🎯⭐ INTERACTIVE LESSON

Inverse Trigonometric Functions

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Inverse Trigonometric Functions - Complete Interactive Lesson

Part 1: Inverse Sine

🔄 Inverse Trigonometric Functions — Principal Values & Restricted Domains

Part 1 of 7

Trig functions are not one-to-one, so to define inverses we must restrict the domain to an interval where the function passes the horizontal line test.

The Three Main Inverse Functions

Function	Notation	Restricted Domain	Range (Principal Values)
$\arcsin x$	$\sin^{-1}x$	$[-1, 1]$	$\left[-\frac{\pi}{2},\, \frac{\pi}{2}\right]$
$\arccos x$	$\cos^{-1}x$	$[-1, 1]$	$[0,$
$\arctan x$	$\tan^{-1}x$	$(-\infty, \infty)$	$($

Key Idea

$\boxed{\text{Inverse trig functions output ANGLES, not ratios}}$

$\arcsin\!\left(\frac{1}{2}\right) = \frac{\pi}{6}$ means "the angle (in the principal range) whose sine is ".

Notation warning: $\sin^{-1}x$ means $\arcsin x$ , NOT $\frac{1}{\sin x}$ (that's ).

📝 Worked Examples

Example 1: Evaluate $\arcsin\!\left(\frac{\sqrt{3}}{2}\right)$

🔍 Why We Restrict the Domain

Without Restriction: Infinitely Many Answers

$\sin\theta = \frac{1}{2}$ has solutions $\theta = 30°, 150°, 390°, 510°, \ldots$ and also

Concept Check 🎯

Inverse Trig Evaluation 🧮

1) $\arcsin\!\left(\frac{\sqrt{2}}{2}\right)$ in degrees = ? (e.g., since )

Domain & Range Matching 🔽

Exit Quiz ✅

Part 2: Inverse Cosine

📈 Graphs of Inverse Trig Functions

Part 2 of 7

Each inverse trig graph is the reflection of the restricted trig graph across the line $y = x$ .

Arcsin Graph: $y = \arcsin x$

Feature	Value
Domain	$[- 1, 1]$

Part 3: Inverse Tangent

🎯 Evaluating Inverse Trig — Exact Values

Part 3 of 7

Evaluating inverse trig functions means finding exact angle values from the unit circle. The key is memorizing outputs for special inputs.

Complete Special-Value Table

$x$	$\arcsin x$	$\arccos x$	$\arctan x$

Part 4: Compositions with Inverses

🔗 Compositions of Trig & Inverse Trig

Part 4 of 7

One of the most important skills is simplifying compositions like $\sin(\arccos x)$ or $\cos(\arctan x)$ .

Two Types of Compositions

Type 1: Trig(InverseTrig) — e.g., $\sin(\arccos \frac{3}{5})$

Part 5: Solving Trig Equations

📐 Inverse Trig with Right Triangles

Part 5 of 7

Inverse trig functions let us find angles in right triangles when we know the sides.

The Setup

Given a right triangle with known side lengths, find an angle $\theta$ :

$\boxed{\theta = \arctan\!\left(\frac{\text{opposite}}{\text{adjacent}}\right) = \arcsin\!\left(\frac{\text{opposite}}{\text{hypotenuse}}\right) = \arccos\!\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)}$

Part 6: Problem-Solving Workshop

🌍 Applications of Inverse Trig

Part 6 of 7

Inverse trig functions appear everywhere in real-world problems — navigation, physics, engineering, and more.

Angle of Elevation & Depression

$\boxed{\text{Angle of Elevation/Depression} = \arctan\!\left(\frac{\text{vertical distance}}{\text{horizontal distance}}\right)}$

Part 7: Review & Applications

🏆 Inverse Trig — Full Synthesis

Part 7 of 7

This part brings together everything from Parts 1–6: domains & ranges, graphs, exact values, compositions, triangle problems, and applications.

