Trigonometric Functions

📐 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

Memory tip: $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$ correspond to $30°, 45°, 60°$ — the denominators $6, 4, 3$ decrease as the angles increase.

🔄 What Is a Radian?

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

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

Key Facts

📝 Worked Examples

Example 1: Convert $225°$ to radians

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

$\boxed{225° = \frac{5\pi}{4}}$

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

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: $3/2$)

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: $6$)

Angle Fundamentals 🔽

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⭕ 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 $(\cos\theta,\;\sin\theta)$.

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

First-Quadrant Key Coordinates

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")

Reference Angle

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

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

1. $\frac{5\pi}{6}$ is in Quadrant II (between $\frac{\pi}{2}$ and $\pi$)
2. Reference angle: $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$
3. From the table: $\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$, $\sin\frac{\pi}{6} = \frac{1}{2}$
4. In Q II: cosine is negative, sine is positive

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

📋 Full Unit Circle — All Four Quadrants

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

1. $\frac{4\pi}{3}$ is in Q III → reference angle $= \frac{4\pi}{3} - \pi = \frac{\pi}{3}$
2. $\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$
3. Sine is negative in Q III

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

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 $\frac{\pi}{4}$, $\sin\frac{\pi}{4} = \sqrt{2}/2$, negative in Q IV → answer: $-\sqrt{2}/2$)

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 $30° = \pi/6$, answer: $1/6$). (e.g., $\frac{2\pi}{3}$ is in Q II: $\pi - \frac{2\pi}{3} = \frac{\pi}{3}$, answer: $1/3$)

Quadrant & Sign Matching 🔽

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🌊 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 $x$:

$\boxed{\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r} \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r}}$

Unit-Circle Connection

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

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

Key Properties Comparison

🔗 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

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

Example: Evaluate $\sin 300°$

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 → $-1/2$)

3) If $\cos\theta = \frac{\sqrt{3}}{2}$, find $\sin(\frac{\pi}{2} - \theta)$ using the cofunction identity. Write as a simplified expression. (e.g., $\sin\alpha = 0.6$: $\cos(\frac{\pi}{2} - \alpha) = \sin\alpha = 0.6$)

Properties Matching 🔽

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🔺 Trigonometric Functions — Tangent, Cotangent, Secant & Cosecant

Part 4 of 7

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

Definitions

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

Complete Comparison Table

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:

Symmetry Properties

📝 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 $= \frac{\pi}{6}$.

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., $\cot\alpha = 3/4$ in Q I: $\csc^2\alpha = 1 + 9/16 = 25/16$, so $\csc\alpha = 5/4$)

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

Function Properties 🔽

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📊 Trigonometric Functions — Graphing Sinusoids

Part 5 of 7

The general sinusoidal form is:

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

Parameter Summary

Quick Reference

📝 Worked Examples — Reading Parameters

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

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

Key Difference: Sine vs Cosine Starting Points

📈 Graphing Tangent & Cotangent

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

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

Period $= \frac{\pi}{2}$. Asymptotes where $2x = \frac{\pi}{2} + n\pi$, i.e., $x = \frac{\pi}{4} + \frac{n\pi}{2}$.

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

Same period formula $\frac{\pi}{|b|}$, but asymptotes where $bx - c = n\pi$ (where $\sin = 0$).

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 $= 2\pi / \pi = 2$)

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 🔽

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🛠️ 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

🌊 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.

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

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 $T(t)$.

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

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., $h(t) = -20\cos(\frac{2\pi}{3}t) + 25$: $h(0.75) = -20\cos(\pi/2) + 25 = 25$)

Modeling Strategy 🔽

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🏆 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

📋 Complete Formula Reference

Definitions & Identities

Graphing

Measurement

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

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., $\sec\alpha = 13/5$ in Q I: $\tan^2\alpha = (13/5)^2 - 1 = 144/25$, so $\tan\alpha = 12/5$)

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 $20/3$). (e.g., $r = 10$, $\theta = 60° = \pi/3$: $s = 10 \cdot \pi/3 = 10\pi/3$, answer: $10/3$)

Comprehensive Matching 🔽

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