📐 Concept diagram

02-09 - Trigonometric Identities

Phase: 2 | Subject: 02-09 Prerequisites: 02-08-trigonometric-functions.md Next subject: 02-10-vectors-basic.md

Learning Objectives

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

1. Recall and apply reciprocal identities
2. Use Pythagorean identities to simplify expressions
3. Apply sum/difference formulas for sine and cosine
4. Use double-angle formulas
5. Solve basic trigonometric equations

Core Content

Reciprocal Identities

$csc(θ) = 1/sin(θ)   sec(θ) = 1/cos(θ)   cot(θ) = 1/tan(θ)
$

Example: If sin(θ) = 3/5, then csc(θ) = 5/3.

Pythagorean Identities

⚠️ THIS IS CRITICAL — sin²(θ) + cos²(θ) = 1 is the single most important trig identity. It is derived directly from the unit circle and Pythagoras, and it underlies nearly every trig simplification and proof you will encounter.

Primary:

$sin²(θ) + cos²(θ) = 1
$

Derived (divide by cos² or sin²):

$tan²(θ) + 1 = sec²(θ)
1 + cot²(θ) = csc²(θ)
$

Example: If sin(θ) = 4/5 and θ is in Quadrant II, find cos(θ).

sin²(θ) + cos²(θ) = 1 (4/5)² + cos²(θ) = 1 16/25 + cos²(θ) = 1 cos²(θ) = 9/25 cos(θ) = ±3/5

Quadrant II: cos is negative. cos(θ) = -3/5.

Sum and Difference Formulas

$sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))
$

Memory tip: Sine sums keep the same sign; cosine sums flip sign.

Double-Angle Formulas

$sin(2θ) = 2sin(θ)cos(θ)

cos(2θ) = cos²(θ) - sin²(θ)
         = 2cos²(θ) - 1
         = 1 - 2sin²(θ)

tan(2θ) = 2tan(θ) / (1 - tan²(θ))
$

The three forms of cos(2θ) are useful depending on what you know: - If you know sin(θ), use: 1 - 2sin²(θ) - If you know cos(θ), use: 2cos²(θ) - 1 - If you know both, use: cos²(θ) - sin²(θ)

Solving Trig Equations

Example: Solve 2sin(θ) = 1 for 0° ≤ θ < 360°

1. sin(θ) = 1/2
2. Reference angle: 30°
3. Sine is positive in Quadrants I and II
4. Solutions: θ = 30°, 150°

General solution: θ = 30° + 360°n or θ = 150° + 360°n (for integer n)

Key Terms

• 02 09 Trigonometric Identities
• Correct: A)
• Correct: B)
• Correct: C)
• Double-Angle Formulas
• Example 1: Simplify using Pythagorean identity
• Example 2: Find exact value
• Example 3: Solve 2cos²(x) - 3cos(x) + 1 = 0, 0 ≤ x < 2π
• Pythagorean Identities
• Reciprocal Identities
• Solving Trig Equations
• Sum and Difference Formulas

Worked Examples

Example 1: Simplify using Pythagorean identity

Simplify: cos²(x) - sin²(x) + 2sin²(x)

1. Group: cos²(x) + (-sin²(x) + 2sin²(x)) = cos²(x) + sin²(x)
2. Using the identity cos²(x) + sin²(x) = 1
3. Result: 1

This shows how combining terms reveals the fundamental Pythagorean identity.

Example 2: Find exact value

Find cos(75°) using sum formula.

cos(75°) = cos(45° + 30°) = cos(45°)cos(30°) - sin(45°)sin(30°) = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4

Example 3: Solve 2cos²(x) - 3cos(x) + 1 = 0, 0 ≤ x < 2π

1. Quadratic in cos(x): let u = cos(x)
2. 2u² - 3u + 1 = 0
3. (2u - 1)(u - 1) = 0
4. u = 1/2 or u = 1
5. cos(x) = 1/2: x = π/3, 5π/3
6. cos(x) = 1: x = 0
7. Solutions: x = 0, π/3, 5π/3

Quiz

Q1: What does the concept of Double-Angle Formulas primarily refer to in this subject?

A) A computational error related to Double-Angle Formulas B) A historical anecdote about Double-Angle Formulas C) The definition and application of Double-Angle Formulas D) A visual representation of Double-Angle Formulas

Correct: C)

• If you chose A: This is incorrect. Double-Angle Formulas is defined as: the definition and application of double-angle formulas. The other options describe different aspects that are not the primary focus.
• If you chose B: This is incorrect. Double-Angle Formulas is defined as: the definition and application of double-angle formulas. The other options describe different aspects that are not the primary focus.
• If you chose C: Double-Angle Formulas is defined as: the definition and application of double-angle formulas. The other options describe different aspects that are not the primary focus. Correct!
• If you chose D: This is incorrect. Double-Angle Formulas is defined as: the definition and application of double-angle formulas. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Pythagorean Identities?

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

Correct: C)

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

Q3: Which statement about Reciprocal Identities is TRUE?

