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Mastering Trigonometric Identities: The Ultimate Guide with Formulas and Tips

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Comprehensive Guide to Trigonometric Identities

What are Trigonometric Identities?

Trigonometric identities are equations involving trigonometric functions that are true for all values within their domains. They form the foundation for solving complex trigonometric problems and have numerous applications in mathematics, physics, engineering, and other fields.

1. Fundamental Trigonometric Identities

Pythagorean Identities

These identities are derived from the Pythagorean theorem as applied to the unit circle.

sin²θ + cos²θ = 1
tan²θ + 1 = sec²θ
1 + cot²θ = csc²θ

Example 1: Verify that sin²(π/3) + cos²(π/3) = 1

sin(π/3) = √3/2 and cos(π/3) = 1/2
sin²(π/3) + cos²(π/3) = (√3/2)² + (1/2)²
= 3/4 + 1/4 = 4/4 = 1
Therefore, sin²(π/3) + cos²(π/3) = 1

Example 2: Use the Pythagorean identity to find cos θ if sin θ = 3/5 and θ is in the first quadrant.

Using sin²θ + cos²θ = 1
cos²θ = 1 - sin²θ = 1 - (3/5)² = 1 - 9/25 = 16/25
cos θ = ±√(16/25) = ±4/5
Since θ is in the first quadrant, cos θ is positive.
Therefore, cos θ = 4/5

Quotient Identities

These identities express tangent and cotangent in terms of sine and cosine.

tan θ = sin θ / cos θ
cot θ = cos θ / sin θ

Example 1: Calculate tan(π/4) using the quotient identity.

sin(π/4) = 1/√2 and cos(π/4) = 1/√2
tan(π/4) = sin(π/4) / cos(π/4) = (1/√2) / (1/√2) = 1

Example 2: If sin θ = 5/13 and cos θ = 12/13, find tan θ using the quotient identity.

tan θ = sin θ / cos θ = (5/13) / (12/13) = (5/13) × (13/12) = 5/12

Reciprocal Identities

These identities express the relationship between trigonometric functions and their reciprocals.

sin θ = 1/csc θ
cos θ = 1/sec θ
tan θ = 1/cot θ
csc θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ

Example 1: If sec θ = 5, find cos θ using the reciprocal identity.

cos θ = 1/sec θ = 1/5 = 0.2

Example 2: Express the expression sin θ + csc θ in terms of sin θ only.

Using csc θ = 1/sin θ
sin θ + csc θ = sin θ + 1/sin θ = (sin²θ + 1)/sin θ

Even-Odd Identities

These identities describe the behavior of trigonometric functions with negative angles.

Odd Functions:
sin(-θ) = -sin θ
tan(-θ) = -tan θ
csc(-θ) = -csc θ

Even Functions:
cos(-θ) = cos θ
sec(-θ) = sec θ
cot(-θ) = -cot θ

Example 1: Simplify sin(-135°) using even-odd identities.

Using sin(-θ) = -sin θ
sin(-135°) = -sin(135°)
sin(135°) = sin(180° - 45°) = sin(45°) = 1/√2
Therefore, sin(-135°) = -(1/√2) = -1/√2

Example 2: Simplify cos(-π/3) using even-odd identities.

Using cos(-θ) = cos θ
cos(-π/3) = cos(π/3) = 1/2

Cofunction Identities

These identities relate a trigonometric function of an angle to the corresponding cofunction of the complement.

sin(π/2 - θ) = cos θ
cos(π/2 - θ) = sin θ
tan(π/2 - θ) = cot θ
cot(π/2 - θ) = tan θ
sec(π/2 - θ) = csc θ
csc(π/2 - θ) = sec θ

Example 1: Verify that sin(π/2 - π/6) = cos(π/6)

sin(π/2 - π/6) = sin(π/3) = √3/2
cos(π/6) = √3/2
Since sin(π/2 - π/6) = cos(π/6), the identity is verified.

Example 2: Use cofunction identities to find tan(π/2 - θ) if tan θ = 3/4.

