🎯⭐ INTERACTIVE LESSON

Trigonometric Functions

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Trigonometric Functions - Complete Interactive Lesson

Part 1: Unit Circle Fundamentals

📐 Trigonometric Functions — Angles & Radian Measure

Part 1 of 7

Trigonometry begins with measuring angles. The radian is the natural unit for angles in calculus and higher math.

Degree ↔ Radian Conversion

$\boxed{\text{radians} = \text{degrees} \times \frac{\pi}{180} \qquad \text{degrees} = \text{radians} \times \frac{180}{\pi}}$

Common Angle Reference Table

Degrees	$0°$	$30°$	$45°$	$60°$	$90°$

Memory tip: $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$ correspond to — the denominators decrease as the angles increase.

🔄 What Is a Radian?

A radian is the angle whose arc length equals the radius.

$\boxed{\theta = \frac{s}{r} \quad \text{or equivalently} \quad s = r\theta}$

📝 Worked Examples

Example 1: Convert $225°$ to radians

$225° \times \frac{\pi}{180} = \frac{225\pi}{180} = \frac{5\pi}{4}$

Concept Check 🎯

Conversion Practice 🧮

1) Convert $315°$ to radians. Express as a fraction of $\pi$ — write just the fraction (e.g., $5/4$ ). (e.g., $270° = 270/180 \cdot \pi = 3/2 \cdot \pi$ , so answer: )

Angle Fundamentals 🔽

Exit Quiz ✅

Part 2: Sine & Cosine Graphs

⭕ Trigonometric Functions — The Unit Circle

Part 2 of 7

The unit circle is a circle with radius $1$ centered at the origin. Every point on it has coordinates $(\cos\theta,\;\sin\theta)$ .

$x^{2} + y^{2} = 1 ⟹ \cos^{2} θ + \sin^{2}$

Part 3: Tangent & Reciprocal Functions

🌊 Trigonometric Functions — Sine & Cosine Definitions

Part 3 of 7

Right-Triangle Definitions

For a right triangle with angle $\theta$ , hypotenuse $r$ , opposite side $y$ , and adjacent side $x$ :

$\sin θ = \frac{opposite}{hypotenuse} =$

Part 4: Amplitude & Period

🔺 Trigonometric Functions — Tangent, Cotangent, Secant & Cosecant

Part 4 of 7

Beyond sine and cosine, four more trig functions are built from ratios.

Definitions

$\boxed{\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \csc\theta = \frac{1}{\sin\theta}}$

Part 5: Phase Shifts

📊 Trigonometric Functions — Graphing Sinusoids

Part 5 of 7

The general sinusoidal form is:

$\boxed{y = a\sin\bigl(b(x - c)\bigr) + d \qquad \text{or} \qquad y = a\cos\bigl(b(x - c)\bigr) + d}$

Part 6: Problem-Solving Workshop

🛠️ Trigonometric Functions — Applied Problems & Modeling

Part 6 of 7

Real-world phenomena — tides, temperature, sound, rotation — follow sinusoidal patterns. This part focuses on translating word problems into trig equations.

Modeling Workflow

Step	Action
1	Identify the maximum and minimum values
2	Compute amplitude: $a = \frac{\text{max} - \text{min}}{2}$

Part 7: Review & Applications

🏆 Trigonometric Functions — Full Synthesis & Review

Part 7 of 7

This final part ties together everything from Parts 1–6: radian measure, the unit circle, all six trig functions, graphing, and modeling.

Master Decision Flowchart

Task	Key Tool
Convert degrees ↔ radians	Multiply by $\frac{\pi}{180}$ or $\frac{180}{\pi}$

Quantity	Symbol	Unit
Arc length	$s$	same as $r$ (meters, cm, etc.)
Radius	$r$	length units
Angle	$\theta$	radians (dimensionless)

Statement	Value
One full revolution	$2\pi$ radians $= 360°$
Half revolution	$\pi$ radians $= 180°$
Quarter revolution	$\frac{\pi}{2}$ radians $= 90°$
$1$ radian	$\approx 57.3°$
$1°$	$\approx 0.01745$ radians

$\boxed{225° = \frac{5\pi}{4}}$

Example 2: Convert $\frac{7\pi}{6}$ to degrees

$\frac{7\pi}{6} \times \frac{180}{\pi} = \frac{7 \times 180}{6} = \frac{1260}{6} = 210°$

Example 3: Arc Length

A circle has radius $10$ cm. Find the arc length subtended by a central angle of $\frac{2\pi}{3}$ .

