5.5.4 Inverse Trig Functions

Inverse trigonometric functions are the inverse operations of the basic trigonometric functions (sine, cosine, tangent, etc.). They are used to find the angle that corresponds to a given trigonometric value. These functions are important in many areas of mathematics, including calculus, geometry, and solving trigonometric equations.

1. Definitions of Inverse Trigonometric Functions:

• Inverse Sine $\sin^{-1} x\ or \ (arcsin x)$: $\sin^{-1} x = \theta \quad \text{if and only if} \quad \sin \theta = x$
• The inverse sine function returns the angle whose sine is $x$.
• Domain: -1 ≤ x ≤ 1
• Range: -π/2 ≤ θ ≤ π/2 (or -90° ≤ θ ≤ 90°)
• Inverse Cosine ($\cos^{-1} x\ or \ (arccos x)$: $\cos^{-1} x = \theta \quad \text{if and only if} \quad \cos \theta = x$
• The inverse cosine function returns the angle whose cosine is $x$.
• Domain: -1 ≤ x ≤ 1
• Range: 0 ≤ θ ≤ π (or 0° ≤ θ ≤ 180°)

• Inverse Tangent ($\tan^{-1} x\ or \ (\arctan x)$: $\tan^{-1} x = \theta \quad \text{if and only if} \quad \tan \theta = x$
• The inverse tangent function returns the angle whose tangent is $x$.
• Domain: -∞ < x < ∞
• Range: -π/2 < θ < π/2 (or -90° < θ < 90°)
• Inverse Cosecant ($\csc^{-1} x\ or\ arccsc x$): $\csc^{-1} x = \theta \quad \text{if and only if} \quad \csc \theta = x$
• Domain: x ≤ -1 or x ≥ 1
• Range: -π/2 ≤ θ ≤ π/2, θ ≠ 0 (or -90° ≤ θ ≤ 90°, θ ≠ 0°)

• Inverse Secant $(\sec^{-1} x\ or \ arcsec x$): $\sec^{-1} x = \theta \quad \text{if and only if} \quad \sec \theta = x$
• Domain: x ≤ -1 or x ≥ 1
• Range: 0 ≤ θ ≤ π, θ ≠ π/2 (or 0° ≤ θ ≤ 180°, θ ≠ 90°)
• Inverse Cotangent ($\cot^{-1} x\ or \ arccot x)$): $\cot^{-1} x = \theta \quad \text{if and only if} \quad \cot \theta = x$
• Domain: -∞ < x < ∞
• Range: 0 < θ < π (or 0° < θ < 180°)

2. Graphs of Inverse Trigonometric Functions:

• Inverse Sine $\ (\arcsin x):$
• The graph of $y = \arcsin x$ is a smooth curve that starts at (-1, -π/2) and ends at (1, π/2).

• Inverse Cosine ($\arccos x$):

• The graph of $y = \arccos x$ is a decreasing curve that starts at (-1, π) and ends at (1, 0).

• Inverse Tangent ($\arctan x$):

• The graph of y = $\arctan x$ is an increasing curve that approaches horizontal asymptotes at y = -π/2 and y = π/2 as x approaches negative and positive infinity, respectively.

  

3. Key Properties and Identities:

• Inverse Function Property: $\sin(\arcsin x) = x, \quad \cos(\arccos x) = x, \quad \tan(\arctan x) = x$ These are true for all x within the domain of the inverse function.
• Composite Function Property: $\arcsin(\sin \theta) = \theta \quad \text{for} \quad -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$ $\arccos(\cos \theta) = \theta \quad \text{for} \quad 0 \leq \theta \leq \pi$ $\arctan(\tan \theta) = \theta \quad \text{for} \quad -\frac{\pi}{2} < \theta < \frac{\pi}{2}$
• Pythagorean Identity: $\arcsin x + \arccos x = \frac{\pi}{2}$
  • This identity shows the relationship between the inverse sine and inverse cosine functions.

4. Example Problems Using Inverse Trigonometric Functions:

Summary:

• Inverse trigonometric functions allow you to find angles from given trigonometric values.
• These functions have specific domains and ranges, which correspond to the possible values of the angle.
• Understanding the graphs, properties, and key identities of these functions is crucial for solving trigonometric equations and simplifying expressions involving angles.

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Inverse Trig Functions

• Only 1-to-1 functions have inverses.

$$\begin{align*} y &= \sin(x) \\ y &= \cos(x) \\ y &= \tan(x) \end{align*}$$

• As they stand, these functions are not 1 to 1, therefore do not have an inverse. We must restrict the domain of these functions for an inverse to exist.

$$y = \sin(x)$$ $$-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$$

• This version of the sine function when its domain is restricted to -π/2 ≤ x ≤ π/2 is 1 to 1, so has an inverse.

$$\begin{align*} y &= \sin^{-1}(x) \\ y &= \arcsin(x) \\ -1 &\leq x \leq 1 \end{align*}$$

• Note: The domain of the inverse function is the range of the original.

  

• To draw an inverse trig function, we reflect the original function through y = x.

  • (Hint: Look at the key points of the original graph and swap x and y coordinates.)

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$$y = \cos x \quad \text{is 1-to-1 between}\\ \quad 0 \leq x \leq \pi$$ $$\begin{align*} y &= \cos^{-1}(x) \\ y &= \arccos(x) \\ -1 &\leq x \leq 1 \end{align*}$$

Graph description:

• The first graph shows $y = \cos x$ between 0 and π, depicting the cosine curve restricted to one period.
• The second graph depicts $y = \cos^{-1}(x)\ or\ y = \arccos(x)$, which is the inverse function of $\cos x$, showing the range from (-1, π) to (1, 0).

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$$y = \tan x \quad \text{is 1-to-1 between} \quad -\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$$ $$\begin{align*} y &= \tan^{-1}(x) \\ y &= \arctan(x) \\ x &\in \mathbb{R} \end{align*}$$

Graph description:

• The third graph shows the function $y = \tan x$ between -π/2 and π/2, illustrating the tangent function's behaviour within these limits.
• The fourth graph displays $y = \tan^{-1}(x)\ or\ y = \arctan(x)$, indicating the tangent's inverse function over all real numbers.

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Hint: Perform the inverse operation of $\arcsin$ to both sides.

$$\sin(\arcsin(x)) = \sin\left(\frac{\pi}{4}\right) \implies x = \frac{\sqrt{2}}{2}$$

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Recap of Solving Trig Equations

Domain: $0 \leq x \leq 2\pi$

$$x = \arcsin(0.2) \approx 0.2014$$

Graph Description:

• The graph shows the sine function, $y = \sin x$, with marked intersections at x = 0.2014 and another at π - 0.2014.
• The first intersection at 0.2014 is where $\sin x = 0.2$.
• The second intersection at π - 0.2014 ≈ 2.940 is where $\sin x = 0.2$ again.

$$x = 0.2014, \, \pi - 0.2014$$ $$x = 0.2014, \, 2.940$$

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