Master Summary

Property	$\arcsin x$	$\arccos x$	$\arctan x$
Domain

(- \frac{π}{2}, \frac{π}{2})

Ask: "What angle $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ has $\sin\theta = \frac{\sqrt{3}}{2}$ ?"

$\theta = \frac{\pi}{3} = 60°$

Example 2: Evaluate $\arccos(-1)$

Ask: "What angle $\theta \in [0, \pi]$ has $\cos\theta = -1$ ?"

$\theta = \pi = 180°$

Example 3: Evaluate $\arctan(-1)$

Ask: "What angle $\theta \in (-\frac{\pi}{2}, \frac{\pi}{2})$ has $\tan\theta = -1$ ?"

$\theta = -\frac{\pi}{4} = -45°$

Example 4: Why $\arcsin(\sin 240°) \neq 240°$

$\sin 240° = -\frac{\sqrt{3}}{2}$ . The principal value of $\arcsin(-\frac{\sqrt{3}}{2})$ is $-60°$ , not $240°$ , because $\arcsin$ outputs must be in $[-90°, 90°]$ .

30°,150°,390°,510°,…

-210°, -330°, \ldots

A function can only return one output. So we pick the interval where each trig function is one-to-one:

Function	Why This Interval?
$\sin$ restricted to $[-\frac{\pi}{2}, \frac{\pi}{2}]$	Sine goes from $-1$ to $1$ (hits every y-value exactly once)
$\cos$ restricted to $[0, \pi]$	Cosine goes from $1$ to $-1$ (hits every y-value exactly once)
$\tan$ restricted to $(-\frac{\pi}{2}, \frac{\pi}{2})$

Quick Reference: Special Angle Outputs

Input $x$	$\arcsin x$	$\arccos x$
$0$	$0°$	$90°$
$\frac{1}{2}$	$30°$	$60°$
$\frac{\sqrt{2}}{2}$
$\frac{\sqrt{3}}{2}$
$1$	$90°$	$0°$

\arcsin(\frac{\sqrt{3}}{2}) = 60

\sin 60° = \frac{\sqrt{3}}{2}

2) $\arccos(0)$ in degrees = ? (e.g., $\arccos(1) = 0$ since $\cos 0° = 1$ )

3) $\arctan(\sqrt{3})$ in degrees = ? (e.g., $\arctan(1) = 45$ since $\tan 45° = 1$ )

Arccos Graph: $y = \arccos x$

Feature	Value
Domain	$[-1, 1]$
Range	$[0, \pi]$
Passes through	$(0, \frac{\pi}{2})$
Decreasing	On the entire domain
Endpoints	$(-1, \pi)$ and $(1, 0)$

Arctan Graph: $y = \arctan x$

Feature	Value
Domain	$(-\infty, \infty)$
Range	$(-\frac{\pi}{2}, \frac{\pi}{2})$
Passes through	$(0, 0)$
Increasing	On the entire domain
Horizontal asymptotes	$y = -\frac{\pi}{2}$ and $y = \frac{\pi}{2}$

🔑 Key Graphing Relationships

Reflection Property

If $(a, b)$ is on $y = \sin x$ (restricted), then $(b, a)$ is on $y = \arcsin x$ .

For example: $(\frac{\pi}{6}, \frac{1}{2})$ on → on

Complementary Identity

$\boxed{\arcsin x + \arccos x = \frac{\pi}{2} \quad \text{for all } x \in [-1,1]}$

This means the arcsin and arccos graphs are "complementary" — at any $x$ -value, their outputs sum to $\frac{\pi}{2}$ .

Symmetry

Function	Symmetry	Meaning
$\arcsin$	Odd: $\arcsin(-x) = -\arcsin x$	Symmetric about origin
$\arccos$	Neither odd nor even

📝 Worked Examples

Example 1: Key Points on the Arcsin Graph

Plot: $(-1, -\frac{\pi}{2}),\; (-\frac{\sqrt{3}}{2}, -\frac{\pi}{3}),\; (-\frac{1}{2}, -\frac{\pi}{6}),\; (0, 0),\; (\frac{1}{2}, \frac{\pi}{6}),\; (\frac{\sqrt{3}}{2}, \frac{\pi}{3}),\; (1, \frac{\pi}{2})$

Connect with a smooth increasing curve from $(-1, -\frac{\pi}{2})$ to $(1, \frac{\pi}{2})$ .