Using tan(π/2 - θ) = cot θ
cot θ = 1/tan θ = 1/(3/4) = 4/3
Therefore, tan(π/2 - θ) = 4/3

2. Sum and Difference Identities

Sine Sum and Difference:
sin(α + β) = sin α cos β + cos α sin β
sin(α - β) = sin α cos β - cos α sin β

Cosine Sum and Difference:
cos(α + β) = cos α cos β - sin α sin β
cos(α - β) = cos α cos β + sin α sin β

Tangent Sum and Difference:
tan(α + β) = (tan α + tan β) / (1 - tan α tan β)
tan(α - β) = (tan α - tan β) / (1 + tan α tan β)

Problem: Find the exact value of sin(75°) using sum formulas.

Solution:

We can write 75° as 45° + 30°
Using sin(α + β) = sin α cos β + cos α sin β
sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°)
sin(45°) = 1/√2, cos(30°) = √3/2, cos(45°) = 1/√2, sin(30°) = 1/2
sin(75°) = (1/√2)(√3/2) + (1/√2)(1/2)
= √3/(2√2) + 1/(2√2)
= (√3 + 1)/(2√2)
= (√3 + 1)/(2√2) × (√2/√2) = (√3 + 1)/(2√2) × (√2/√2) = (√3 + 1)√2/4

Answer: sin(75°) = (√3 + 1)√2/4

Problem: Verify the identity: cos(α - β) - cos(α + β) = 2sin α sin β

Solution:

Start with the left side: cos(α - β) - cos(α + β)
Using the cosine difference formula: cos(α - β) = cos α cos β + sin α sin β
Using the cosine sum formula: cos(α + β) = cos α cos β - sin α sin β
Substituting: [cos α cos β + sin α sin β] - [cos α cos β - sin α sin β]
= cos α cos β + sin α sin β - cos α cos β + sin α sin β
= 2sin α sin β
Therefore, cos(α - β) - cos(α + β) = 2sin α sin β is verified.

3. Double-Angle Identities

Sine Double-Angle:
sin(2θ) = 2sin θ cos θ

Cosine Double-Angle:
cos(2θ) = cos²θ - sin²θ
cos(2θ) = 2cos²θ - 1
cos(2θ) = 1 - 2sin²θ

Tangent Double-Angle:
tan(2θ) = 2tan θ / (1 - tan²θ)

Problem: If sin θ = 3/5 and θ is in the first quadrant, find sin(2θ).

Solution:

Using sin(2θ) = 2sin θ cos θ
We know sin θ = 3/5, but we need cos θ.
Using sin²θ + cos²θ = 1
cos²θ = 1 - sin²θ = 1 - (3/5)² = 1 - 9/25 = 16/25
Since θ is in first quadrant, cos θ = 4/5
sin(2θ) = 2sin θ cos θ = 2(3/5)(4/5) = 2(12/25) = 24/25

Answer: sin(2θ) = 24/25

Problem: Express 4cos²θ - 3 in terms of cos(2θ).

Solution:

Using cos(2θ) = 2cos²θ - 1
Solving for cos²θ: cos²θ = (cos(2θ) + 1)/2
4cos²θ - 3 = 4((cos(2θ) + 1)/2) - 3
= 2(cos(2θ) + 1) - 3
= 2cos(2θ) + 2 - 3
= 2cos(2θ) - 1

Answer: 4cos²θ - 3 = 2cos(2θ) - 1

4. Half-Angle Identities

Sine Half-Angle:
sin(θ/2) = ±√((1 - cos θ)/2)

Cosine Half-Angle:
cos(θ/2) = ±√((1 + cos θ)/2)

Tangent Half-Angle:
tan(θ/2) = (1 - cos θ)/sin θ = sin θ/(1 + cos θ)

The correct sign for the half-angle formulas depends on the quadrant where θ/2 lies:

  • If θ/2 is in the first quadrant (0° to 90°), both sin(θ/2) and cos(θ/2) are positive.
  • If θ/2 is in the second quadrant (90° to 180°), sin(θ/2) is positive and cos(θ/2) is negative.
  • If θ/2 is in the third quadrant (180° to 270°), both sin(θ/2) and cos(θ/2) are negative.
  • If θ/2 is in the fourth quadrant (270° to 360°), sin(θ/2) is negative and cos(θ/2) is positive.

