Inverse Trigonometric Functions | Malone Math

The Central Idea

The inverse trigonometric functions reverse sine, cosine, and tangent on carefully chosen intervals.

The three principal inverse functions are

\[ \boxed{ y=\arcsin x, \qquad y=\arccos x, \qquad y=\arctan x }. \]

They may also be written

\[ \boxed{ y=\sin^{-1}x, \qquad y=\cos^{-1}x, \qquad y=\tan^{-1}x }. \]

Their outputs are angles.

What an Inverse Function Does

If a function \(f\) maps an input \(a\) to an output \(b\),

\[ f(a)=b, \]

then its inverse reverses the relationship:

\[ f^{-1}(b)=a. \]

For example, since

\[ \sin\left(\frac{\pi}{6}\right)=\frac12, \]

we have

\[ \boxed{ \arcsin\left(\frac12\right)=\frac{\pi}{6} }. \]

Inverse Does Not Mean Reciprocal

The notation

\[ \sin^{-1}x \]

means inverse sine.

It does not mean

\[ \frac1{\sin x}. \]

The reciprocal of sine is cosecant:

\[ \boxed{ \frac1{\sin x}=\csc x }. \]

Similarly,

\[ \frac1{\cos x}=\sec x \]

and

\[ \frac1{\tan x}=\cot x. \]

Why Trigonometric Functions Need Restricted Domains

A function must be one-to-one in order to have an inverse function.

Sine, cosine, and tangent repeat periodically, so over their full natural domains they assign the same output to many different inputs.

For example,

\[ \sin\left(\frac{\pi}{6}\right) = \sin\left(\frac{5\pi}{6}\right) = \frac12. \]

If inverse sine were allowed to return both angles, it would not be a function.

Principal Values

To create inverse trigonometric functions, we choose a restricted interval on which each original function is one-to-one.

The inverse then returns the unique angle from that interval.

This selected output is called the principal value.

Restricted Domain for Sine

To define inverse sine, sine is restricted to

\[ \boxed{ -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} }. \]

On this interval, sine increases from \(-1\) to \(1\) and is one-to-one.

Domain and Range of Arcsine

Since the restricted sine function has outputs from \(-1\) to \(1\), inverse sine has domain

\[ \boxed{[-1,1]}. \]

Its outputs are the restricted sine angles:

\[ \boxed{ \operatorname{range}(\arcsin) = \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] }. \]

Meaning of Arcsine

The statement

\[ y=\arcsin x \]

means

\[ \sin y=x \]

with

\[ -\frac{\pi}{2}\leq y\leq\frac{\pi}{2}. \]

The range condition is part of the definition.

Example: \(\arcsin(1/2)\)

We seek the angle in

\[ \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \]

whose sine equals \(1/2\).

Since

\[ \sin\left(\frac{\pi}{6}\right)=\frac12, \]

we obtain

\[ \boxed{ \arcsin\left(\frac12\right)=\frac{\pi}{6} }. \]

Example: \(\arcsin(-1/2)\)

We need the principal angle whose sine is negative.

Within the arcsine range, that angle is

\[ -\frac{\pi}{6}. \]

Therefore,

\[ \boxed{ \arcsin\left(-\frac12\right) = -\frac{\pi}{6} }. \]

Restricted Domain for Cosine

To define inverse cosine, cosine is restricted to

\[ \boxed{0\leq\theta\leq\pi}. \]

On this interval, cosine decreases continuously from \(1\) to \(-1\), so it is one-to-one.

Domain and Range of Arccosine

Arccosine has domain

\[ \boxed{[-1,1]} \]

and range

\[ \boxed{ \operatorname{range}(\arccos) = [0,\pi] }. \]

Meaning of Arccosine

The statement

\[ y=\arccos x \]

means

\[ \cos y=x \]

with

\[ 0\leq y\leq\pi. \]

Example: \(\arccos(1/2)\)

Since

\[ \cos\left(\frac{\pi}{3}\right)=\frac12 \]

and

\[ \frac{\pi}{3}\in[0,\pi], \]

we have

\[ \boxed{ \arccos\left(\frac12\right)=\frac{\pi}{3} }. \]

Example: \(\arccos(-1/2)\)

Within the principal cosine interval,

\[ \cos\left(\frac{2\pi}{3}\right)=-\frac12. \]

Therefore,

\[ \boxed{ \arccos\left(-\frac12\right)=\frac{2\pi}{3} }. \]

Restricted Domain for Tangent

Tangent is one-to-one on

\[ \boxed{ -\frac{\pi}{2} < \theta < \frac{\pi}{2} }. \]

The endpoints are excluded because tangent is undefined there.