Example 2: Using the Complementary Identity

Find $\arccos(\frac{1}{2})$ given that $\arcsin(\frac{1}{2}) = \frac{\pi}{6}$ :

$\arccos\!\left(\frac{1}{2}\right) = \frac{\pi}{2} - \arcsin\!\left(\frac{1}{2}\right) = \frac{\pi}{2} - \frac{\pi}{6} = \frac{\pi}{3}$

Example 3: Negative Input Symmetry

$\arctan(-\sqrt{3}) = -\arctan(\sqrt{3}) = -\frac{\pi}{3}$

We used the odd-function property: $\arctan(-x) = -\arctan x$ .

Graph Features 🎯

Using Graphing Properties 🧮

All answers in degrees.

1) If $\arcsin(\frac{\sqrt{3}}{2}) = 60°$ , then $\arccos(\frac{\sqrt{3}}{2})$ = ? (e.g., $\arcsin(\frac{1}{2}) = 30°$ gives $\arccos(\frac{1}{2}) = 90° - 30° = 60°$ )

2) $\arctan(-1)$ = ? (e.g., $\arctan(-\sqrt{3}) = -60°$ by the odd-function property)

3) $\arcsin(-\frac{1}{2})$ = ? (e.g., $\arcsin(-\frac{\sqrt{2}}{2}) = -45°$ by the odd-function property)

Graph Identification 🔽

Arctan Special Values

$x$	$\arctan x$
$-\sqrt{3}$	$-\frac{\pi}{3}$
$-1$	$-\frac{\pi}{4}$
$-\frac{\sqrt{3}}{3}$
$0$	$0$
$\frac{\sqrt{3}}{3}$
$1$	$\frac{\pi}{4}$
$\sqrt{3}$	$\frac{\pi}{3}$

🧠 Evaluation Strategy

Step-by-Step Process

$\boxed{\text{1. Identify the function → 2. Recall its range → 3. Find the angle in that range}}$

Example 1: $\arcsin\!\left(-\frac{\sqrt{2}}{2}\right)$

Function: $\arcsin$ → range is $[-\frac{\pi}{2}, \frac{\pi}{2}]$

Example 2: $\arccos\!\left(-\frac{1}{2}\right)$

Function: $\arccos$ → range is $[0, \pi]$
Need $\theta$ with $\cos\theta = -\frac{1}{2}$ and

Example 3: Undefined Inputs

$\arcsin(2)$ is undefined — there's no angle whose sine equals $2$ (sine only outputs $[-1, 1]$ ).

$\arccos(-3)$ is also undefined for the same reason.

$\arctan(100)$ IS defined — tangent can take any real value, so arctan accepts all reals.

📝 More Practice

Example 4: Converting Between Degrees and Radians

$\arcsin(\frac{1}{2}) = 30° = \frac{\pi}{6}$ rad

Always be aware of whether the problem asks for degrees or radians!

Example 5: Tricky Negative Values

$\arccos(-\frac{\sqrt{3}}{2})$ :

We know $\cos 30° = \frac{\sqrt{3}}{2}$

Pattern for Negative Inputs

Function	Negative Input Formula
$\arcsin(-x)$	$= -\arcsin(x)$
$\arccos(-x)$

Exact Value Quiz 🎯

Compute Exact Values 🧮

Give answers in degrees.

1) $\arcsin(-1)$ = ? (e.g., $\arcsin(1) = 90$ since $\sin 90° = 1$ )

2) $\arccos(-\frac{1}{2})$ = ? (e.g., $\arccos(\frac{1}{2}) = 60$ since $\cos 60° = \frac{1}{2}$ )

3) $\arctan(-\sqrt{3})$ = ? (e.g., $\arctan(\sqrt{3}) = 60$ since )

Quick Evaluation 🔽

Strategy: Draw a right triangle from the inverse trig value.

