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!

Calculator: Trigonometric Functions

Make sure angle mode is set correctly!

Check indicator: D = Degrees, R = Radians

Function Keys
\(\sin(x)\) sin
\(\cos(x)\) cos
\(\tan(x)\) tan
\(\sin^{-1}(x)\) SHIFT + sin

Example: \(\sin(30°) = 0.5\)

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

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

Key Takeaways

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: Mock Exam 04

Session 04-05 is exam only - no lecture

  • 90 minutes working time plus 15 minutes reading time, 50 points
  • Covers all of Section 04: polynomial, power, rational, exponential, logarithmic and trigonometric functions
  • Allowed: non-programmable calculator, drawing instruments, monolingual dictionary
  • Not allowed: notes, formula sheets, any electronic devices
  • Counts toward the 60% of test points needed for exam admission

Complete Tasks 04-04 and the Mock 04 prep worksheet - the mock exam builds on them!