Domain and Range of Arctangent

Tangent takes every real output value on its restricted interval.

Therefore arctangent has domain

\[ \boxed{(-\infty,\infty)} \]

and range

\[ \boxed{ \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) }. \]

Meaning of Arctangent

The statement

\[ y=\arctan x \]

means

\[ \tan y=x \]

with

\[ -\frac{\pi}{2}

Example: \(\arctan(1)\)

Since

\[ \tan\left(\frac{\pi}{4}\right)=1, \]

we have

\[ \boxed{ \arctan(1)=\frac{\pi}{4} }. \]

Example: \(\arctan(-1)\)

Since

\[ \tan\left(-\frac{\pi}{4}\right)=-1, \]

and the angle lies in the principal arctangent interval,

\[ \boxed{ \arctan(-1)=-\frac{\pi}{4} }. \]

Principal Ranges at a Glance

\[ \boxed{ \begin{array}{c|c|c} \text{Function} & \text{Domain} & \text{Range}\\ \hline \arcsin x & [-1,1] & \left[-\frac{\pi}{2},\frac{\pi}{2}\right]\\[6pt] \arccos x & [-1,1] & [0,\pi]\\[6pt] \arctan x & (-\infty,\infty) & \left(-\frac{\pi}{2},\frac{\pi}{2}\right) \end{array} }. \]

Exact Arcsine Values

Important exact values include

\[ \boxed{ \arcsin(0)=0 }, \] \[ \boxed{ \arcsin\left(\frac12\right)=\frac{\pi}{6} }, \] \[ \boxed{ \arcsin\left(\frac{\sqrt2}{2}\right)=\frac{\pi}{4} }, \] \[ \boxed{ \arcsin\left(\frac{\sqrt3}{2}\right)=\frac{\pi}{3} }, \]

and

\[ \boxed{\arcsin(1)=\frac{\pi}{2}}. \]

Exact Arccosine Values

Important exact values include

\[ \boxed{\arccos(1)=0}, \] \[ \boxed{ \arccos\left(\frac{\sqrt3}{2}\right)=\frac{\pi}{6} }, \] \[ \boxed{ \arccos\left(\frac{\sqrt2}{2}\right)=\frac{\pi}{4} }, \] \[ \boxed{ \arccos\left(\frac12\right)=\frac{\pi}{3} }, \]

and

\[ \boxed{\arccos(0)=\frac{\pi}{2}}. \]

Exact Arctangent Values

Important exact values include

\[ \boxed{\arctan(0)=0}, \] \[ \boxed{ \arctan\left(\frac{\sqrt3}{3}\right)=\frac{\pi}{6} }, \] \[ \boxed{\arctan(1)=\frac{\pi}{4}}, \]

and

\[ \boxed{ \arctan(\sqrt3)=\frac{\pi}{3} }. \]

Negative Exact Values

Because arcsine and arctangent have ranges symmetric about zero,

\[ \boxed{\arcsin(-x)=-\arcsin x} \]

and

\[ \boxed{\arctan(-x)=-\arctan x}. \]

For example,

\[ \arctan(-\sqrt3) = \boxed{-\frac{\pi}{3}}. \]

Arccosine Handles Negative Inputs Differently

Arccosine returns values in

\[ [0,\pi]. \]

Therefore negative cosine inputs produce principal angles in Quadrant II.

For example,

\[ \boxed{ \arccos\left(-\frac{\sqrt2}{2}\right) = \frac{3\pi}{4} }. \]

Inverse Functions Reverse Their Originals

When an input is in the appropriate domain,

\[ \boxed{\sin(\arcsin x)=x} \]

for

\[ -1\leq x\leq1. \]

Similarly,

\[ \boxed{\cos(\arccos x)=x} \]

for

\[ -1\leq x\leq1, \]

and

\[ \boxed{\tan(\arctan x)=x} \]

for every real \(x\).

Composition in the Other Direction Requires Care

The expression

\[ \arcsin(\sin\theta) \]

equals \(\theta\) only when

\[ \theta\in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]. \]

Outside that interval, arcsine returns the principal angle with the same sine value rather than the original angle.