Problem: Find the exact value of sin(15°) using the half-angle formula.

Solution:

We can write 15° as 30°/2
Using sin(θ/2) = ±√((1 - cos θ)/2)
sin(15°) = sin(30°/2) = √((1 - cos(30°))/2)
cos(30°) = √3/2
sin(15°) = √((1 - √3/2)/2)
= √((2 - √3)/4)
= √(2 - √3)/2
Since 15° is in the first quadrant, we take the positive value.

Answer: sin(15°) = √(2 - √3)/2

Problem: Solve the equation 2sin²(θ/2) = 1 - cos θ for 0 ≤ θ < 2π.

Solution:

Using the half-angle formula: sin²(θ/2) = (1 - cos θ)/2
Substituting into the equation: 2((1 - cos θ)/2) = 1 - cos θ
The equation becomes: 1 - cos θ = 1 - cos θ
This is an identity, true for all values of θ.
Therefore, all values of θ in the range 0 ≤ θ < 2π are solutions.

Answer: All values of θ in the range 0 ≤ θ < 2π are solutions.

5. Product-to-Sum Identities

Product-to-Sum Formulas:
sin α sin β = ½[cos(α - β) - cos(α + β)]
cos α cos β = ½[cos(α - β) + cos(α + β)]
sin α cos β = ½[sin(α + β) + sin(α - β)]
cos α sin β = ½[sin(α + β) - sin(α - β)]

Problem: Express sin(3x) sin(2x) as a sum or difference of cosines.

Solution:

Using sin α sin β = ½[cos(α - β) - cos(α + β)]
sin(3x) sin(2x) = ½[cos(3x - 2x) - cos(3x + 2x)]
= ½[cos(x) - cos(5x)]

Answer: sin(3x) sin(2x) = ½[cos(x) - cos(5x)]

6. Sum-to-Product Identities

Sum-to-Product Formulas:
sin α + sin β = 2sin((α + β)/2)cos((α - β)/2)
sin α - sin β = 2cos((α + β)/2)sin((α - β)/2)
cos α + cos β = 2cos((α + β)/2)cos((α - β)/2)
cos α - cos β = -2sin((α + β)/2)sin((α - β)/2)

Problem: Express sin(5x) + sin(3x) as a product.

Solution:

Using sin α + sin β = 2sin((α + β)/2)cos((α - β)/2)
sin(5x) + sin(3x) = 2sin((5x + 3x)/2)cos((5x - 3x)/2)
= 2sin(4x)cos(x)

Answer: sin(5x) + sin(3x) = 2sin(4x)cos(x)

7. Strategies for Proving Trigonometric Identities

Key Strategies:

  1. Work with one side at a time until it matches the other side
  2. Convert all functions to sine and cosine, then simplify
  3. Look for opportunities to use fundamental identities (Pythagorean, double-angle, etc.)
  4. Factor expressions when possible
  5. Find common denominators for fractions
  6. Multiply by a strategic form of 1 (e.g., (sin θ/sin θ) or (cos θ/cos θ))

Problem: Prove that (1 + tan²θ)/(1 + cot²θ) = tan²θ

Solution:

Start with the left side: (1 + tan²θ)/(1 + cot²θ)
Using the identity 1 + tan²θ = sec²θ: (sec²θ)/(1 + cot²θ)
Using the identity 1 + cot²θ = csc²θ: (sec²θ)/(csc²θ)
Using secant and cosecant reciprocal identities: (1/cos²θ)/(1/sin²θ)
= (1/cos²θ) × (sin²θ/1)
= sin²θ/cos²θ
= (sin θ/cos θ)²
= tan²θ
Therefore, (1 + tan²θ)/(1 + cot²θ) = tan²θ is proven.

Problem: Prove that (sin θ - sin θ cos²θ)/(sin θ) = sin²θ

Solution:

Start with the left side: (sin θ - sin θ cos²θ)/(sin θ)
Factor out sin θ in the numerator: sin θ(1 - cos²θ)/(sin θ)
Simplify by canceling sin θ: (1 - cos²θ)
Using the identity sin²θ + cos²θ = 1: sin²θ
Therefore, (sin θ - sin θ cos²θ)/(sin θ) = sin²θ is proven.

