📐 Concept diagram

02-08 - Trigonometric Functions

Phase: 2 | Subject: 02-08 Prerequisites: 02-07-unit-circle-and-radians.md Next subject: 02-09-trigonometric-identities.md

Learning Objectives

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

1. Sketch graphs of sin(x), cos(x), and tan(x)
2. Identify amplitude, period, phase shift, and vertical shift
3. Apply transformations to trig functions
4. Find the equation of a transformed trig function from its graph
5. Understand the properties of inverse trig functions

Core Content

Graphs of Basic Trig Functions

sin(x)

• Period: 2π
• Amplitude: 1
• Range: [-1, 1]
• Zeroes at: 0, π, 2π, ...
• Max at π/2, min at 3π/2

cos(x)

• Period: 2π
• Amplitude: 1
• Range: [-1, 1]
• Zeroes at: π/2, 3π/2, ...
• Max at 0, min at π

tan(x)

• Period: π
• No amplitude (unbounded)
• Vertical asymptotes at ±π/2, ±3π/2, ...
• Zeroes at: 0, π, 2π, ...

Transformations: y = a·sin(b(x - c)) + d

⚠️ THIS IS CRITICAL — understanding how a, b, c, d transform any periodic function is essential for signal processing, Fourier analysis, and modelling real-world oscillations.

Example: y = 2·sin(3x) - 1

• Amplitude: 2
• Period: 2π/3
• No phase shift
• Vertical shift: down 1
• Range: [-3, 1]

Example: y = -cos(2(x - π/4))

• Amplitude: 1
• Reflection across x-axis (because of -)
• Period: 2π/2 = π
• Phase shift: right π/4

Finding the Equation from a Graph

Steps: 1. Find the midline (vertical shift d) = (max + min)/2 2. Find amplitude a = (max - min)/2 3. Find the period from the graph, then b = 2π/period 4. Find phase shift by seeing where a "standard" point (like a zero crossing going up) has moved

Inverse Trig Functions

sin⁻¹(x), cos⁻¹(x), tan⁻¹(x) give the ANGLE whose trig function equals x.

Important: To be functions (one output per input), we restrict their domains:

Key Terms

• 02 08 Trigonometric Functions
• Correct: A)
• Correct: C)
• Example 1: Sketch y = 3·cos(2x)
• Example 2: Find equation from a graph
• Example 3: Equations from key features
• Example: y = -cos(2(x - π/4))
• Example: y = 2·sin(3x) - 1
• Finding the Equation from a Graph
• Function
• Graphs of Basic Trig Functions
• Inverse Trig Functions

Worked Examples

Example 1: Sketch y = 3·cos(2x)

• Amplitude: 3
• Period: 2π/2 = π
• No phase shift
• Max at x = 0, 0 at x = π/4, min at x = π/2
• Range: [-3, 3]

Example 2: Find equation from a graph

Graph has max 4, min -2, period π, starts at max when x = π/6.

1. Midline d = (4 + (-2))/2 = 1
2. Amplitude a = (4 - (-2))/2 = 3
3. Period π → b = 2π/π = 2
4. Phase shift: cos starts at max at x = 0, but here max is at π/6. Shift right π/6.
5. Equation: y = 3·cos(2(x - π/6)) + 1

Example 3: Equations from key features

A cosine function oscillates between y = -1 and y = 5 with period π. Its minimum occurs at x = π/3. Find the equation.

1. Midline d = (5 + (-1))/2 = 2
2. Amplitude a = (5 - (-1))/2 = 3
3. Period π → b = 2π/π = 2
4. Cosine normally has its maximum at x = 0. Here the minimum is at π/3. Using a negative amplitude (-3) flips the function so its "max" becomes a min. With -3·cos(2x), the minimum is at x = 0. Shifted right π/3: c = π/3.
5. Equation: y = -3·cos(2(x - π/3)) + 2

Verify: at x = π/3: y = -3·cos(0) + 2 = -1 ✓. At x = π/3 + π/2 = 5π/6: y = -3·cos(π) + 2 = 5 ✓.

Quiz

Q1: What does the concept of Finding the Equation from a Graph primarily refer to in this subject?

A) A visual representation of Finding the Equation from a Graph B) The definition and application of Finding the Equation from a Graph C) A computational error related to Finding the Equation from a Graph D) A historical anecdote about Finding the Equation from a Graph

Correct: B)

• If you chose A: This is incorrect. Finding the Equation from a Graph is defined as: the definition and application of finding the equation from a graph. The other options describe different aspects that are not the primary focus.
• If you chose B: Finding the Equation from a Graph is defined as: the definition and application of finding the equation from a graph. The other options describe different aspects that are not the primary focus. Correct!
• If you chose C: This is incorrect. Finding the Equation from a Graph is defined as: the definition and application of finding the equation from a graph. The other options describe different aspects that are not the primary focus.
• If you chose D: This is incorrect. Finding the Equation from a Graph is defined as: the definition and application of finding the equation from a graph. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Function?

A) It is primarily a historical notation system B) It is used to function in mathematical analysis C) It is used only in advanced research contexts D) It replaces all other methods in this domain

Correct: B)

• If you chose A: This is incorrect. Function serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: Function serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose C: This is incorrect. Function serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: This is incorrect. Function serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about Graphs of Basic Trig Functions is TRUE?

