Trigonometry

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Secant, Cosecant, and Cotangent

These three new functions are defined as reciprocals (meaning "one divided by") of the original three:

Secant (sec):

• sec θ = 1/cos θ
• Secant is the reciprocal of cosine

Cosecant (cosec or csc):

• cosec θ = 1/sin θ
• Cosecant is the reciprocal of sine

Cotangent (cot):

• cot θ = 1/tan θ = cos θ/sin θ
• Cotangent is the reciprocal of tangent

Important note: These functions are undefined (meaning they don't exist) when the denominator equals zero. For example:

• sec θ is undefined when cos θ = 0 (at θ = 90°, 270°, etc.)
• cosec θ is undefined when sin θ = 0 (at θ = 0°, 180°, 360°, etc.)
• cot θ is undefined when sin θ = 0 (at θ = 0°, 180°, 360°, etc.)

Example 1: Find the value of sec 60°

Step 1: Remember that cos 60° = 1/2

Step 2: Use the definition sec θ = 1/cos θ

sec 60° = 1/(1/2) = 2

Example 2: Find the value of cosec 30°

Step 1: Remember that sin 30° = 1/2

Step 2: Use the definition cosec θ = 1/sin θ

cosec 30° = 1/(1/2) = 2

Identity 2: cosec²θ = 1 + cot²θ

Derivation: Start with: sin²θ + cos²θ = 1

Divide every term by sin²θ: sin²θ/sin²θ + cos²θ/sin²θ = 1/sin²θ

Simplify (remember cot θ = cos θ/sin θ and cosec θ = 1/sin θ): 1 + cot²θ = cosec²θ

Therefore: cosec²θ = 1 + cot²θ

Example 3: Solve the equation: 2 sec²θ - tan θ = 5 for 0° ≤ θ ≤ 360°

Step 1: Use the identity sec²θ = 1 + tan²θ to replace sec²θ

2(1 + tan²θ) - tan θ = 5

Step 2: Expand and rearrange: 2 + 2tan²θ - tan θ = 5 2tan²θ - tan θ - 3 = 0

Step 3: This is a quadratic equation. Let x = tan θ: 2x² - x - 3 = 0

Step 4: Factorize: (2x - 3)(x + 1) = 0

Step 5: Solve for x: x = 3/2 or x = -1

Step 6: Replace x with tan θ: tan θ = 3/2 or tan θ = -1

Step 7: Find θ:

• When tan θ = 3/2: θ = 56.3° or θ = 236.3° (add 180° for second solution)
• When tan θ = -1: θ = 135° or θ = 315°

Solutions: θ = 56.3°, 135°, 236.3°, 315°

4. Compound Angle Formulae

These formulae allow you to expand expressions like sin(A + B) or cos(A - B).

Addition Formulae:

• 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)

Example 4: Simplify cos(x - 30°) - 3 sin(x - 60°)

Step 1: Expand cos(x - 30°) using cos(A - B) = cos A cos B + sin A sin B: cos(x - 30°) = cos x cos 30° + sin x sin 30° = cos x × (√3/2) + sin x × (1/2) = (√3/2)cos x + (1/2)sin x

Step 2: Expand sin(x - 60°) using sin(A - B) = sin A cos B - cos A sin B: sin(x - 60°) = sin x cos 60° - cos x sin 60° = sin x × (1/2) - cos x × (√3/2) = (1/2)sin x - (√3/2)cos x

Step 3: Substitute into the original expression: cos(x - 30°) - 3 sin(x - 60°) = [(√3/2)cos x + (1/2)sin x] - 3[(1/2)sin x - (√3/2)cos x] = (√3/2)cos x + (1/2)sin x - (3/2)sin x + (3√3/2)cos x

Step 4: Collect like terms: = [(√3/2) + (3√3/2)]cos x + [(1/2) - (3/2)]sin x = (4√3/2)cos x - (2/2)sin x = 2√3 cos x - sin x

5. Double Angle Formulae

When A = B in the compound angle formulae, we get the double angle formulae. These are very useful shortcuts.