Type 2: InverseTrig(Trig) — e.g., $\arcsin(\sin \frac{5\pi}{6})$

Strategy: Check if the angle is in the principal range. If not, find the equivalent angle.

Type 1 — The Right Triangle Method

$\boxed{\text{Let } \theta = \arccos x \text{, draw triangle, find desired ratio}}$

If $\theta = \arccos\!\left(\frac{3}{5}\right)$ , then $\cos\theta = \frac{3}{5}$ .

Draw a right triangle: adjacent = $3$ , hypotenuse = $5$ , so opposite = $\sqrt{25 - 9} = 4$ .

$\sin(\arccos \tfrac{3}{5}) = \sin\theta = \frac{4}{5}$

📝 Worked Examples

Example 1: $\tan(\arcsin \frac{5}{13})$

Let $\theta = \arcsin \frac{5}{13}$ , so $\sin\theta = \frac{5}{13}$ .

Right triangle: opposite = $5$ , hypotenuse = $13$ , adjacent = $\sqrt{169 - 25} = 12$

$\tan(\arcsin \tfrac{5}{13}) = \frac{5}{12}$

Example 2: $\cos(\arctan 2)$

Let $\theta = \arctan 2$ , so $\tan\theta = \frac{2}{1}$ .

Right triangle: opposite = $2$ , adjacent = $1$ , hypotenuse = $\sqrt{4 + 1} = \sqrt{5}$

$\cos(\arctan 2) = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}$

Example 3: General Formula $\sin(\arccos x)$

Let $\theta = \arccos x$ . Then $\cos\theta = x = \frac{x}{1}$ .

Adjacent = $x$ , hypotenuse = $1$ , opposite = $\sqrt{1 - x^2}$

$\boxed{\sin(\arccos x) = \sqrt{1 - x^2}}$

🔄 Type 2: InverseTrig(Trig)

The Cancellation Rules

These only work when the angle is in the principal range:

Expression	Simplifies to	Condition
$\arcsin(\sin\theta)$	$\theta$	$\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$
$\arccos(\cos\theta)$	$\theta$	$\theta \in [0, \pi]$
$\arctan(\tan\theta)$	$\theta$	$\theta \in (-\frac{\pi}{2}, \frac{\pi}{2})$

Example 4: $\arcsin(\sin \frac{7\pi}{6})$

$\frac{7\pi}{6}$ is NOT in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ , so we can't just cancel.

$\sin \frac{7\pi}{6} = -\frac{1}{2}$

$\arcsin(-\frac{1}{2}) = -\frac{\pi}{6}$

Example 5: $\arccos(\cos(-\frac{\pi}{3}))$

$-\frac{\pi}{3}$ is NOT in $[0, \pi]$ , so we can't cancel.

$\cos(-\frac{\pi}{3}) = \frac{1}{2}$

$\arccos(\frac{1}{2}) = \frac{\pi}{3}$

Composition Quiz 🎯

Evaluate Compositions 🧮

Write answers as simplified fractions or integers.

1) $\tan(\arccos \frac{3}{5})$ = ? (e.g., $\sin(\arccos \frac{4}{5}) = \frac{3}{5}$ using a 3-4-5 triangle)

2) $\arcsin(\sin 210°)$ in degrees = ? (e.g., $\arcsin(\sin 150°) = 30°$ since $\sin 150° = \frac{1}{2}$ and )

3) $\cos(\arctan \frac{3}{4})$ = ? Write as a decimal like 0.8 (e.g., $\cos(\arctan 1) = \frac{\sqrt{2}}{2} \approx 0.707$ )

True or False 🔽

θ=arctan(adjacentopposite)=arcsin(hypotenuseopposite)=arccos(hypotenuseadjacent)

Example: Triangle with sides 5, 12, 13

For the angle opposite the side of length 5:

$\theta = \arcsin\!\left(\frac{5}{13}\right) = \arctan\!\left(\frac{5}{12}\right) \approx 22.6°$

For the angle opposite the side of length 12:

$\alpha = \arcsin\!\left(\frac{12}{13}\right) = \arctan\!\left(\frac{12}{5}\right) \approx 67.4°$

Verify: $22.6° + 67.4° + 90° = 180°$ ✓

🔢 Algebraic Expressions with Inverse Trig

Writing Trig Ratios as Algebraic Expressions

Problem: Write $\sin(\arctan \frac{x}{3})$ as an algebraic expression in $x$ .

