Session 04-04 - Introduction to Trigonometric Functions

Section 04: Advanced Functions

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz - 10 Minutes

Review from Session 04-03

Work individually for 5 minutes, then we discuss

  1. Evaluate: \(e^{\ln(5)}\)

  2. If an investment grows from €1000 to €2000 in 8 years with continuous compounding, what is the annual rate \(r\)? (Use \(A = Pe^{rt}\))

  3. Solve: \(2^{3x-1} = 64\)

  4. A bacteria population doubles every 4 hours. If you start with 100 bacteria, write the exponential model \(N(t)\) where \(t\) is in hours.

Homework Discussion - 15 Minutes

Your questions from Tasks 04-03

Focus on exponential functions and applications

  • Challenges with exponential growth and decay models
  • Compound interest calculations (discrete vs. continuous)
  • Half-life and doubling time problems
  • Comparing exponential vs. polynomial growth rates
  • Real-world modeling (population, finance, radioactive decay)

Trigonometry introduces periodic (repeating) behavior - a new type of function compared to the always-increasing exponentials! Instead of continuous growth, we’ll see cycles and oscillations.

Learning Objectives

Today’s Goals

By the end of this session, you will be able to:

  • Understand angles in degrees and radians
  • Define sine, cosine, and tangent using the unit circle
  • Calculate exact values for special angles
  • Sketch basic trigonometric function graphs
  • Identify amplitude, period, and phase shifts
  • Apply trigonometry to real-world periodic phenomena

Angles and Their Measurement

Degrees vs. Radians

Two ways to measure angles

  • Full rotation = 360°
  • Right angle = 90°
  • Straight angle = 180°
  • Historical: Based on ancient calendars
  • Full rotation = \(2\pi\) radians
  • Right angle = \(\frac{\pi}{2}\) radians
  • Straight angle = \(\pi\) radians
  • Radians make calculus formulas simpler!
  • They’re the “natural” unit for mathematics

Conversion: \(180° = \pi \text{ radians}\), \(1° = \frac{\pi}{180}\) radians and \(1 \text{ radian} = \frac{180°}{\pi}\)

Why Radians Are Natural

The arc length connection

For a circle with radius \(r\) and central angle \(\theta\) (in radians):

\[\text{Arc length } s = r\theta\]

Why is this great?

  • If \(\theta = 1\) radian, then arc length = radius (that’s the definition!)
  • For a full circle: \(s = r \cdot 2\pi = 2\pi r\) (the circumference formula!)
  • No conversion factors needed - it just works!

The Unit Circle

Defining the Unit Circle I

The unit circle is a circle with:

  • Center at the origin (0, 0)
  • Radius = 1
  • Equation: \(x^2 + y^2 = 1\)

For any angle \(\theta\) from the positive x-axis:

  • The point on the circle is \((\cos\theta, \sin\theta)\)
  • This is the fundamental definition!

Every point on the unit circle can be written as \((cos θ, sin θ)\) for some angle θ!

Defining the Unit Circle II

Think-Pair-Share: Unit Circle Practice

2 minutes individual, 3 minutes pairs, 2 minutes class discussion

Find the coordinates on the unit circle

For each angle, find the point (cos θ, sin θ):

  1. θ = π/2
  2. θ = π
  3. θ = 3π/2
  4. θ = 2π

Discuss: What pattern do you notice as we go around the circle?

The Sine and Cosine Functions

Sine Function Graph

Domain: All real numbers, range: [-1, 1], period: 2π

Cosine Function Graph

Domain: All real numbers, range: [-1, 1], period: 2π -> Shifted by π/2

Negative Transformations

Understanding -sin(x) and -cos(x)

Multiplying by -1 creates a reflection across the x-axis!

The Tangent Function

Definition and Graph

The ratio that creates asymptotes

Why Tangent Matters

Understanding slopes and angles

The tangent function has a special geometric meaning:

\[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\]

  • Navigation & Surveying: Finding heights of buildings or mountains
  • Physics: Calculating angles of projectile motion or inclined planes
  • Computer Graphics: Rotating objects and calculating viewing angles

However, it is likely not important for the FSP and thus we won’t go into too much detail here!