$s = r\theta = 10 \cdot \frac{2\pi}{3} = \frac{20\pi}{3} \approx 20.94\text{ cm}$

Example 4: Sector Area

The area of a sector with angle $\theta$ (radians):

$\boxed{A = \frac{1}{2}r^2\theta}$

For $r = 6$ and $\theta = \frac{\pi}{4}$ : $A = \frac{1}{2}(36)\left(\frac{\pi}{4}\right) = \frac{9\pi}{2} \approx 14.14\text{ sq units}$

2) A circle has radius $12$ cm. Find the arc length for a central angle of $\frac{\pi}{3}$ radians. Write the answer in terms of $\pi$ as a number (e.g., if $s = 5\pi$ , write $5$ ). (e.g., $r = 10$ , $\theta = \pi/2$ : $s = 10 \cdot \pi/2 = 5\pi$ , so answer: $5$ )

3) Find the area of a sector with $r = 4$ and $\theta = \frac{\pi}{2}$ . Write the answer in terms of $\pi$ as a number. (e.g., $r = 6$ , $\theta = \pi/3$ : $A = \frac{1}{2}(36)(\pi/3) = 6\pi$ , so answer: $6$ )

x^{2} + y^{2} = 1 ⟹ cos^{2} θ + sin^{2} θ = 1

First-Quadrant Key Coordinates

$\theta$	$0$	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$
$\cos\theta$	$1$	$\frac{\sqrt{3}}{2}$

Pattern: cosine values descend $1, \frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{1}{2}, 0$ while sine values ascend $0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1$ — they mirror each other!

🧭 Signs by Quadrant & Reference Angles

ASTC Rule ("All Students Take Calculus")

Quadrant	Positive Functions	Sign of $(\cos,\sin)$
I ( $0$ to $\frac{\pi}{2}$ )	All	$(+, +)$
II ( $\frac{\pi}{2}$ to $\pi$ )	Sine	$(-, +)$
III ( $\pi$ to $\frac{3\pi}{2}$ )	Tangent	$(-, -)$
IV ( $\frac{3\pi}{2}$ to $2\pi$ )	Cosine	$(+, -)$

Reference Angle

The reference angle $\theta_R$ is the acute angle between the terminal side and the $x$ -axis:

Quadrant	Reference Angle Formula
I	$\theta_R = \theta$
II	$\theta_R = \pi - \theta$

Worked Example: Find $\cos\frac{5\pi}{6}$ and $\sin\frac{5\pi}{6}$

$\frac{5\pi}{6}$ is in Quadrant II (between $\frac{\pi}{2}$ and )

$\boxed{\cos\frac{5\pi}{6} = -\frac{\sqrt{3}}{2}, \quad \sin\frac{5\pi}{6} = \frac{1}{2}}$

📋 Full Unit Circle — All Four Quadrants

Angle $\theta$	$(\cos\theta, \sin\theta)$		Angle $\theta$	$(\cos\theta, \sin\theta)$
$0$	$(1, 0)$		$\pi$	$(-1, 0)$

Worked Example: Find $\sin\frac{4\pi}{3}$

$\frac{4\pi}{3}$ is in Q III → reference angle $= \frac{4\pi}{3} - \pi = \frac{\pi}{3}$

$\boxed{\sin\frac{4\pi}{3} = -\frac{\sqrt{3}}{2}}$

Unit Circle Check 🎯

Unit Circle Computations 🧮

All answers should be exact. Use sqrt for square roots and fractions (e.g., sqrt(3)/2).