Solution:

Let $\theta = \arctan \frac{x}{3}$ , so $\tan\theta = \frac{x}{3}$
Right triangle: opposite = $x$ , adjacent = $3$
Hypotenuse = $\sqrt{x^2 + 9}$
$\sin\theta = \frac{x}{\sqrt{x^2 + 9}}$

$\boxed{\sin\!\left(\arctan \frac{x}{3}\right) = \frac{x}{\sqrt{x^2 + 9}}}$

Common General Formulas

Expression	Algebraic Form
$\sin(\arccos x)$	$\sqrt{1 - x^2}$

✏️ Solving for Missing Angles

Example 1: Ladder Problem

A 20-foot ladder leans against a wall with its base 8 feet from the wall. Find the angle with the ground.

Adjacent = $8$ , hypotenuse = $20$ .

$\theta = \arccos\!\left(\frac{8}{20}\right) = \arccos(0.4) \approx 66.4°$

Example 2: Finding Both Acute Angles

In a right triangle with legs $a = 7$ and $b = 24$ :

$\alpha = \arctan\!\left(\frac{7}{24}\right) \approx 16.3°$ $β = \arctan ⁣$

Check: $16.3° + 73.7° = 90°$ ✓ (The acute angles in a right triangle sum to $90°$ .)

Example 3: Using a Known Hypotenuse

Right triangle with opposite $= 6$ , hypotenuse $= 10$ .

$\theta = \arcsin\!\left(\frac{6}{10}\right) = \arcsin(0.6) \approx 36.9°$

This is a 3-4-5 triangle scaled by 2 (sides 6, 8, 10), and $\arcsin(0.6) = 36.87°$ .

Triangle & Algebra Quiz 🎯

Solving Triangles 🧮

Round to the nearest degree.

1) Right triangle: opposite = 3, adjacent = 4. Find angle $\theta$ in degrees. (e.g., if opposite = 5, adjacent = 12, then $\theta = \arctan(\frac{5}{12}) \approx 23°$ )

2) Right triangle: opposite = 7, hypotenuse = 25. Find angle $\theta$ in degrees. (e.g., if opp = 5, hyp = 13, then $\theta = \arcsin(\frac{5}{13}) \approx 23°$ )

3) Right triangle: adjacent = 9, hypotenuse = 15. Find angle $\theta$ in degrees. (e.g., if adj = 4, hyp = 5, then $\theta = \arccos(\frac{4}{5}) \approx 37°$ )

Choose the Right Expression 🔽

Example 1: Angle of Elevation

A 6-foot person looks up at the top of a 50-foot building from 80 feet away. What is the angle of elevation?

Vertical distance = $50 - 6 = 44$ ft, horizontal distance = $80$ ft.

$\theta = \arctan\!\left(\frac{44}{80}\right) = \arctan(0.55) \approx 28.8°$

Example 2: Angle of Depression

A drone at 200 feet altitude spots a target 500 feet away horizontally. The angle of depression is:

$\theta = \arctan\!\left(\frac{200}{500}\right) = \arctan(0.4) \approx 21.8°$

🧭 Navigation & Bearings

Example 3: Finding Direction

A ship sails 15 km east and 8 km north. What bearing has it traveled?

$\theta = \arctan\!\left(\frac{15}{8}\right) \approx 61.9°$

Bearing: approximately $\text{N } 62° \text{ E}$ (or $062°$ in compass notation).