Double Angle Formulae:

• sin 2A = 2 sin A cos A
• cos 2A = cos²A - sin²A
  • Also: cos 2A = 2cos²A - 1 (using sin²A = 1 - cos²A)
  • Also: cos 2A = 1 - 2sin²A (using cos²A = 1 - sin²A)
• tan 2A = (2 tan A)/(1 - tan²A)

Example 5: If sin A = 3/5 where A is acute, find the value of sin 2A and cos 2A.

Step 1: Find cos A using sin²A + cos²A = 1: (3/5)² + cos²A = 1 9/25 + cos²A = 1 cos²A = 16/25 cos A = 4/5 (positive because A is acute)

Step 2: Calculate sin 2A using sin 2A = 2 sin A cos A: sin 2A = 2 × (3/5) × (4/5) = 24/25

Step 3: Calculate cos 2A using cos 2A = cos²A - sin²A: cos 2A = (4/5)² - (3/5)² = 16/25 - 9/25 = 7/25

Example 6: Solve tan θ + cot θ = 4 for 0° ≤ θ ≤ 180°

Step 1: Replace cot θ with 1/tan θ: tan θ + 1/tan θ = 4

Step 2: Multiply through by tan θ: tan²θ + 1 = 4 tan θ

Step 3: Rearrange: tan²θ - 4 tan θ + 1 = 0

Step 4: Use the quadratic formula (a = 1, b = -4, c = 1): tan θ = [4 ± √(16 - 4)]/2 = [4 ± √12]/2 = [4 ± 2√3]/2 = 2 ± √3

Step 5: Find θ:

• When tan θ = 2 + √3 ≈ 3.732: θ = 75°
• When tan θ = 2 - √3 ≈ 0.268: θ = 15°

Solutions: θ = 15°, 75°

Method for a sin θ + b cos θ = R sin(θ + α)

Step 1: Expand R sin(θ + α) using the compound angle formula: R sin(θ + α) = R(sin θ cos α + cos θ sin α) = R cos α sin θ + R sin α cos θ

Step 2: Compare coefficients with a sin θ + b cos θ:

• R cos α = a ... (equation 1)
• R sin α = b ... (equation 2)

Step 3: Find R by squaring both equations and adding: R²cos²α + R²sin²α = a² + b² R²(cos²α + sin²α) = a² + b² R² = a² + b² R = √(a² + b²)

Step 4: Find α by dividing equation 2 by equation 1: (R sin α)/(R cos α) = b/a tan α = b/a α = tan⁻¹(b/a)

Note: Check the signs of a and b to determine which form to use:

• If a > 0, b > 0: use R sin(θ + α) or R cos(θ - α)
• If a > 0, b < 0: use R sin(θ - α) or R cos(θ + α)
• If a < 0, b > 0: use R cos(θ - α) or -R sin(θ + α)
• If a < 0, b < 0: use R cos(θ + α) or -R sin(θ - α)

Example 7: Express 3 sin θ + 4 cos θ in the form R sin(θ + α), where R > 0 and 0° < α < 90°.

Step 1: Find R: R = √(3² + 4²) = √(9 + 16) = √25 = 5

Step 2: Find α: tan α = 4/3 α = tan⁻¹(4/3) = 53.1°

Answer: 3 sin θ + 4 cos θ = 5 sin(θ + 53.1°)

Example 8: Solve the equation 3 cos θ + 2 sin θ = 1 for 0° ≤ θ ≤ 360°

Step 1: Express 3 cos θ + 2 sin θ in R-form. Since we have cos first, use R cos(θ - α):

R cos(θ - α) = R cos θ cos α + R sin θ sin α = 2 sin θ + 3 cos θ

Comparing: R cos α = 3 and R sin α = 2

Step 2: Find R: R = √(3² + 2²) = √13

Step 3: Find α: tan α = 2/3 α = 33.7°

Step 4: Rewrite the equation: √13 cos(θ - 33.7°) = 1 cos(θ - 33.7°) = 1/√13 cos(θ - 33.7°) = 0.277

Step 5: Solve: θ - 33.7° = 73.9° or θ - 33.7° = 360° - 73.9° = 286.1° θ = 107.6° or θ = 319.8°

Solutions: θ = 107.6°, 319.8°