1) $\cos\frac{7\pi}{6} = \;?$ Write the exact value including the sign. (e.g., $\cos\frac{2\pi}{3}$ : Q II, ref angle $\frac{\pi}{3}$ , $\cos\frac{\pi}{3} = 1/2$ , negative in Q II → answer: $-1/2$ )

2) $\sin\frac{11\pi}{6} = \;?$ Write as a fraction. (e.g., $\sin\frac{7\pi}{4}$ : Q IV, ref angle , , negative in Q IV → answer: )

3) What is the reference angle of $\frac{5\pi}{4}$ ? Express as a fraction of $\pi$ — write just the fraction (e.g., for $210°$ the ref angle is $30 ° = π /$ , answer: ). (e.g., is in Q II: , answer: )

Quadrant & Sign Matching 🔽

sin θ = \frac{opposite}{hypotenuse} = \frac{y}{r} cos θ = \frac{adjacent}{hypotenuse} = \frac{x}{r}

Unit-Circle Connection

On the unit circle ( $r = 1$ ), this simplifies to:

$\sin\theta = y \qquad \cos\theta = x$

Key Properties Comparison

Property	$\sin\theta$	$\cos\theta$
Domain	$(-\infty, \infty)$	$(-\infty, \infty)$
Range	$[-1, 1]$	$[-1, 1]$
Period	$2\pi$	$2\pi$
At $\theta = 0$	$0$	$1$
At $\theta = \frac{\pi}{2}$	$1$	$0$
Symmetry	Odd: $\sin(-\theta) = -\sin\theta$	Even: $\cos(-\theta) = \cos\theta$

🔗 Fundamental Identities

Pythagorean Identity

$\boxed{\sin^2\theta + \cos^2\theta = 1}$

This gives us two useful rearrangements:

$\sin^2\theta = 1 - \cos^2\theta$
$\cos^2\theta = 1 - \sin^2\theta$

Cofunction Identities

$\sin\theta = \cos\left(\frac{\pi}{2} - \theta\right) \qquad \cos\theta = \sin\left(\frac{\pi}{2} - \theta\right)$

Cofunctions of complementary angles are equal. For example: $\sin 30° = \cos 60° = \frac{1}{2}$ .

Worked Example: Using the Pythagorean Identity

Given $\sin\theta = \frac{3}{5}$ and $\theta$ is in Quadrant II, find $\cos\theta$ .

$\cos^2\theta = 1 - \sin^2\theta = 1 - \frac{9}{25} = \frac{16}{25}$

$\cos\theta = \pm\frac{4}{5}$

In Q II, cosine is negative: $\boxed{\cos\theta = -\frac{4}{5}}$

📝 Evaluating Sine & Cosine — Strategy

Step-by-Step Method

Identify the quadrant of $\theta$
Find the reference angle $\theta_R$
Look up $\sin\theta_R$ and $\cos\theta_R$ from the first-quadrant table
Apply the correct sign based on the quadrant

Example: Evaluate $\sin 300°$

Step	Work
1. Quadrant	$300°$ is in Q IV ( $270° < 300° < 360°$ )
2. Reference angle	$360° - 300° = 60°$

Example: Evaluate $\cos(-\frac{\pi}{4})$

Using the even property: $\cos(-\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$

No need to find reference angles — the even/odd properties are a shortcut!

Sine & Cosine Check 🎯

Computation Practice 🧮

1) Given $\sin\theta = \frac{8}{17}$ and $\theta$ is in Q I, find $\cos\theta$ . Write as a fraction (e.g., $3/5$ ). (e.g., $\sin\alpha = 5/13$ in Q I: $\cos\alpha = \sqrt{1 - 25/169} = \sqrt{144/169} = 12/13$ )

2) Evaluate $\cos 240°$ . Write as a fraction (e.g., $-1/2$ ). (e.g., $\cos 120°$ : Q II, ref $60°$ , $\cos 60° = 1/2$ , negative in Q II → )

3) If $\cos\theta = \frac{\sqrt{3}}{2}$ , find using the cofunction identity. Write as a simplified expression. (e.g., : )

Properties Matching 🔽

Complete Comparison Table

Function	Definition	Period	Domain Restriction	Range
$\tan\theta$	$\frac{\sin\theta}{\cos\theta}$	$\pi$	$\theta \neq \frac{\pi}{2} + n\pi$	$(-\infty, \infty)$
$\cot\theta$	$\frac{\cos\theta}{\sin\theta}$
$\sec\theta$	$\frac{1}{\cos\theta}$	$2\pi$
$\csc\theta$	$\frac{1}{\sin\theta}$	$2\pi$

Key insight: Tangent and secant share the same domain restrictions (undefined where $\cos\theta = 0$ ). Cotangent and cosecant share theirs (undefined where $\sin\theta = 0$ ).