Example: \(\arcsin(\sin(5\pi/6))\)

First,

\[ \sin\left(\frac{5\pi}{6}\right)=\frac12. \]

Then

\[ \arcsin\left(\frac12\right)=\frac{\pi}{6}. \]

Therefore,

\[ \boxed{ \arcsin\left( \sin\left(\frac{5\pi}{6}\right) \right) = \frac{\pi}{6} }. \]

It does not return \(5\pi/6\) because that angle lies outside the principal arcsine range.

Arccosine Composition

The composition

\[ \arccos(\cos\theta) \]

equals \(\theta\) when

\[ \boxed{0\leq\theta\leq\pi}. \]

Outside that interval, the result is the principal angle in \([0,\pi]\) with the same cosine value.

Example: \(\arccos(\cos(5\pi/3))\)

First,

\[ \cos\left(\frac{5\pi}{3}\right)=\frac12. \]

Then

\[ \arccos\left(\frac12\right)=\frac{\pi}{3}. \]

Therefore,

\[ \boxed{ \arccos\left( \cos\left(\frac{5\pi}{3}\right) \right) = \frac{\pi}{3} }. \]

Arctangent Composition

The composition

\[ \arctan(\tan\theta) \]

equals \(\theta\) only when

\[ \boxed{ -\frac{\pi}{2} < \theta < \frac{\pi}{2} }. \]

Otherwise, arctangent returns a coterminally related principal angle in that interval.

Example: \(\arctan(\tan(3\pi/4))\)

Since

\[ \tan\left(\frac{3\pi}{4}\right)=-1, \]

we obtain

\[ \arctan(-1)=-\frac{\pi}{4}. \]

Therefore,

\[ \boxed{ \arctan\left( \tan\left(\frac{3\pi}{4}\right) \right) = -\frac{\pi}{4} }. \]

Inverse Trigonometric Graphs

The graph of an inverse function is obtained by reflecting the graph of the corresponding restricted original function across the line

\[ y=x. \]

Therefore the graphs of arcsine, arccosine, and arctangent are reflections of the appropriately restricted sine, cosine, and tangent graphs.

Graph of Arcsine

The arcsine graph has domain

\[ [-1,1] \]

and range

\[ \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]. \]

It passes through

\[ (-1,-\frac{\pi}{2}), \qquad (0,0), \qquad (1,\frac{\pi}{2}) \]

and increases across its domain.

Graph of Arccosine

The arccosine graph has domain

\[ [-1,1] \]

and range

\[ [0,\pi]. \]

It passes through

\[ (-1,\pi), \qquad (0,\frac{\pi}{2}), \qquad (1,0) \]

and decreases across its domain.

Graph of Arctangent

Arctangent has domain

\[ (-\infty,\infty) \]

and range

\[ \left( -\frac{\pi}{2}, \frac{\pi}{2} \right). \]

Its graph passes through the origin and increases continuously.

It approaches the horizontal lines

\[ y=\frac{\pi}{2} \]

and

\[ y=-\frac{\pi}{2} \]

without reaching them.

Horizontal Asymptotes of Arctangent

\[ x\to\infty, \]

we have

\[ \arctan x\to\frac{\pi}{2}. \]

\[ x\to-\infty, \]

we have

\[ \arctan x\to-\frac{\pi}{2}. \]

Thus

\[ \boxed{ y=\pm\frac{\pi}{2} } \]

are horizontal asymptotes of the arctangent graph.

Calculator Use

Scientific calculators typically provide inverse trigonometric operations using keys such as

\[ \sin^{-1}, \qquad \cos^{-1}, \qquad \tan^{-1}. \]

These keys return principal values.

The calculator angle mode determines whether the output is displayed in degrees or radians.

Example: Calculator Arcsine

Suppose

\[ \sin\theta=0.42 \]

and we want the principal angle.

Then

\[ \theta=\arcsin(0.42). \]

In degree mode,

\[ \boxed{\theta\approx24.8^\circ}. \]

Example: Calculator Arctangent

Suppose

\[ \tan\theta=\frac{7}{11}. \]

Then

\[ \theta=\arctan\left(\frac7{11}\right). \]

In degree mode,

\[ \boxed{\theta\approx32.5^\circ}. \]

Inverse Trigonometry in Right Triangles

In a right triangle, inverse trigonometric functions recover acute angles from side ratios.

\[ \sin\theta=\frac{O}{H}, \]

then

\[ \boxed{ \theta=\arcsin\left(\frac{O}{H}\right) }. \]

Similarly,

\[ \boxed{ \theta=\arccos\left(\frac{A}{H}\right) } \]

\[ \boxed{ \theta=\arctan\left(\frac{O}{A}\right) }. \]

Example: Angle from a \(5\)-\(12\)-\(13\) Triangle

Let \(\theta\) be opposite the side of length \(5\) and adjacent to the side of length \(12\).

