Glossary

6.1 Inverse Sine, Cosine, and Tangent Functions

3 min read•july 25, 2024

Inverse trigonometric functions flip the script, finding angles from trig ratios. They're essential for solving triangles and real-world problems. , , and each have unique domains and ranges, reflecting their original function's behavior.

These functions pop up in navigation, engineering, and even satellite dish alignment. Knowing how to evaluate and graph them opens doors to practical problem-solving. They're the key to unlocking angles when you only have side ratios to work with.

Inverse Trigonometric Functions

Inverse trigonometric functions: definitions and properties

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  • Inverse sine function (arcsine) reverses sine function finding angle from sine value
    • Notation: sin⁡−1x\sin^{-1}x or arcsin⁡x\arcsin x used interchangeably
    • : [−1,1][-1, 1] restricted to unit circle values
    • : [−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}] or [−90°,90°][-90°, 90°] quadrants I and IV
  • Inverse cosine function (arccosine) determines angle from cosine value
    • Notation: cos⁡−1x\cos^{-1}x or arccos⁡x\arccos x both acceptable
    • Domain: [−1,1][-1, 1] limited to unit circle
    • Range: [0,π][0, \pi] or [0°,180°][0°, 180°] quadrants I and II
  • Inverse tangent function (arctangent) finds angle from tangent ratio
    • Notation: tan⁡−1x\tan^{-1}x or arctan⁡x\arctan x commonly used
    • Domain: All real numbers, no restrictions
    • Range: (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2}) or (−90°,90°)(-90°, 90°) open interval

Evaluation of inverse trigonometric functions

  • Use scientific calculator for precise values (TI-84, Casio fx-991EX)
  • Convert between radians and degrees for angle measure flexibility
    • 1 radian=180°π1 \text{ radian} = \frac{180°}{\pi} approximately 57.3°
    • 1°=π180 radians1° = \frac{\pi}{180} \text{ radians} about 0.0175 radians
  • Evaluate common angles without calculator for quick mental math
    • sin⁡−1(0)=0\sin^{-1}(0) = 0 sine of 0° is 0
    • cos⁡−1(1)=0\cos^{-1}(1) = 0 cosine of 0° is 1
    • tan⁡−1(1)=π4\tan^{-1}(1) = \frac{\pi}{4} or 45°45° tangent of 45° is 1
    • sin⁡−1(12)=π6\sin^{-1}(\frac{1}{2}) = \frac{\pi}{6} or 30°30° sine of 30° is 12\frac{1}{2}
    • cos⁡−1(22)=π4\cos^{-1}(\frac{\sqrt{2}}{2}) = \frac{\pi}{4} or 45°45° cosine of 45° is 22\frac{\sqrt{2}}{2}

Graphs of inverse trigonometric functions

  • Inverse sine function graph shows angle output for sine input
    • Shape: Increasing, concave down for x>0x > 0, concave up for x<0x < 0
    • x-intercept: (0,0)(0, 0) origin point
    • Endpoints: (−1,−π2)(-1, -\frac{\pi}{2}) and (1,π2)(1, \frac{\pi}{2}) vertical asymptotes
  • Inverse cosine function graph displays angle result from cosine input
    • Shape: Decreasing, concave up for x<0x < 0, concave down for x>0x > 0
    • y-intercept: (1,0)(1, 0) right side of graph
    • Endpoints: (−1,π)(-1, \pi) and (1,0)(1, 0) vertical asymptotes
  • Inverse tangent function graph shows angle output for tangent input
    • Shape: Increasing, always concave down for x>0x > 0, always concave up for x<0x < 0
    • Origin: (0,0)(0, 0) center point
    • Horizontal asymptotes: y=π2y = \frac{\pi}{2} and y=−π2y = -\frac{\pi}{2} approaching but never reaching

Applications of inverse trigonometric functions

  • Use inverse functions to find unknown angles in right triangles
  • Steps for solving right triangle problems:
    1. Identify known sides and angles from given information
    2. Determine appropriate inverse function based on available data
    3. Apply function to find unknown angle using trigonometric ratios
    4. Use Pythagorean theorem if needed for missing side lengths
  • Example applications demonstrate real-world usage:
    • Navigation and bearings calculate ship or aircraft headings
    • Inclined planes determine angle of ramps or slopes
    • Elevation and depression angles measure vertical positioning (telescopes, surveying)
    • Satellite dish alignment for optimal signal reception
    • Robotics arm movement and positioning

Key Terms to Review (18)