🔗 Pythagorean Identity Family

Dividing $\sin^2\theta + \cos^2\theta = 1$ by $\cos^2\theta$ or $\sin^2\theta$ produces two more identities:

Divide by	Result
$\cos^2\theta$	$\boxed{\tan^2\theta + 1 = \sec^2\theta}$

Symmetry Properties

Function	Odd/Even	Identity
$\tan\theta$	Odd	$\tan(-\theta) = -\tan\theta$
$\cot\theta$

📝 Worked Examples

Example 1: Evaluate all six trig functions at $\theta = \frac{5\pi}{6}$

$\frac{5\pi}{6}$ is in Q II, reference angle $= \frac{\pi}{6}$ .

Function	Reference Value	Sign in Q II	Result
$\sin$	$\frac{1}{2}$	$+$

Example 2: Given $\tan\theta = -\frac{3}{4}$ and $\theta$ in Q II, find $\sec\theta$

Using $\sec^2\theta = 1 + \tan^2\theta = 1 + \frac{9}{16} = \frac{25}{16}$

$\sec\theta = \pm\frac{5}{4}$ . In Q II, cosine is negative, so secant is negative:

$\boxed{\sec\theta = -\frac{5}{4}}$

Concept Check 🎯

Trig Function Computations 🧮

1) Evaluate $\tan\frac{5\pi}{3}$ . Write as a simplified expression using sqrt if needed. (e.g., $\tan\frac{2\pi}{3}$ : Q II, ref $\frac{\pi}{3}$ , $\tan 60° = \sqrt{3}$ , negative in Q II → $-\sqrt{3}$ )

2) Given $\cot\theta = \frac{5}{12}$ and $\theta$ in Q III, find $\csc\theta$ . Write as a fraction with sign. (e.g., in Q I: , so )

3) Evaluate $\sec\frac{\pi}{3}$ . Write as an integer. (e.g., $\sec 0 = 1/\cos 0 = 1/1 = 1$ )

Function Properties 🔽

Parameter	Name	Formula / Effect
$a$	Amplitude	$
$b$	Frequency	Period $= \frac{2\pi}{
$c$	Phase shift	Shifts graph right by $c$ (left if $c < 0$ )
$d$	Vertical shift	Midline at $y = d$ ; range becomes $[d -

Measurement	Formula
Period	$\frac{2\pi}{
Amplitude	$
Maximum	$d +
Minimum	$d -
Midline	$y = d$

📝 Worked Examples — Reading Parameters

Example 1: $y = 3\sin(2x) - 1$

Parameter	Value	Meaning
$a = 3$	Amplitude $= 3$	Height stretches by factor of $3$
$b = 2$	Period $= \frac{2\pi}{2} = \pi$	Completes one cycle in units
$c = 0$	No phase shift	Starts at the origin
$d = -1$	Midline at $y = -1$	Range: $[-4, 2]$

Example 2: $y = -2\cos\left(\frac{\pi}{3}(x - 1)\right) + 5$

Parameter	Value	Meaning
$a = -2$	Amplitude $= 2$ , reflected	Starts at a minimum instead of maximum
$b = \frac{\pi}{3}$

Key Difference: Sine vs Cosine Starting Points

Function	Starts at midline	Goes to...
$\sin(x)$	Midline ( $y = 0$ )	Up (if $a > 0$ )

📈 Graphing Tangent & Cotangent

Tangent: $y = a\tan(bx - c) + d$

Feature	Value
Period	$\frac{\pi}{
Vertical asymptotes	$bx - c = \pm\frac{\pi}{2} + n\pi$
No amplitude	Range is $(-\infty, \infty)$
Passes through midline	At the center of each period

Example: $y = \tan(2x)$

Period $= \frac{\pi}{2}$ . Asymptotes where $2x = \frac{\pi}{2} + n\pi$ , i.e., .