Example 4: Surveying

A surveyor stands at point A and measures:

Distance to point B: 120 meters
Height difference: 35 meters

Angle: $\theta = \arcsin\!\left(\frac{35}{120}\right) \approx 17.0°$

Example 5: Physics — Launch Angle

A projectile needs to reach a target at the same height, 200 m away, with initial speed 50 m/s.

The range formula gives: $R = \frac{v^2 \sin(2\theta)}{g}$

$200 = \frac{2500 \sin(2\theta)}{9.8} \implies \sin(2\theta) = 0.784$

$2\theta = \arcsin(0.784) \approx 51.6° \implies \theta \approx 25.8°$

🔧 Solving Inverse Trig Equations

Example 6: Solve $2\arcsin(x) = \frac{\pi}{3}$

$\arcsin(x) = \frac{\pi}{6}$ $x = \sin\!\left(\frac{\pi}{6}\right) = \frac{1}{2}$

Example 7: Solve $\arctan(2x - 1) = \frac{\pi}{4}$

$2x - 1 = \tan\!\left(\frac{\pi}{4}\right) = 1$

Key Strategy for Solving

$\boxed{\text{Isolate the inverse trig function, then apply the corresponding trig function to both sides}}$

If $\arcsin(\text{expr}) = \theta$ , then $\text{expr} = \sin\theta$ .

If $\arccos(\text{expr}) = \theta$ , then $\text{expr} = \cos\theta$ .

If $\arctan(\text{expr}) = \theta$ , then $\text{expr} = \tan\theta$ .

Applications Quiz 🎯

Solve Equations 🧮

1) Solve $\arcsin(x) = \frac{\pi}{6}$ . What is $x$ ? Write as a decimal. (e.g., If $\arccos(x) = \frac{\pi}{3}$ , then $x = \cos(\frac{\pi}{3}) = 0.5$ )

2) Solve $\arctan(x) = \frac{\pi}{4}$ . What is $x$ ? (e.g., If $\arctan(x) = 0$ , then )

3) A tree casts a 40-foot shadow when the sun's elevation is 50°. Tree height = $40\tan(50°) \approx$ ? feet. Round to nearest integer. (e.g., $40\tan(45°) = 40$ since $\tan 45° = 1$ )

Application Matching 🔽

$\boxed{\arcsin x + \arccos x = \frac{\pi}{2}}$

$\arcsin(-x) = -\arcsin x, \quad \arctan(-x) = -\arctan x$

$\arccos(-x) = \pi - \arccos x$

📝 Mixed Review Problems

Problem 1: Exact Value

$\arccos\!\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4}$ because $\cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2}$ and $\frac{3\pi}{4} \in [0, \pi]$ .

Problem 2: Composition

$\sin(\arctan \frac{3}{4})$ : Triangle with opp = 3, adj = 4, hyp = 5. Answer: $\frac{3}{5}$ .

Problem 3: InverseTrig(Trig)

$\arccos(\cos \frac{7\pi}{4})$ : $\cos\frac{7\pi}{4} = \frac{\sqrt{2}}{2}$ . .

Problem 4: Equation

Solve $\arcsin(2x - 1) = -\frac{\pi}{6}$ : $2 x - 1 = \sin (- \frac{π}{6}) = - \frac{}{}$

Problem 5: Application

Lighthouse 150 ft tall, boat 400 ft away. Angle of depression: $\arctan\!\left(\frac{150}{400}\right) = \arctan(0.375) \approx 20.6°$

⚠️ Common Mistakes to Avoid

Mistake	Why It's Wrong	Correct
$\arcsin(\sin 200°) = 200°$	$200°$ not in $[-90°, 90°]$	Find equivalent angle in range
$\sin^{-1}(0.5) = \frac{1}{\sin(0.5)}$	means inverse, not reciprocal
$\arccos(-0.5) = -60°$	$\arccos$ range is $[0°, 180°]$ , never negative	$\arccos (- 0.5) = 120 °$
Forgetting to rationalize	$\frac{1}{\sqrt{5}}$ should be
Using wrong triangle sides	Confusing which sides are opp/adj/hyp	Always label relative to the angle