Then

\[ \tan\theta=\frac5{12}. \]

Therefore,

\[ \theta = \arctan\left(\frac5{12}\right) \approx \boxed{22.6^\circ}. \]

One Ratio Can Be Used in Several Ways

In a \(5\)-\(12\)-\(13\) triangle,

\[ \sin\theta=\frac5{13}, \] \[ \cos\theta=\frac{12}{13}, \]

and

\[ \tan\theta=\frac5{12}. \]

Therefore,

\[ \arcsin\left(\frac5{13}\right), \qquad \arccos\left(\frac{12}{13}\right), \qquad \arctan\left(\frac5{12}\right) \]

all return the same acute angle, apart from rounding differences.

Solving a Basic Trigonometric Equation with an Inverse

Suppose

\[ \sin x=0.3. \]

Applying inverse sine gives the principal solution

\[ x=\arcsin(0.3). \]

But because sine is periodic and takes the same positive value in two quadrants during one revolution, this principal value is not automatically the complete solution set.

Principal Value versus All Solutions

Inverse trigonometric functions return one principal angle.

A trigonometric equation may have many solutions because the original functions repeat.

For example,

\[ \sin x=\frac12 \]

has principal inverse value

\[ \arcsin\left(\frac12\right)=\frac{\pi}{6}, \]

but during one revolution sine also equals \(1/2\) at

\[ \frac{5\pi}{6}. \]

Periodicity then generates infinitely many solutions.

Finding the Other Sine Angle

\[ \alpha=\arcsin(a) \]

is an acute principal angle for a positive sine value, then the other angle in one revolution is

\[ \boxed{\pi-\alpha}. \]

This follows from unit-circle symmetry.

Finding Cosine Solutions from a Principal Angle

\[ \alpha=\arccos(a), \]

then \(\alpha\) is already the principal angle in

\[ [0,\pi]. \]

Within one full revolution, the other angle with the same cosine value is often obtained by reflection across the \(x\)-axis:

\[ \boxed{2\pi-\alpha}, \]

when \(\alpha\) is not an axis angle.

Tangent Solutions Repeat after \(\pi\)

\[ \alpha=\arctan(a), \]

then every tangent solution can be generated by

\[ \boxed{x=\alpha+k\pi,\qquad k\in\mathbb Z}. \]

This reflects tangent's period of \(\pi\).

Inverse Trigonometry and Quadrant Information

Sometimes a problem provides a trigonometric value together with a quadrant.

The inverse function gives a principal reference-related angle, while the quadrant determines the actual desired angle.

This is especially useful when recovering directions outside the principal ranges.

Example: Positive Sine in Quadrant II

Suppose

\[ \sin\theta=0.6 \]

and \(\theta\) lies in Quadrant II.

First find the principal acute angle:

\[ \alpha=\arcsin(0.6)\approx36.87^\circ. \]

Quadrant II gives

\[ \theta=180^\circ-\alpha. \]

Therefore,

\[ \boxed{\theta\approx143.13^\circ}. \]

Example: Negative Cosine in Quadrant III

Suppose

\[ \cos\theta=-0.4 \]

and \(\theta\) is known to lie in Quadrant III.

A calculator gives

\[ \arccos(-0.4)\approx113.58^\circ, \]

which is the principal arccosine angle in Quadrant II.

For a Quadrant-III angle with the same reference magnitude, the desired angle is

\[ 360^\circ-113.58^\circ \approx \boxed{246.42^\circ}. \]

Example: Negative Tangent in Quadrant IV

Suppose

\[ \tan\theta=-2 \]

and \(\theta\) lies in Quadrant IV.

The principal arctangent is

\[ \arctan(-2)\approx-63.43^\circ. \]

A positive coterminal angle is

\[ 360^\circ-63.43^\circ = \boxed{296.57^\circ}. \]

Finding Trigonometric Values of Inverse Expressions

Expressions such as

\[ \cos(\arcsin x) \]

can often be simplified using a right triangle or the Pythagorean identity.