Cotangent: $y = a\cot(bx - c) + d$

Same period formula $\frac{\pi}{|b|}$ , but asymptotes where $bx - c = n\pi$ (where ).

Tangent goes from $-\infty$ to $+\infty$ (increasing) between asymptotes.
Cotangent goes from $+\infty$ to $-\infty$ (decreasing) between asymptotes.

Graph Reading Check 🎯

Graph Analysis 🧮

1) Find the amplitude of $y = -7\cos(4x) + 1$ . Write a positive number. (e.g., $y = -3\sin(x)$ has amplitude $|-3| = 3$ )

2) Find the period of $y = \sin(\frac{\pi}{2}x)$ . Write as an integer. (e.g., $y = \sin(\pi x)$ : period )

3) A sinusoid has maximum $10$ and minimum $2$ . What is the midline? Write as an integer. (e.g., max $8$ , min $4$ : midline $= (8+4)/2 = 6$ )

Graphing Concepts 🔽

🌊 Example 1: Tidal Height

The tide at a harbor has a high of $12$ ft at 6:00 AM and a low of $4$ ft at 12:00 PM. Model the height $h(t)$ where $t$ is hours after midnight.

Parameter	Calculation	Value
Amplitude	$\frac{12 - 4}{2}$	$a = 4$
Midline	$\frac{12 + 4}{2}$	$d = 8$
Period	High to low is half-period: $12 - 6 = 6$ hrs, so $T = 12$	$b = \frac{2\pi}{12} = \frac{\pi}{6}$
Phase shift	Maximum at $t = 6$	Use cosine starting at max: $c = 6$

$\boxed{h(t) = 4\cos\left(\frac{\pi}{6}(t - 6)\right) + 8}$

Check: $h(6) = 4\cos(0) + 8 = 12$ ✓ $h(12) = 4\cos(\pi) + 8 = 4$ ✓

🌡️ Example 2: Monthly Temperature

A city's average monthly temperature ranges from $28°F$ in January ( $t = 1$ ) to $82°F$ in July ( $t = 7$ ). Write a model $T(t)$ .

Parameter	Calculation	Value
Amplitude	$\frac{82 - 28}{2}$	$a = 27$
Midline

$\boxed{T(t) = -27\cos\left(\frac{\pi}{6}(t - 1)\right) + 55}$

Why negative cosine? Cosine normally starts at a maximum. We need it to start at a minimum, so we negate it.

🎡 Example 3: Ferris Wheel

A Ferris wheel of radius $20$ m has its center $25$ m above ground and takes $3$ minutes per revolution. A rider boards at the bottom.

Bottom height $= 25 - 20 = 5$ m. Top height $= 25 + 20 = 45$ m.

Starting at the bottom (minimum) with period $T = 3$ :

$\boxed{h(t) = -20\cos\left(\frac{2\pi}{3}t\right) + 25}$

Check: $h(0) = -20(1) + 25 = 5$ m ✓ (bottom) $h(1.5) = -20(-1) + 25 = 45$ m ✓ (top)

Modeling Check 🎯

Modeling Practice 🧮

1) A sound wave oscillates between $+3$ and $-3$ with a period of $0.01$ seconds. What is $b$ (the angular frequency)? Write as a multiple of $\pi$ — give just the coefficient (e.g., if $b = 5\pi$ , write $5$ ). (e.g., period $= 0.5$ s: $b = 2\pi/0.5 = 4\pi$ , answer: $4$ )

2) A spring oscillates between heights $2$ cm and $14$ cm. What is the midline (in cm)? Write as an integer. (e.g., min $= 5$ , max $= 11$ : midline $= (5+11)/2 = 8$ )

3) A Ferris wheel has radius $30$ m, center $35$ m high, period $4$ min, rider starts at the bottom. What is the rider's height at $t = 1$ min? Write as an integer. (e.g., $h(t) = -20\cos(\frac{2\pi}{3}t) + 25$ : )