Arccosine: Arccosine is the inverse function of cosine, denoted as $$\text{arccos}(x)$$, which returns the angle whose cosine is a given number. This function is essential for finding angles in right triangles when the length of the adjacent side and hypotenuse are known, and it connects to various aspects of trigonometric relationships, helping to solve problems involving angles and distances in both theoretical and practical applications.
Arcsine: Arcsine is the inverse function of the sine function, denoted as $$\arcsin(x)$$ or sometimes $$\sin^{-1}(x)$$. It gives the angle whose sine is the given number, allowing you to find an angle when you know the ratio of the opposite side to the hypotenuse in a right triangle. This connection makes it crucial for solving various problems related to angles, triangles, and trigonometric equations.
Arctangent: Arctangent is the inverse function of the tangent function, allowing you to find an angle when you know the ratio of the opposite side to the adjacent side in a right triangle. This function is essential for determining angles based on specific trigonometric ratios and connects deeply with concepts like inverse trigonometric functions, right triangle relationships, and evaluating trigonometric functions. It is denoted as $$ ext{arctan}(x)$$ or sometimes $$ an^{-1}(x)$$, providing a way to express angles in relation to their tangent values.
Cos^(-1)(x): The notation cos^(-1)(x) represents the inverse cosine function, which returns the angle whose cosine is x. This function is essential for solving problems where the cosine value is known, and you need to find the corresponding angle. The range of cos^(-1)(x) is restricted to the interval [0, π], ensuring that each value of x between -1 and 1 corresponds to one unique angle.
D/dx[cos^(-1)(x)]: The expression d/dx[cos^(-1)(x)] represents the derivative of the inverse cosine function, also known as arccosine. This derivative plays a crucial role in calculus, particularly in understanding how the inverse trigonometric functions relate to their original functions and their rates of change. The derivative of the inverse cosine function is especially useful for solving problems involving angles, distances, and various applications in physics and engineering.
D/dx[sin^(-1)(x)]: The derivative of the inverse sine function, represented as $$\frac{d}{dx}[sin^{-1}(x)]$$, provides the rate of change of the angle whose sine is x. This derivative is crucial for understanding how the inverse sine function behaves, particularly in terms of its range and domain, which are essential features when dealing with inverse trigonometric functions.
Domain: In mathematics, the domain refers to the complete set of possible values that an independent variable can take in a given function. Understanding the domain is crucial for analyzing functions, especially in trigonometry where certain functions are only defined for specific inputs, affecting their graphs and equations.
Finding angles in a right triangle: Finding angles in a right triangle involves determining the measures of the angles given certain side lengths or other angle measures. This process is heavily reliant on trigonometric functions, particularly the inverse functions, which allow us to calculate unknown angles based on the ratios of the sides of the triangle. Understanding how to find these angles is crucial for solving problems in geometry, physics, and various fields that utilize triangle properties.
Graph of Arccosine: The graph of arccosine, also known as the inverse cosine function, is a representation of the relationship between angles and their corresponding cosine values. It is defined for the range of angles from $0$ to $ rac{ ext{π}}{2}$ and from $ rac{ ext{π}}{2}$ to $ ext{π}$, producing outputs that range from $-1$ to $1$. This graph helps visualize how the arccosine function can be used to determine angles based on their cosine values, making it a crucial component of trigonometric functions and their inverses.
Graph of arcsine: The graph of arcsine is a visual representation of the inverse sine function, denoted as $$y = \arcsin(x)$$, which gives the angle whose sine is x. This graph maps the range of the arcsine function, which is limited to angles between $$-\frac{\pi}{2}$$ and $$\frac{\pi}{2}$$ radians (or -90° to 90°), making it distinct from the regular sine function that extends infinitely in both positive and negative directions. The graph has a specific shape that reflects the properties of inverse functions, including being one-to-one and having a restricted domain from -1 to 1, since sine values fall within this interval.
Graph of Arctangent: The graph of arctangent, or the inverse tangent function, is a visual representation of the values of the function $$y = \tan^{-1}(x)$$, which indicates the angle whose tangent is x. It is defined for all real numbers and has a characteristic S-shape, with horizontal asymptotes at $$y = \frac{-\pi}{2}$$ and $$y = \frac{\pi}{2}$$. The graph demonstrates how the arctangent function maps inputs from the entire x-axis to outputs between these two asymptotes.
Principal Value: Principal value refers to the specific angle or output that an inverse trigonometric function returns when determining the angle corresponding to a given trigonometric ratio. This concept is essential when dealing with inverse sine, cosine, and tangent functions, as it ensures a single, standardized output from these functions, facilitating the solving of equations involving trigonometric identities and ratios.
Range: The range of a function is the set of all possible output values it can produce, based on its domain. Understanding the range helps in determining the behavior of functions, especially in relation to inverse operations, periodicity, and transformations.
Sin(x) = y implies arcsin(y) = x: This statement explains the relationship between the sine function and its inverse, the arcsine function. It indicates that if the sine of an angle $x$ equals a value $y$, then the angle $x$ can be obtained by taking the arcsine of $y$. Understanding this connection is crucial for solving problems involving angles and their corresponding sine values, particularly in the context of inverse functions.
Sin^(-1)(x): The expression sin^(-1)(x), also known as the inverse sine function or arcsin(x), represents the angle whose sine is x. This function takes a value from the range of -1 to 1 and returns an angle in radians or degrees within a specific interval, typically from -π/2 to π/2. Understanding this function is essential for grasping the properties of inverse trigonometric functions and how they relate to their original counterparts.
Tan^(-1)(x): The term tan^(-1)(x), also known as the inverse tangent function, is used to determine the angle whose tangent is x. This function allows you to reverse the process of finding the tangent of an angle, providing a way to calculate angles from given ratios. The range of this function is limited to angles between -π/2 and π/2 radians, which reflects the properties of inverse functions and their restrictions.
Y = cos^(-1)(x): The expression y = cos^(-1)(x) defines the inverse cosine function, also known as arccosine. This function returns the angle whose cosine is x, where x is in the range of [-1, 1]. Inverse functions like this are essential for finding angles in trigonometry and are connected to the concept of solving triangles and modeling periodic phenomena.
Y = sin^(-1)(x): The expression y = sin^(-1)(x) represents the inverse sine function, also known as arcsine. It is used to find the angle whose sine value is x, where x must be in the range of -1 to 1. This function essentially 'reverses' the sine operation, allowing us to determine angles based on known sine values.
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