Amplitude and Period

Transformations of Sine and Cosine I

Modifying the basic wave

General form: \(y = A\sin(B(x - C)) + D\)

  • A: Amplitude (height of wave)
  • B: Affects period (Period = 2π/B)
  • C: Phase shift (horizontal shift)
  • D: Vertical shift

You already know the order from functions!

Apply transformations in this order: horizontal shift, horizontal stretch/compress, vertical stretch/compress, vertical shift.

Transformations of Sine and Cosine II

Spot the Error: Trigonometry Mistakes

Can you identify the errors? Work with your neighbor

Time allocation: 5 minutes to find errors, 5 minutes to discuss

Student work:

  1. “Since sin(30°) = 0.5, then sin(60°) = 1”

  2. “tan(90°) = sin(90°)/cos(90°) = 1/0 = ∞”

  3. “The period of sin(3x) is 6π”

  4. “cos²(x) + sin²(x) = 1 only when x = 0”

Reflect

Quickly think about these questions

  • How are sine and cosine related to the unit circle?
  • Why do we use radians instead of degrees in calculus?
  • Think of real-world examples of periodic behavior
  • Can you name three phenomena that oscillate?

Break - 10 Minutes

Real-World Applications

Sound Waves I

Music is trigonometry

A pure musical tone: \(y = A\sin(2\pi ft)\)

  • A = amplitude (volume)
  • f = frequency (pitch)
  • t = time

Question: What happens if we increase the frequency?

Example: Middle A (440 Hz) \[y = \sin(2\pi \cdot 440 \cdot t) = \sin(880\pi t)\]

Sound Waves II

Comparing frequencies

Notice: Doubling the frequency halves the period! The 880 Hz wave completes two cycles in the same time as 440 Hz completes one.

Seasonal Patterns I

Temperature variation

Average daily temperature in many locations: \[T(d) = A\sin\left(\frac{2\pi}{365}(d - C)\right) + T_{avg}\]

where:

  • d = day of year
  • A = amplitude (half the difference between summer/winter)
  • \(T_{avg}\) = average yearly temperature
  • C = phase shift (adjusts when the peak occurs - typically 80-110 days)

Seasonal Patterns II

Hamburg’s temperature model

Using real Hamburg climate data (Weather Spark):

  • Average annual temperature: \(T_{avg} = 9.2°C\)
  • Warmest month (July): ~18°C
  • Coldest months (Jan/Feb): ~1°C
  • Amplitude: \(A = 8.5°C\)

\[T(d) = 8.5\sin\left(\frac{2\pi}{365}(d - 105)\right) + 9.2\]

Seasonal Patterns III

The sine function provides an great fit to Hamburg’s real climate data!

Guided Practice - 35 Minutes

Task 1: Analyzing Function Properties

Work alone for 5 minutes, then discuss for 3 minutes

For \(y = 3\sin(2x) - 1\), find:

  1. Amplitude
  2. Period
  3. Vertical shift
  4. Range

Task 2: Tidal Heights

Work alone for 5 minutes, then discuss for 3 minutes

The water depth in a harbor varies with the tides. At high tide, the water is 12 meters deep. At low tide, it is 4 meters deep. High tide occurs at noon, and the tide cycle repeats every 12 hours.

Write a function d(t) for the water depth t hours after noon.

Hint: What is the average depth? What is the amplitude?