Modeling Strategy 🔽

📋 Complete Formula Reference

Definitions & Identities

Formula	Category
$\sin\theta = \frac{\text{opp}}{\text{hyp}}$ , $\cos\theta = \frac{\text{adj}}{\text{hyp}}$	Right triangle
$\tan\theta = \frac{\sin\theta}{\cos\theta}$ , $\cot θ = \frac{\cos θ}{\sin}$
$\sec\theta = \frac{1}{\cos\theta}$ , $\csc\theta = \frac{1}{\sin\theta}$
$\sin^2\theta + \cos^2\theta = 1$	Pythagorean
$\tan^2\theta + 1 = \sec^2\theta$	Pythagorean
$1 + \cot^2\theta = \csc^2\theta$	Pythagorean
$\sin(-\theta) = -\sin\theta$	Odd function
$\cos(-\theta) = \cos\theta$	Even function
$\sin\theta = \cos(\frac{\pi}{2} - \theta)$	Cofunction

Graphing

Form	Period
$y = a\sin(b(x-c)) + d$	$\frac{2\pi}{
$y = a\cos(b(x-c)) + d$

Measurement

Quantity	Formula
Arc length	$s = r\theta$
Sector area	$A = \frac{1}{2}r^2\theta$

📝 Multi-Step Worked Problem

A lighthouse beam rotates once every $10$ seconds. The beam hits a wall $200$ m away. The closest point on the wall to the lighthouse is directly east. Model the position $x$ (in meters north/south of the closest point) of the beam on the wall.

Setup: Let $\theta = \frac{2\pi}{10}t = \frac{\pi}{5}t$ be the angle at time $t$ .

From trigonometry: $x = 200\tan\theta = 200\tan\left(\frac{\pi}{5}t\right)$

Time $t$	Angle $\theta$	$\tan\theta$	Position $x$
$0$ s

Key insight: This is a tangent model because the position can grow to $\pm\infty$ (the beam sweeps past the wall's endpoint). Tangent is the right function when the output is unbounded.

Comprehensive Review 🎯

Cross-Topic Computations 🧮

1) Convert $\frac{11\pi}{6}$ to degrees. Write as an integer. (e.g., $\frac{3\pi}{4} \times \frac{180}{\pi} = \frac{3 \times 180}{4} = 135$ )

2) Given $\sec\theta = -\frac{5}{3}$ with $\theta$ in Q II, find $\tan\theta$ . Write as a fraction with sign. (e.g., in Q I: , so )

3) A wind turbine blade is $40$ m long and sweeps through $150°$ . What arc length does the tip trace (in meters)? Write as a simplified fraction times $\pi$ — give the coefficient as a fraction (e.g., if $s = \frac{20}{3}\pi$ , write ). (e.g., , : , answer: )

Comprehensive Matching 🔽

Reference angle:

\pi - \frac{5\pi}{6} = \frac{\pi}{6}

From the table:

\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}

\sin\frac{\pi}{6} = \frac{1}{2}

In Q II: cosine is negative, sine is positive

\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}

Sine is negative in Q III

\sin\frac{\pi}{4} = \sqrt{2}/2

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

\sin(\frac{\pi}{2} - \theta)

\cos(\frac{\pi}{2} - \alpha) = \sin\alpha = 0.6

\csc^2\alpha = 1 + 9/16 = 25/16

x = \frac{\pi}{4} + \frac{n\pi}{2}

h(0.75) = -20\cos(\pi/2) + 25 = 25

cot θ = \frac{c o s θ}{s i n θ}

y = a cos (b (x - c)) + d

\tan^2\alpha = (13/5)^2 - 1 = 144/25

\theta = 60° = \pi/3

s = 10 \cdot \pi/3 = 10\pi/3

Trigonometric Functions

Trigonometric Functions - Complete Interactive Lesson

Part 1: Unit Circle Fundamentals

📐 Trigonometric Functions — Angles & Radian Measure

Degree ↔ Radian Conversion

Common Angle Reference Table

🔄 What Is a Radian?