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

Correct: A)

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

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

A) The inverse of the correct answer B) An unrelated numerical value C) 4/5 (positive in QI). D) A different result from a common mistake

Correct: C)

• If you chose A: This is incorrect. The worked examples show that the result is 4/5 (positive in QI).. The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is 4/5 (positive in QI).. The other options represent common errors.
• If you chose C: The worked examples show that the result is 4/5 (positive in QI).. The other options represent common errors. Correct!
• If you chose D: This is incorrect. The worked examples show that the result is 4/5 (positive in QI).. The other options represent common errors.

Q5: How are Reciprocal Identities and Solving Trig Equations related?

A) Reciprocal Identities is the inverse of Solving Trig Equations B) Reciprocal Identities is a special case of Solving Trig Equations C) Reciprocal Identities and Solving Trig Equations are completely unrelated topics D) Reciprocal Identities and Solving Trig Equations are closely related concepts

Correct: D)

• If you chose A: This is incorrect. Both Reciprocal Identities and Solving Trig Equations are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both Reciprocal Identities and Solving Trig Equations are covered in this subject as interconnected topics.
• If you chose C: This is incorrect. Both Reciprocal Identities and Solving Trig Equations are covered in this subject as interconnected topics.
• If you chose D: Both Reciprocal Identities and Solving Trig Equations are covered in this subject as interconnected topics. Correct!

Q6: What is a common pitfall when working with Sum and Difference Formulas?

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

Correct: B)

• If you chose A: This is incorrect. Students often confuse Sum and Difference Formulas with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: Students often confuse Sum and Difference Formulas with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose C: This is incorrect. Students often confuse Sum and Difference Formulas with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: This is incorrect. Students often confuse Sum and Difference Formulas with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply Example 1: Simplify Using Pythagorean Identity?

A) Use Example 1: Simplify Using Pythagorean Identity only in pure mathematics contexts B) Avoid Example 1: Simplify Using Pythagorean Identity unless explicitly instructed C) Apply Example 1: Simplify Using Pythagorean Identity to solve problems in this subject's domain D) Example 1: Simplify Using Pythagorean Identity is not practically useful

Correct: C)

• If you chose A: This is incorrect. Example 1: Simplify Using Pythagorean Identity is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: This is incorrect. Example 1: Simplify Using Pythagorean Identity is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: Example 1: Simplify Using Pythagorean Identity is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose D: This is incorrect. Example 1: Simplify Using Pythagorean Identity is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. If sin(θ) = 3/5 and θ is acute, find cos(θ). Answer: cos²(θ) = 1 - 9/25 = 16/25. cos(θ) = 4/5 (positive in QI).

2. Exact value: sin(15°) using difference formula. Answer: sin(45° - 30°) = sin(45°)cos(30°) - cos(45°)sin(30°) = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4

3. Simplify: sec²(x) - tan²(x) Answer: 1 (Pythagorean identity)

4. Solve sin(x) = √3/2 for 0 ≤ x < 2π Answer: x = π/3, 2π/3

5. If tan(θ) = 2 and θ is acute, find sin(2θ). Answer: sin(2θ) = 2tan(θ)/(1+tan²(θ)) = 4/5. Or: tan = 2/1, so opposite = 2, adjacent = 1, hypotenuse = √5. sin(θ) = 2/√5, cos(θ) = 1/√5. sin(2θ) = 2(2/√5)(1/√5) = 4/5.

Summary

Key takeaways:

• Reciprocal: csc = 1/sin, sec = 1/cos, cot = 1/tan
• Pythagorean: sin² + cos² = 1 (most important!)
• Sum/difference: sin(A±B), cos(A±B) formulas
• Double-angle: sin(2θ), cos(2θ), tan(2θ)
• Three forms of cos(2θ) for different known values
• Trig equations may have multiple solutions in [0, 2π)

Pitfalls

• Getting the cos sum/difference sign wrong: cos(A + B) = cos(A)cos(B) - sin(A)sin(B) (minus sign). cos(A - B) = cos(A)cos(B) + sin(A)sin(B) (plus sign). The cosine formulas FLIP the sign — this is one of the most memorised-then-forgotten details in trig.
• Forgetting ± when solving from a squared identity: If sin²(θ) = 1/4, then sin(θ) = ±1/2. The quadrant determines the sign, but you must consider both possibilities. Dropping the ± loses solutions.
• Choosing the wrong form of cos(2θ): cos(2θ) has three equivalent forms (cos²θ - sin²θ, 2cos²θ - 1, 1 - 2sin²θ). Picking the wrong one for the problem means more work. Match the form to what you know: use 2cos²θ - 1 when you know cos θ; use 1 - 2sin²θ when you know sin θ.
• Missing solutions in trig equations: Sine is positive in QI and QII, cos is positive in QI and QIV, tan is positive in QI and QIII. After finding the principal value, always check if there's a second solution in [0, 2π) based on ASTC.
• Confusing reciprocal with inverse: csc(θ) = 1/sin(θ) is a reciprocal function. sin⁻¹(x) is the inverse function (arcsine). These are completely different things. csc(x) ≠ sin⁻¹(x).

Next Steps

Next up: 02-10-vectors-basic.md