🔺Trigonometry Unit 6 – Inverse Trigonometric Functions

Key Concepts

• Inverse trigonometric functions are the reverse of the standard trigonometric functions (sine, cosine, tangent)
• Denoted by the prefix "arc" or the superscript "-1" (e.g., arcsin or sin$^{-1}$)
• Used to find the angle measure when given the ratio of sides in a right triangle
• Restricted domains ensure the functions are one-to-one and have unique inverses
• Graphs of inverse trigonometric functions are reflections of the original functions across the line $y=x$
• Useful in solving equations involving trigonometric expressions
• Have specific identities and relationships that simplify calculations and problem-solving
• Applied in various fields such as physics, engineering, and navigation

Defining Inverse Trig Functions

• Inverse sine function (arcsin or sin$^{-1}$) returns the angle whose sine is the input value
  • Defined as $\arcsin(x) = \theta$ if and only if $\sin(\theta) = x$ and $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$
• Inverse cosine function (arccos or cos$^{-1}$) returns the angle whose cosine is the input value
  • Defined as $\arccos(x) = \theta$ if and only if $\cos(\theta) = x$ and $0 \leq \theta \leq \pi$
• Inverse tangent function (arctan or tan$^{-1}$) returns the angle whose tangent is the input value
  • Defined as $\arctan(x) = \theta$ if and only if $\tan(\theta) = x$ and $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$
• Inverse cosecant (arccsc or csc$^{-1}$), inverse secant (arcsec or sec$^{-1}$), and inverse cotangent (arccot or cot$^{-1}$) are less commonly used but follow similar definitions
• Example: If $\sin(\theta) = \frac{1}{2}$ and $0 \leq \theta \leq \frac{\pi}{2}$, then $\theta = \arcsin(\frac{1}{2}) = \frac{\pi}{6}$

Domain and Range

• The domain of an inverse trigonometric function is the range of its corresponding trigonometric function
  • Domain of arcsin and arccos is $[-1, 1]$
  • Domain of arctan is $(-\infty, \infty)$
• The range of an inverse trigonometric function is a restricted interval to ensure a one-to-one relationship
  • Range of arcsin is $[-\frac{\pi}{2}, \frac{\pi}{2}]$
  • Range of arccos is $[0, \pi]$
  • Range of arctan is $(-\frac{\pi}{2}, \frac{\pi}{2})$
• Restricting the range prevents multiple angles from having the same trigonometric ratio
• Example: $\arcsin(0.5)$ is defined, but $\arcsin(2)$ is undefined because $2$ is outside the domain of arcsin

Graphs and Properties

• Graphs of inverse trigonometric functions are reflections of the original functions across the line $y=x$
  • The graph of $y = \arcsin(x)$ is the reflection of $y = \sin(x)$ restricted to the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$
  • The graph of $y = \arccos(x)$ is the reflection of $y = \cos(x)$ restricted to the interval $[0, \pi]$
  • The graph of $y = \arctan(x)$ is the reflection of $y = \tan(x)$ restricted to the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$
• Inverse trigonometric functions are continuous and monotonic on their domains
  • arcsin and arctan are increasing functions
  • arccos is a decreasing function
• Inverse trigonometric functions have vertical asymptotes at the endpoints of their domains
  • arcsin and arccos have vertical asymptotes at $x = -1$ and $x = 1$
  • arctan has vertical asymptotes at $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$

Solving Equations

• Inverse trigonometric functions are used to solve equations involving trigonometric expressions
• To solve an equation like $\sin(x) = a$, apply the inverse sine function to both sides: $x = \arcsin(a)$
  • Remember to consider the restricted range of the inverse function and any additional solutions outside the range
• When solving equations with multiple trigonometric functions, isolate one function before applying the inverse
• Example: To solve $\cos(2x) = \frac{1}{2}$ for $0 \leq x \leq \pi$:
  • First, isolate the cosine function: $2x = \arccos(\frac{1}{2})$
  • Then, solve for $x$: $x = \frac{1}{2}\arccos(\frac{1}{2}) = \frac{\pi}{6}$ or $x = \frac{5\pi}{6}$ (since $\cos(\frac{\pi}{3}) = \frac{1}{2}$ and $\frac{\pi}{3}$ is within the range of arccos)

Identities and Relationships

• Inverse trigonometric functions have identities and relationships that simplify calculations and problem-solving
• Pythagorean identities: $\arcsin(x) + \arccos(x) = \frac{\pi}{2}$ and $\arctan(x) + \arctan(\frac{1}{x}) = \frac{\pi}{2}$ for $x > 0$
• Composition identities: $\sin(\arcsin(x)) = x$ for $-1 \leq x \leq 1$ and $\arcsin(\sin(x)) = x$ for $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ (similar identities hold for cosine and tangent)
• Sum and difference formulas: $\arcsin(x) \pm \arcsin(y) = \arcsin(x\sqrt{1-y^2} \pm y\sqrt{1-x^2})$ and $\arctan(x) \pm \arctan(y) = \arctan(\frac{x \pm y}{1 \mp xy})$
• Example: Simplify $\arccos(\frac{1}{2}) + \arcsin(\frac{1}{2})$
  • Using the Pythagorean identity, $\arccos(\frac{1}{2}) + \arcsin(\frac{1}{2}) = \frac{\pi}{2}$

Applications in Real-World Problems

• Inverse trigonometric functions have applications in various fields, such as physics, engineering, and navigation
• Used to calculate angles in triangles when given side lengths or ratios
  • Example: In a right triangle with hypotenuse 5 and opposite side 3, the angle opposite the side of length 3 is $\arcsin(\frac{3}{5})$
• Employed in solving problems involving vectors, oscillations, and waves
  • Example: If a pendulum's position is given by $x(t) = 2\sin(3t)$, the times at which the pendulum is at its maximum displacement can be found using $t = \frac{1}{3}\arcsin(1) + \frac{2\pi n}{3}$, where $n$ is an integer
• Applied in navigation to calculate bearings and courses
  • Example: If a ship travels 10 km east and then 5 km north, the bearing from the starting point to the endpoint is $\arctan(\frac{5}{10}) = 26.6°$ east of north

Common Pitfalls and Tips

• Remember that inverse trigonometric functions have restricted domains and ranges
  • Be cautious when applying inverse functions to expressions outside their domains
• When solving equations, consider the possibility of multiple solutions due to the periodicity of trigonometric functions
  • Example: The equation $\sin(x) = \frac{1}{2}$ has solutions $x = \frac{\pi}{6} + 2\pi n$ and $x = \frac{5\pi}{6} + 2\pi n$, where $n$ is an integer
• Use identities and relationships to simplify expressions and solve problems more efficiently
  • Example: To find $\arcsin(\frac{\sqrt{3}}{2})$, use the composition identity: $\arcsin(\frac{\sqrt{3}}{2}) = \arcsin(\sin(\frac{\pi}{3})) = \frac{\pi}{3}$
• When working with angles in radians, be mindful of the radian measure of common angles (e.g., $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$)
• Double-check your calculator's mode (degrees or radians) when evaluating inverse trigonometric functions to avoid errors
• Practice solving a variety of problems to develop a strong understanding of inverse trigonometric functions and their applications