📝 Worked Examples

Example 1: Convert 225°225°225° to radians

Part 2: Sine & Cosine Graphs

⭕ Trigonometric Functions — The Unit Circle

Part 3: Tangent & Reciprocal Functions

🌊 Trigonometric Functions — Sine & Cosine Definitions

Right-Triangle Definitions

Part 4: Amplitude & Period

🔺 Trigonometric Functions — Tangent, Cotangent, Secant & Cosecant

Definitions

Part 5: Phase Shifts

📊 Trigonometric Functions — Graphing Sinusoids

Part 6: Problem-Solving Workshop

🛠️ Trigonometric Functions — Applied Problems & Modeling

Modeling Workflow

Part 7: Review & Applications

🏆 Trigonometric Functions — Full Synthesis & Review

Master Decision Flowchart

Key Facts

Example 2: Convert 7π6\frac{7\pi}{6}67π​ to degrees

Example 3: Arc Length

Example 4: Sector Area

First-Quadrant Key Coordinates

🧭 Signs by Quadrant & Reference Angles

ASTC Rule ("All Students Take Calculus")

Reference Angle

Worked Example: Find cos⁡5π6\cos\frac{5\pi}{6}cos65π​ and sin⁡5π6\sin\frac{5\pi}{6}sin6

📋 Full Unit Circle — All Four Quadrants

Worked Example: Find sin⁡4π3\sin\frac{4\pi}{3}sin34π​

Unit-Circle Connection

Key Properties Comparison

🔗 Fundamental Identities

Pythagorean Identity

Cofunction Identities

Worked Example: Using the Pythagorean Identity

📝 Evaluating Sine & Cosine — Strategy

Step-by-Step Method

Example: Evaluate sin⁡300°\sin 300°sin300°

Example: Evaluate cos⁡(−π4)\cos(-\frac{\pi}{4})cos(−4π​)

Complete Comparison Table

🔗 Pythagorean Identity Family

Symmetry Properties

📝 Worked Examples

Example 1: Evaluate all six trig functions at θ=5π6\theta = \frac{5\pi}{6}θ=65π​

Example 2: Given tan⁡θ=−34\tan\theta = -\frac{3}{4}tanθ=−43​ and θ\thetaθ in Q II, find sec⁡θ\sec\thetasec

Parameter Summary

Quick Reference

📝 Worked Examples — Reading Parameters

Example 1: y=3sin⁡(2x)−1y = 3\sin(2x) - 1y=3sin(2x)−1

Example 2: y=−2cos⁡(π3(x−1))+5y = -2\cos\left(\frac{\pi}{3}(x - 1)\right) + 5y=−2cos(3π​(x−1))

Key Difference: Sine vs Cosine Starting Points

📈 Graphing Tangent & Cotangent

Tangent: y=atan⁡(bx−c)+dy = a\tan(bx - c) + dy=atan(bx−c)+d

Example: y=tan⁡(2x)y = \tan(2x)y=tan(2x)

Cotangent: y=acot⁡(bx−c)+dy = a\cot(bx - c) + dy=acot(bx−c)+d

🌊 Example 1: Tidal Height

🌡️ Example 2: Monthly Temperature

🎡 Example 3: Ferris Wheel

📋 Complete Formula Reference

Definitions & Identities

Graphing

Measurement

📝 Multi-Step Worked Problem

Example 1: Convert $225°$ to radians

Example 2: Convert $\frac{7\pi}{6}$ to degrees

Worked Example: Find $\cos\frac{5\pi}{6}$ and $\sin\frac{5\pi}{6}$

Worked Example: Find $\sin\frac{4\pi}{3}$

Example: Evaluate $\sin 300°$

Example: Evaluate $\cos(-\frac{\pi}{4})$

Example 1: Evaluate all six trig functions at $\theta = \frac{5\pi}{6}$

Example 2: Given $\tan\theta = -\frac{3}{4}$ and $\theta$ in Q II, find $\sec\theta$

Example 1: $y = 3\sin(2x) - 1$

Example 2: $y = -2\cos\left(\frac{\pi}{3}(x - 1)\right) + 5$

Tangent: $y = a\tan(bx - c) + d$

Example: $y = \tan(2x)$

Cotangent: $y = a\cot(bx - c) + d$