Task 3: Matching Graphs

Work in pairs for 5 minutes, then discuss for 3 minutes

Match each equation to its description:

Equations:

  1. \(y = 2\sin(x)\)

  2. \(y = \sin(2x)\)

  3. \(y = \sin(x) + 2\)

  4. \(y = \sin\left(x - \frac{\pi}{2}\right)\)

Descriptions:

  1. Is shifted up 2 units
  2. Has amplitude 2
  3. Completes two cycles in 2π
  4. Looks like the cosine function

Task 4: Blood Pressure Modeling

Work alone for 7 minutes, then discuss for 4 minutes

A person’s blood pressure oscillates with each heartbeat. Suppose a person has: a maximum pressure (systolic): 120 mmHg, minimum pressure (diastolic): 80 mmHg and a heart rate: 72 beats per minute.

Questions:

  1. What is the amplitude of the blood pressure oscillation?
  2. What is the period in minutes?
  3. Write a function P(t) for blood pressure at time t minutes, assuming pressure starts at maximum.

Task 5: Ferris Wheel Challenge

Work alone for 5 minutes, then discuss for 3 minutes

A Ferris wheel with radius 20 meters completes one rotation every 4 minutes. The bottom of the wheel is 2 meters above ground. Write a function for the height of a rider at time t (in minutes), starting at the bottom.

Hints to consider:

  • What is the center height of the wheel?
  • What is the amplitude of the up-and-down motion?
  • What is the period of rotation?
  • Which function starts at the bottom: sine or cosine?

Coffee Break - 15 Minutes

Inverse Trigonometric Functions

Brief Addition

Going backwards

Sometimes we need to find the angle:

  • If sin(θ) = 0.5, what is θ?
  • Answer: θ = arcsin(0.5) = π/6 (or 30°)

Question: But wait! Doesn’t sin(150°) also equal 0.5?

Yes! That’s why we need restrictions…

Restricted Inverse

The inverse functions

  • arcsin(x) or sin⁻¹(x): gives angle whose sine is x
  • arccos(x) or cos⁻¹(x): gives angle whose cosine is x
  • arctan(x) or tan⁻¹(x): gives angle whose tangent is x

To make inverses work, we restrict the output ranges (also called principal values):

  • arcsin: [-π/2, π/2] (from -90° to 90°)
  • arccos: [0, π] (from 0° to 180°)
  • arctan: (-π/2, π/2) (from -90° to 90°, not including endpoints)

Visualizing Inverse Relationships

How sine and its inverse relate

Inverse functions are reflections across the line y = x.

Challenge Problem

Combining Waves

Work individually for 8 minutes, then discuss for 4 minutes

Consider two sound waves where the combined wave is: \(y = y_1 + y_2\).

  • Wave 1: \(y_1 = 2\sin(3x)\)
  • Wave 2: \(y_2 = \sin(3x + \pi)\)
  1. Use the fact that \(\sin(x + \pi) = -\sin(x)\) to simplify y in terms of a single sine function.
  2. What is the amplitude and period of the combined wave?
  3. What happens if Wave 2 had amplitude 2: \(y_2 = 2\sin(3x + \pi) = -2\sin(3x)\)?

Key Concepts

Summary

You’ve learned

  • Angle measurement in degrees and radians
  • The unit circle and its significance
  • Sine, cosine, and tangent functions
  • Graphing and transformations
  • Real-world periodic phenomena

Final Assessment

5 minutes - Individual work

Quick Check:

  1. Convert 45° to radians

  2. What is the period of y = sin(4x)?

  3. What is the amplitude of y = -3cos(x) + 2?

Looking Ahead

Next Session Preview

New Function Types

  • Rational functions: \(f(x) = \frac{p(x)}{q(x)}\)
  • Radical functions beyond square root
  • Reciprocal transformations

Key Concepts

  • Asymptotic behavior
  • Domain restrictions
  • Holes vs. asymptotes
  • End behavior analysis

Complete Tasks 04-04: Practice with angles, exact values, graphing, and real-world applications

Final Thought

Why Trigonometry?

Trigonometry is everywhere

From your heartbeat to the tides, from music to earthquakes, trigonometry describes the rhythms of our world.

Your phone’s GPS? Triangulation with satellites Weather prediction? Modeling atmospheric waves Computer graphics? Rotation matrices Medical imaging? Fourier transforms