A) Graphs of Basic Trig Functions is a fundamental concept covered in this subject B) Graphs of Basic Trig Functions is an advanced topic beyond this subject's scope C) Graphs of Basic Trig Functions is mentioned only as a historical footnote D) Graphs of Basic Trig Functions is not related to this subject

Correct: A)

• If you chose A: Graphs of Basic Trig Functions is a fundamental concept covered in this subject. This subject covers Graphs of Basic Trig Functions as part of its core content. Correct!
• If you chose B: This is incorrect. Graphs of Basic Trig Functions is a fundamental concept covered in this subject. This subject covers Graphs of Basic Trig Functions as part of its core content.
• If you chose C: This is incorrect. Graphs of Basic Trig Functions is a fundamental concept covered in this subject. This subject covers Graphs of Basic Trig Functions as part of its core content.
• If you chose D: This is incorrect. Graphs of Basic Trig Functions is a fundamental concept covered in this subject. This subject covers Graphs of Basic Trig Functions as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) An unrelated numerical value B) A different result from a common mistake C) The inverse of the correct answer D) π/2

Correct: D)

• If you chose A: This is incorrect. The worked examples show that the result is π/2. The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is π/2. The other options represent common errors.
• If you chose C: This is incorrect. The worked examples show that the result is π/2. The other options represent common errors.
• If you chose D: The worked examples show that the result is π/2. The other options represent common errors. Correct!

Q5: How are Graphs of Basic Trig Functions and Inverse Trig Functions related?

A) Graphs of Basic Trig Functions and Inverse Trig Functions are closely related concepts B) Graphs of Basic Trig Functions is a special case of Inverse Trig Functions C) Graphs of Basic Trig Functions and Inverse Trig Functions are completely unrelated topics D) Graphs of Basic Trig Functions is the inverse of Inverse Trig Functions

Correct: A)

• If you chose A: Both Graphs of Basic Trig Functions and Inverse Trig Functions are covered in this subject as interconnected topics. Correct!
• If you chose B: This is incorrect. Both Graphs of Basic Trig Functions and Inverse Trig Functions are covered in this subject as interconnected topics.
• If you chose C: This is incorrect. Both Graphs of Basic Trig Functions and Inverse Trig Functions are covered in this subject as interconnected topics.
• If you chose D: This is incorrect. Both Graphs of Basic Trig Functions and Inverse Trig Functions are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with Sin(X)?

A) Sin(X) is always computed the same way in all contexts B) Sin(X) has no common misconceptions C) The main error with Sin(X) is using it when it is not needed D) A common mistake is confusing Sin(X) with a similar concept

Correct: D)

• If you chose A: This is incorrect. Students often confuse Sin(X) with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: This is incorrect. Students often confuse Sin(X) with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: This is incorrect. Students often confuse Sin(X) with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: Students often confuse Sin(X) with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!

Q7: When should you apply Cos(X)?

A) Avoid Cos(X) unless explicitly instructed B) Cos(X) is not practically useful C) Use Cos(X) only in pure mathematics contexts D) Apply Cos(X) to solve problems in this subject's domain

Correct: D)

• If you chose A: This is incorrect. Cos(X) is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: This is incorrect. Cos(X) is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: This is incorrect. Cos(X) is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: Cos(X) is a practical tool used throughout this subject to solve relevant problems. Correct!

Practice Problems

1. Period of y = sin(4x) Answer: 2π/4 = π/2

2. Amplitude of y = -2·cos(x) + 5 Answer: |-2| = 2

3. Sketch description: y = 0.5·sin(3x) Answer: Amplitude 0.5, period 2π/3, range [-0.5, 0.5]

4. Range of y = 2·cos(x) - 1 Answer: [-3, 1]

5. If tan⁻¹(1) = θ, what is θ? Answer: π/4 (45°)

6. Period of y = tan(2x) Answer: π/2

Summary

Key takeaways:

• sin, cos have period 2π; tan has period π
• y = a·f(b(x-c)) + d: a=amplitude, b=period, c=phase, d=vertical shift
• Amplitude = |a|; Period = 2π/|b| (sin/cos) or π/|b| (tan)
• Inverse trig functions return angles, restricted to principal values
• Always check for reflections when a < 0

Pitfalls

• Confusing amplitude with vertical shift: Amplitude is |a| (the multiplier on the trig function), not the +d (vertical shift). In y = 4·sin(x) + 2, the amplitude is 4 and the midline is y = 2. The range is [d - |a|, d + |a|].
• Getting the period wrong for tan: sin and cos have period 2π/|b|, but tan has period π/|b|. Using 2π/|b| for tan doubles the correct period. Similarly, using π/|b| for sin or cos halves it.
• Not factoring before identifying phase shift: For y = sin(2x + π), you must rewrite as sin(2(x + π/2)) to see the true phase shift (left π/2). Treating 2x + π as a shift of π without factoring is a very common error.
• Horizontal shift direction confusion: f(x - c) shifts RIGHT by c. f(x + c) shifts LEFT by c. Because the sign inside is opposite to the direction, many students get this backwards.
• Using the wrong inverse trig function range: sin⁻¹ returns values in [-π/2, π/2], cos⁻¹ in [0, π], tan⁻¹ in (-π/2, π/2). Using the wrong range when solving equations can lose or generate invalid solutions.

Next Steps

Next up: 02-09-trigonometric-identities.md