8. Applications of Trigonometric Identities

Calculus Applications

Example: Integration Using Identities

Evaluate ∫sin²x dx

Using the identity sin²x = (1 - cos(2x))/2
∫sin²x dx = ∫(1 - cos(2x))/2 dx
= (1/2)∫dx - (1/2)∫cos(2x)dx
= (x/2) - (1/2)(sin(2x)/2) + C
= (x/2) - (sin(2x)/4) + C

Example: Differentiation Using Identities

Find d/dx[sin(2x) cos(3x)]

Using the product rule: d/dx[sin(2x) cos(3x)] = cos(2x) · 2 · cos(3x) + sin(2x) · (-sin(3x)) · 3
= 2cos(2x)cos(3x) - 3sin(2x)sin(3x)
Alternatively, using the product-to-sum formula first:
sin(2x) cos(3x) = (1/2)[sin(2x+3x) + sin(2x-3x)] = (1/2)[sin(5x) + sin(-x)] = (1/2)[sin(5x) - sin(x)]
Then differentiating: d/dx[(1/2)(sin(5x) - sin(x))] = (1/2)(5cos(5x) - cos(x))

Simplification Applications

Example: Simplifying Complex Expressions

Simplify (sin x + cos x)²

(sin x + cos x)² = sin²x + 2sin x cos x + cos²x
Using sin²x + cos²x = 1: = 1 + 2sin x cos x
Using the identity 2sin x cos x = sin(2x): = 1 + sin(2x)

Example: Finding Exact Values

Find the exact value of cos(15°) using identities.

15° = 45° - 30°
Using cos(α - β) = cos α cos β + sin α sin β
cos(15°) = cos(45° - 30°) = cos(45°)cos(30°) + sin(45°)sin(30°)
= (1/√2)(√3/2) + (1/√2)(1/2)
= √3/(2√2) + 1/(2√2)
= (√3 + 1)/(2√2)
= (√3 + 1)√2/4

Solving Trigonometric Equations

Example: Using Identities to Solve Equations

Solve sin(2x) = sin x for 0 ≤ x < 2π

Since sin(2x) = sin x, either:
  1. 2x = x + 2nπ, or
  2. 2x = π - x + 2nπ
From case 1: 2x = x + 2nπ ⟹ x = 2nπ
From case 2: 2x = π - x + 2nπ ⟹ 3x = π + 2nπ ⟹ x = (π + 2nπ)/3
For 0 ≤ x < 2π, the solutions are:
  • From case 1: x = 0, x = 2π (but 2π is outside our range)
  • From case 2: x = π/3, x = π, x = 5π/3
Therefore, the solutions are x = 0, π/3, π, 5π/3

Example: Using Double-Angle Formula

Solve cos(2x) + cos x = 0 for 0 ≤ x < 2π

Using the double-angle formula: cos(2x) = 2cos²x - 1
Substituting: (2cos²x - 1) + cos x = 0
2cos²x + cos x - 1 = 0
Setting u = cos x: 2u² + u - 1 = 0
Using the quadratic formula: u = (-1 ± √(1 + 8))/4 = (-1 ± 3)/4
u = 1/2 or u = -1
Therefore, cos x = 1/2 or cos x = -1
When cos x = 1/2, x = ±π/3 + 2nπ, which gives x = π/3, 5π/3 in our range
When cos x = -1, x = π + 2nπ, which gives x = π in our range
The solutions are x = π/3, π, 5π/3

9. Identity Verifier

Use this tool to verify basic trigonometric identities for specific angle values:

10. Trigonometric Identities Quiz

Test Your Knowledge

1. Which of the following is a Pythagorean identity?

2. If sin θ = 4/5 and θ is in the first quadrant, what is cos(2θ)?

3. The identity sin(-θ) = -sin θ illustrates that sine is:

4. Which identity would be most helpful to evaluate sin(75°)?

5. Simplify the expression: (sin²θ)(1 + cot²θ)

6. The expression cos α cos β - sin α sin β is equivalent to:

7. What is cos(-x) in terms of cos(x)?

8. Which identity is used to convert sin²θ to a function of 2θ?

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