Comprehensive Quiz 🎯

Mixed Skill Check 🧮

1) $\tan(\arcsin \frac{8}{17})$ = ? Write as a fraction. (e.g., $\tan(\arcsin \frac{3}{5}) = \frac{3}{4}$ using a 3-4-5 triangle)

2) Solve $\arccos(x) = \frac{2\pi}{3}$ . What is $x$ ? Write as a decimal. (e.g., $\arccos (x) =$ gives )

3) $\arcsin(\frac{\sqrt{3}}{2}) + \arccos(\frac{\sqrt{3}}{2})$ in degrees = ? (e.g., )

Final Review 🔽

Final Exit Quiz ✅

(\frac{1}{2}, \frac{\pi}{6})

\arccos(-x) = \pi - \arccos x

Need

\theta

with

\sin\theta = -\frac{\sqrt{2}}{2}

and

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

\theta = -\frac{\pi}{4}

✓

\theta \in [0, \pi]

\theta = \frac{2\pi}{3}

✓

For the negative input, the angle must be in Quadrant II:

180° - 30° = 150°

\arccos(-\frac{\sqrt{3}}{2}) = 150° = \frac{5\pi}{6}

= \pi - \arccos(x)

\tan 60° = \sqrt{3}

\arcsin \frac{1}{2} = 30°

β = arctan (\frac{24}{7}) \approx 73.7°

2x = 2 \implies x = 1

\arccos(\frac{\sqrt{2}}{2}) = \frac{\pi}{4}

2 x - 1 = sin (- \frac{π}{6}) = - \frac{1}{2}

2x = \frac{1}{2} \implies x = \frac{1}{4}

arccos (- 0.5) = 120°

arccos (x) = \frac{π}{3}

x = \cos\frac{\pi}{3} = 0.5

\arcsin(\frac{1}{2}) + \arccos(\frac{1}{2}) = 30° + 60° = 90°

$-1$	$-\frac{\pi}{2}$	$\pi$	—
$-\frac{\sqrt{3}}{2}$	$-\frac{\pi}{3}$	$\frac{5\pi}{6}$	—
$-\frac{\sqrt{2}}{2}$
$-\frac{1}{2}$	$-\frac{\pi}{6}$
$0$	$0$	$\frac{\pi}{2}$	$0$
$\frac{1}{2}$	$\frac{\pi}{6}$
$\frac{\sqrt{2}}{2}$
$\frac{\sqrt{3}}{2}$
$1$	$\frac{\pi}{2}$	$0$	$\frac{\pi}{4}$

Inverse Trigonometric Functions

Inverse Trigonometric Functions - Complete Interactive Lesson

Part 1: Inverse Sine

🔄 Inverse Trigonometric Functions — Principal Values & Restricted Domains

The Three Main Inverse Functions

Key Idea

📝 Worked Examples

Example 1: Evaluate arcsin⁡ ⁣(32)\arcsin\!\left(\frac{\sqrt{3}}{2}\right)arcsin(23​

🔍 Why We Restrict the Domain

Without Restriction: Infinitely Many Answers

Part 2: Inverse Cosine

📈 Graphs of Inverse Trig Functions

Arcsin Graph: y=arcsin⁡xy = \arcsin xy=arcsinx

Part 3: Inverse Tangent

🎯 Evaluating Inverse Trig — Exact Values

Complete Special-Value Table

Part 4: Compositions with Inverses

🔗 Compositions of Trig & Inverse Trig

Two Types of Compositions

Part 5: Solving Trig Equations

📐 Inverse Trig with Right Triangles

The Setup

Part 6: Problem-Solving Workshop

🌍 Applications of Inverse Trig

Angle of Elevation & Depression

Part 7: Review & Applications

🏆 Inverse Trig — Full Synthesis

Master Summary

Example 2: Evaluate arccos⁡(−1)\arccos(-1)arccos(−1)