Let

\[ \theta=\arcsin x. \]

Then

\[ \sin\theta=x. \]

Example: Simplify \(\cos(\arcsin x)\)

Let

\[ \theta=\arcsin x. \]

Then

\[ \sin\theta=x \]

and

\[ -\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}. \]

Thus cosine is nonnegative.

From

\[ \sin^2\theta+\cos^2\theta=1, \]

we get

\[ \cos\theta=\sqrt{1-x^2}. \]

Therefore,

\[ \boxed{ \cos(\arcsin x)=\sqrt{1-x^2} } \]

for

\[ -1\leq x\leq1. \]

Example: Simplify \(\sin(\arccos x)\)

Let

\[ \theta=\arccos x. \]

Then

\[ \cos\theta=x \]

with

\[ 0\leq\theta\leq\pi. \]

Sine is nonnegative on this interval.

Therefore,

\[ \boxed{ \sin(\arccos x)=\sqrt{1-x^2} }. \]

Example: Simplify \(\tan(\arcsin x)\)

Let

\[ \theta=\arcsin x. \]

Then

\[ \sin\theta=x \]

and

\[ \cos\theta=\sqrt{1-x^2}. \]

Therefore,

\[ \tan\theta = \frac{\sin\theta}{\cos\theta} = \boxed{ \frac{x}{\sqrt{1-x^2}} }. \]

Example: Simplify \(\sin(\arctan x)\)

Let

\[ \theta=\arctan x. \]

Then

\[ \tan\theta=x. \]

Think of a right triangle with opposite side \(x\), adjacent side \(1\), and hypotenuse

\[ \sqrt{x^2+1}. \]

Therefore,

\[ \boxed{ \sin(\arctan x) = \frac{x}{\sqrt{x^2+1}} }. \]

Example: Simplify \(\cos(\arctan x)\)

Using the same triangle,

\[ A=1 \]

and

\[ H=\sqrt{x^2+1}. \]

Because arctangent returns angles between

\[ -\frac{\pi}{2} \qquad\text{and}\qquad \frac{\pi}{2}, \]

cosine is positive.

Thus,

\[ \boxed{ \cos(\arctan x) = \frac1{\sqrt{x^2+1}} }. \]

Inverse Trigonometric Functions and Measurement

Inverse trigonometry is especially useful when lengths, slopes, component ratios, or coordinate differences are known and a direction is needed.

Applications include:

surveying;
navigation;
engineering slopes;
vector directions;
angles of elevation and depression;
computer graphics and geometry;
physical orientation problems.

Example: Angle of Inclination

Suppose a road rises \(7\) meters for every \(80\) meters of horizontal run.

Then

\[ \tan\theta=\frac7{80}. \]

Therefore,

\[ \theta = \arctan\left(\frac7{80}\right) \approx \boxed{5.0^\circ}. \]

Example: Direction of a Vector

Suppose a vector has horizontal component \(12\) and vertical component \(5\), both positive.

Its acute direction angle satisfies

\[ \tan\theta=\frac5{12}. \]

Thus,

\[ \theta = \arctan\left(\frac5{12}\right) \approx \boxed{22.6^\circ}. \]

Quadrant Matters for Vector Direction

The ratio

\[ \frac{y}{x} \]

alone may not uniquely identify a direction because tangent has the same value in opposite quadrants.

For example,

\[ \frac{5}{12} \]

can describe a Quadrant-I direction, while

\[ \frac{-5}{-12} \]

produces the same tangent ratio in Quadrant III.

Coordinate signs or other quadrant information must therefore be retained.

Degrees versus Radians

Inverse trigonometric functions return angle measures.

A calculator may report those angles in degrees or radians depending on its mode.

For example,

\[ \arctan(1) \]

may appear as

\[ 45^\circ \]

in degree mode or

\[ \frac{\pi}{4} \]

in exact radian form.

Exact versus Approximate Inverse Values

If an input matches a known special-angle value, use an exact answer when possible.

For example,

\[ \arcsin\left(\frac{\sqrt2}{2}\right) = \boxed{\frac{\pi}{4}}. \]

For a non-special value such as

\[ \arcsin(0.37), \]

a calculator approximation is appropriate.

