0) = 0 and π.
  • Example 3: Find the value of sin^(-1)(-1/2).
    • Since the sine function is negative in the third and fourth quadrants, we look for an angle in these quadrants whose sine is -1/2.
    • The angle θ in the third quadrant that satisfies sin(θ) = -1/2 is 210 degrees or 7π/6.
    • The angle θ in the fourth quadrant that satisfies sin(θ) = -1/2 is 330 degrees or 11π/6.
    • Therefore, sin^(-1)(-1/2) = 210 degrees or 7π/6, 330 degrees or 11π/6.
  • Example 4: Find the value of sin^(-1)(√3/2).
    • Since the sine function is positive in the first and second quadrants, we look for an angle in these quadrants whose sine is √3/2.
    • The angle θ in the first quadrant that satisfies sin(θ) = √3/2 is 60 degrees or π/3.
    • The angle θ in the second quadrant that satisfies sin(θ) = √3/2 is 120 degrees or 2π/3.
    • Therefore, sin^(-1)(√3/2) = 60 degrees or π/3, 120 degrees or 2π/3.
  • Example 5: Find the value of sin^(-1)(-√3/2).
    • Since the sine function is negative in the third and fourth quadrants, we look for an angle in these quadrants whose sine is -√3/2.
    • The angle θ in the third quadrant that satisfies sin(θ) = -√3/2 is 240 degrees or 4π/3.
    • The angle θ in the fourth quadrant that satisfies sin(θ) = -√3/2 is 300 degrees or 5π/3.
    • Therefore, sin^(-1)(-√3/2) = 240 degrees or 4π/3, 300 degrees or 5π/3.
  1. Inverse Trigonometric Functions - Domain range and graph of tangent inverse
  • The inverse of the tangent function is called the tangent inverse or arctangent function.
  • It is denoted as tan^(-1)(x) or arctan(x).
  • The domain of the tangent inverse function is (-∞, ∞).
  • The range of the tangent inverse function is (-π/2, π/2).
  • The graph of the tangent inverse function is a reflection of the graph of the tangent function about the line y = x.
  • The tangent function is defined for all real numbers (-∞, ∞).
  • The domain of tangent inverse is (-∞, ∞).
  • The range of the tangent inverse function is (-π/2, π/2).
  • The graph of the tangent inverse function is a reflection of the graph of the tangent function about the line y = x.
  • The graph of the function y = tan^(-1)(x) is a curve that is symmetric about the line y = x.
  • The graph starts at (-∞, -π/2) and approaches (∞, π/2) as x approaches -∞ and ∞.
  • The graph is increasing in the interval (-∞, ∞).
  • Example: Find the value of tan^(-1)(1).
    • The angle θ that satisfies tan(θ) = 1 is 45 degrees or π/4.
    • Therefore, tan^(-1)(1) = 45 degrees or π/4.
  • Example: Find the value of tan^(-1)(√3).
    • The angle θ that satisfies tan(θ) = √3 is 60 degrees or π/3.
    • Therefore, tan^(-1)(√3) = 60 degrees or π/3.
  1. Inverse Trigonometric Functions - Examples
  • Example 1: Find the value of tan^(-1)(0).
    • The angle θ that satisfies tan(θ) = 0 is 0 degrees or 0.
    • Therefore, tan^(-1)(0) = 0 degrees or 0.
  • Example 2: Find the value of tan^(-1)(√3/3).
    • The angle θ that satisfies tan(θ) = √3/3 is 30 degrees or π/6.
    • Therefore, tan^(-1)(√3/3) = 30 degrees or π/6.
  • Example 3: Find the value of tan^(-1)(-√3).
    • The angle θ that satisfies tan(θ) = -√3 is -60 degrees or -π/3.
    • Therefore, tan^(-1)(-√3) = -60 degrees or -π/3.
  • Example 4: Find the value of tan^(-1)(2).
    • The angle θ that satisfies tan(θ) = 2 is 63.43 degrees or 1.11 radians (approx.).
    • Therefore, tan^(-1)(2) = 63.43 degrees or 1.11 radians (approx.).
  • Example 5: Find the value of tan^(-1)(-1).
    • The angle θ that satisfies tan(θ) = -1 is -45 degrees or -π/4.
    • Therefore, tan^(-1)(-1) = -45 degrees or -π/4.
  1. Inverse Trigonometric Functions - Domain range and graph of cotangent inverse
  • The inverse of the cotangent function is called the cotangent inverse or arccotangent function.
  • It is denoted as cot^(-1)(x) or arccot(x).
  • The domain of the cotangent inverse function is (-∞, ∞).
  • The range of the cotangent inverse function is (0, π).
  • The graph of the cotangent inverse function is a reflection of the graph of the cotangent function about the line y = x.
  • The cotangent function is defined for all real numbers (-∞, ∞).
  • The domain of cotangent inverse is (-∞, ∞).
  • The range of the cotangent inverse function is (0, π).
  • The graph of the cotangent inverse function is a reflection of the graph of the cotangent function about the line y = x.
  • The graph of the function y = cot^(-1)(x) is a curve that is symmetric about the line y = x.
  • The graph starts at (-∞, π/2) and approaches (∞, 0) as x approaches -∞ and ∞.
  • The graph is decreasing in the interval (-∞, ∞).
  • Example: Find the value of cot^(-1)(1).
    • The angle θ that satisfies cot(θ) = 1 is 45 degrees or π/4.
    • Therefore, cot^(-1)(1) = 45 degrees or π/4.
  • Example: Find the value of cot^(-1)(-1).
    • The angle θ that satisfies cot(θ) = -1 is 135 degrees or 3π/4.
    • Therefore, cot^(-1)(-1) = 135 degrees or 3π/4.
  1. Inverse Trigonometric Functions - Examples
  • Example 1: Find the value of cot^(-1)(0).
    • The angle θ that satisfies cot(θ) = 0 is 90 degrees or π/2.