Example 3: Evaluate arctan⁡(−1)\arctan(-1)arctan(−1)

Example 4: Why arcsin⁡(sin⁡240°)≠240°\arcsin(\sin 240°) \neq 240°arcsin(sin240°)=240°

Quick Reference: Special Angle Outputs

Arccos Graph: y=arccos⁡xy = \arccos xy=arccosx

Arctan Graph: y=arctan⁡xy = \arctan xy=arctanx

🔑 Key Graphing Relationships

Reflection Property

Complementary Identity

Symmetry

📝 Worked Examples

Example 1: Key Points on the Arcsin Graph

Example 2: Using the Complementary Identity

Example 3: Negative Input Symmetry

Arctan Special Values

🧠 Evaluation Strategy

Step-by-Step Process

Example 1: arcsin⁡ ⁣(−22)\arcsin\!\left(-\frac{\sqrt{2}}{2}\right)arcsin(−22​​)

Example 2: arccos⁡ ⁣(−12)\arccos\!\left(-\frac{1}{2}\right)arccos(−21​)

Example 3: Undefined Inputs

📝 More Practice

Example 4: Converting Between Degrees and Radians

Example 5: Tricky Negative Values

Pattern for Negative Inputs

Type 1 — The Right Triangle Method

📝 Worked Examples

Example 1: tan⁡(arcsin⁡513)\tan(\arcsin \frac{5}{13})tan(arcsin135​)

Example 2: cos⁡(arctan⁡2)\cos(\arctan 2)cos(arctan2)

Example 3: General Formula sin⁡(arccos⁡x)\sin(\arccos x)sin(arccosx)

🔄 Type 2: InverseTrig(Trig)

The Cancellation Rules

Example 4: arcsin⁡(sin⁡7π6)\arcsin(\sin \frac{7\pi}{6})arcsin(sin67π​)

Example 5: arccos⁡(cos⁡(−π3))\arccos(\cos(-\frac{\pi}{3}))arccos(cos(−3π​))

Example: Triangle with sides 5, 12, 13

🔢 Algebraic Expressions with Inverse Trig

Writing Trig Ratios as Algebraic Expressions

Common General Formulas

✏️ Solving for Missing Angles

Example 1: Ladder Problem

Example 2: Finding Both Acute Angles

Example 3: Using a Known Hypotenuse

Example 1: Angle of Elevation

Example 2: Angle of Depression

🧭 Navigation & Bearings

Example 3: Finding Direction

Example 4: Surveying

Example 5: Physics — Launch Angle

🔧 Solving Inverse Trig Equations

Example 6: Solve 2arcsin⁡(x)=π32\arcsin(x) = \frac{\pi}{3}2arcsin(x)=3π​

Example 7: Solve arctan⁡(2x−1)=π4\arctan(2x - 1) = \frac{\pi}{4}arctan(2x−1)=4π​

Key Strategy for Solving

Key Identities

Example 1: Evaluate $\arcsin\!\left(\frac{\sqrt{3}}{2}\right)$

Arcsin Graph: $y = \arcsin x$

Example 2: Evaluate $\arccos(-1)$

Example 3: Evaluate $\arctan(-1)$

Example 4: Why $\arcsin(\sin 240°) \neq 240°$

Arccos Graph: $y = \arccos x$

Arctan Graph: $y = \arctan x$

Example 1: $\arcsin\!\left(-\frac{\sqrt{2}}{2}\right)$

Example 2: $\arccos\!\left(-\frac{1}{2}\right)$

Example 1: $\tan(\arcsin \frac{5}{13})$

Example 2: $\cos(\arctan 2)$

Example 3: General Formula $\sin(\arccos x)$

Example 4: $\arcsin(\sin \frac{7\pi}{6})$

Example 5: $\arccos(\cos(-\frac{\pi}{3}))$

Example 6: Solve $2\arcsin(x) = \frac{\pi}{3}$

Example 7: Solve $\arctan(2x - 1) = \frac{\pi}{4}$