Common Error: Treating Inverse Sine as Cosecant

The expressions

\[ \sin^{-1}x \]

and

\[ \csc x \]

are not equivalent.

Inverse sine returns an angle.

Cosecant returns a reciprocal trigonometric value.

Common Error: Expecting Every Original Angle Back

It is not always true that

\[ \arcsin(\sin\theta)=\theta. \]

This equality requires

\[ \theta\in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]. \]

Outside that interval, arcsine returns the corresponding principal value.

Common Error: Ignoring the Principal Range

Consider

\[ \arccos\left(-\frac12\right). \]

Both

\[ \frac{2\pi}{3} \]

and

\[ \frac{4\pi}{3} \]

have cosine \(-1/2\).

But arccosine must return a value in

\[ [0,\pi]. \]

Therefore,

\[ \boxed{ \arccos\left(-\frac12\right) = \frac{2\pi}{3} }. \]

Common Error: Giving Arcsine Inputs Outside \([-1,1]\)

Sine never exceeds \(1\) or falls below \(-1\).

Therefore, over the real numbers,

\[ \arcsin(1.4) \]

is undefined.

The same restriction applies to arccosine.

Common Error: Assuming Arctangent Has Input Restrictions Like Arcsine

Tangent can produce every real number on its principal interval.

Therefore,

\[ \arctan x \]

is defined for every real \(x\).

Common Error: Forgetting Calculator Angle Mode

If a problem asks for an answer in degrees but the calculator is set to radians, the numerical result may be mathematically valid but expressed in the wrong unit.

Always confirm the intended angle unit before evaluating inverse trigonometric functions numerically.

Common Error: Using Only a Principal Value for a Periodic Equation

Applying

\[ x=\arcsin(a) \]

returns one principal angle.

It does not automatically list every solution of

\[ \sin x=a. \]

Equation solving must also account for unit-circle symmetry and periodicity.

Common Error: Losing the Sign in Composite Simplification

When simplifying an expression such as

\[ \sqrt{1-\sin^2\theta}, \]

the result is

\[ |\cos\theta|, \]

not always \(\cos\theta\).

In inverse-function compositions, principal-range restrictions often tell us which sign is valid.

A Reliable Procedure for Evaluating an Inverse Trigonometric Function

Identify whether the function is arcsine, arccosine, or arctangent.
Check that the input lies in the inverse function's domain.
Recall the required principal-value range.
Look for a known exact special-angle value.
Otherwise use a calculator in the correct angle mode.
Check that the returned angle lies in the correct principal range.

A Reliable Procedure for Solving for an Angle in a Right Triangle

Choose the unknown acute angle.
Identify the two known side roles.
Form the appropriate sine, cosine, or tangent ratio.
Apply the matching inverse trigonometric function.
Evaluate in the required angle unit.
Check that the result is acute and geometrically reasonable.

A Reliable Procedure for Composite Inverse Expressions

Introduce an angle variable for the inverse expression.
Translate the inverse statement into an ordinary trigonometric equation.
Record the principal range of the introduced angle.
Use a right triangle or Pythagorean identity to find missing quantities.
Use the principal range to choose the correct sign.
Evaluate the outer trigonometric function.

Practice 1: Exact Arcsine

Evaluate exactly:

\[ \arcsin\left(\frac{\sqrt3}{2}\right). \]

Practice 2: Exact Arccosine

Evaluate exactly:

\[ \arccos\left(-\frac{\sqrt2}{2}\right). \]

Practice 3: Exact Arctangent

Evaluate exactly:

\[ \arctan(-1). \]

Practice 4: Domain Check

\[ \arcsin\left(\frac54\right) \]

defined over the real numbers?

Practice 5: Composition

Simplify:

\[ \sin(\arcsin 0.4). \]

Practice 6: Principal-Value Composition

Evaluate:

\[ \arcsin\left(\sin\frac{5\pi}{6}\right). \]

Practice 7: Another Principal-Value Composition

Evaluate:

\[ \arccos\left(\cos\frac{5\pi}{3}\right). \]

Practice 8: Right-Triangle Angle

A right triangle has opposite side \(8\) and adjacent side \(15\).

Find the acute angle \(\theta\) to the nearest tenth of a degree.

Practice 9: Composite Expression

Simplify:

\[ \cos(\arcsin x). \]

Practice 10: Composite Arctangent Expression

Simplify:

\[ \sin(\arctan x). \]

Practice 11: Quadrant Adjustment

Suppose

\[ \sin\theta=0.8 \]

and

\[ 90^\circ<\theta<180^\circ. \]

Find \(\theta\) to the nearest tenth of a degree.

