Reciprocal & Inverse Trigonometric Functions (Edexcel A-Level Mathematics): Revision Notes

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: $-1 \leq x \leq 1$
• Range: $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$ (or $-90^\circ \leq \theta \leq 90^\circ$)
• 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: $-1 \leq x \leq 1$
• Range: $0 \leq \theta \leq \pi$ (or $0^\circ \leq \theta \leq 180^\circ$)

• 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: $-\infty < x < \infty$
• Range: $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$ (or $-90^\circ < \theta < 90^\circ$)
• Inverse Cosecant ($\csc^{-1} x\ or\ arccsc x$): $\csc^{-1} x = \theta \quad \text{if and only if} \quad \csc \theta = x$
• Domain: $x \leq -1$ or $x \geq 1$
• Range: $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}, \theta \neq 0$ (or $-90^\circ \leq \theta \leq 90^\circ, \theta \neq 0^\circ$)

• Inverse Secant $(\sec^{-1} x\ or \ arcsec x$): $\sec^{-1} x = \theta \quad \text{if and only if} \quad \sec \theta = x$
• Domain: $x \leq -1$ or $x \geq 1$
• Range: $0 \leq \theta \leq \pi, \theta \neq \frac{\pi}{2}$ (or $0^\circ \leq \theta \leq 180^\circ, \theta \neq 90^\circ$)
• Inverse Cotangent ($\cot^{-1} x\ or \ arccot x)$): $\cot^{-1} x = \theta \quad \text{if and only if} \quad \cot \theta = x$
• Domain: $-\infty < x < \infty$
• Range: $0 < \theta < \pi$ (or $0^\circ < \theta < 180^\circ$)

2. Graphs of Inverse Trigonometric Functions:

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

• Inverse Cosine ($\arccos x$):

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

• Inverse Tangent ($\arctan x$):

• The graph of y = $\arctan x$ is an increasing curve that approaches horizontal asymptotes at $y = -\frac{\pi}{2}$ and $y = \frac{\pi}{2}$ 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 $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ 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 $\pi$, 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 ($-1, \pi$) 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 $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, 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 $\pi - 0.2014$.
• The first intersection at $0.2014$ is where $\sin x = 0.2$.
• The second intersection at $\pi - 0.2014 \approx 2.940$ is where $\sin x = 0.2$ again.

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

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