Practice 12: Identify the Error

A student writes

\[ \sin^{-1}x=\frac1{\sin x}. \]

Explain the error and write the correct reciprocal relationship.

Practice Answers

Practice 1

\[ \boxed{ \arcsin\left(\frac{\sqrt3}{2}\right) = \frac{\pi}{3} }. \]

Practice 2

Arccosine returns values in \([0,\pi]\). The angle there with cosine \(-\sqrt2/2\) is

\[ \boxed{ \arccos\left(-\frac{\sqrt2}{2}\right) = \frac{3\pi}{4} }. \]

Practice 3

\[ \boxed{ \arctan(-1) = -\frac{\pi}{4} }. \]

Practice 4

No. The real domain of arcsine is

\[ [-1,1]. \]

Since

\[ \frac54>1, \]

we have

\[ \boxed{ \arcsin\left(\frac54\right) \text{ is not real} }. \]

Practice 5

\[ \boxed{ \sin(\arcsin0.4)=0.4 }. \]

Practice 6

\[ \sin\left(\frac{5\pi}{6}\right)=\frac12. \]

Therefore,

\[ \boxed{ \arcsin\left(\sin\frac{5\pi}{6}\right) = \frac{\pi}{6} }. \]

Practice 7

\[ \cos\left(\frac{5\pi}{3}\right)=\frac12. \]

Therefore,

\[ \boxed{ \arccos\left(\cos\frac{5\pi}{3}\right) = \frac{\pi}{3} }. \]

Practice 8

\[ \tan\theta=\frac8{15}. \] \[ \theta = \arctan\left(\frac8{15}\right) \approx \boxed{28.1^\circ}. \]

Practice 9

Let

\[ \theta=\arcsin x. \]

Then

\[ \sin\theta=x. \]

Since \(\theta\) lies in the principal arcsine range, cosine is nonnegative. Thus,

\[ \boxed{ \cos(\arcsin x)=\sqrt{1-x^2} }. \]

Practice 10

Let

\[ \theta=\arctan x. \]

Use a right triangle with opposite side \(x\), adjacent side \(1\), and hypotenuse

\[ \sqrt{x^2+1}. \]

Therefore,

\[ \boxed{ \sin(\arctan x) = \frac{x}{\sqrt{x^2+1}} }. \]

Practice 11

The principal acute angle is

\[ \alpha=\arcsin(0.8)\approx53.1^\circ. \]

Since \(\theta\) lies in Quadrant II,

\[ \theta=180^\circ-\alpha. \]

Therefore,

\[ \boxed{\theta\approx126.9^\circ}. \]

Practice 12

The notation

\[ \sin^{-1}x \]

means inverse sine, which returns an angle.

The reciprocal of sine is cosecant:

\[ \boxed{ \frac1{\sin x}=\csc x }. \]

What to Remember

The principal inverse trigonometric functions are

\[ \boxed{ \arcsin x,\qquad \arccos x,\qquad \arctan x }. \]

Their principal ranges are

\[ \boxed{ \arcsin x: \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] }, \] \[ \boxed{ \arccos x: [0,\pi] }, \]

and

\[ \boxed{ \arctan x: \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) }. \]

Inverse trigonometric functions return principal angles.
\(\sin^{-1}x\) means inverse sine, not cosecant.
Arcsine and arccosine accept real inputs only from \([-1,1]\).
Arctangent accepts every real input.
\(\sin(\arcsin x)=x\), \(\cos(\arccos x)=x\), and \(\tan(\arctan x)=x\) on their inverse domains.
The reverse compositions require the original angle to lie in the corresponding principal range.
Inverse functions return one principal value, not every solution of a periodic equation.
Quadrant information can be used to recover angles outside a principal range.
Right triangles and Pythagorean identities simplify compositions such as \(\cos(\arcsin x)\).
Calculator angle mode determines whether numerical inverse values are reported in degrees or radians.

Where We Go Next

Inverse trigonometric functions recover angles from ratios and coordinate information, making them essential whenever an unknown direction must be determined.

The next lesson, Laws of Sines and Cosines, extends triangle solving beyond right triangles and uses trigonometric relationships to determine unknown sides and angles in general triangles.