Trigonometric functions

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Core Functions and Their Properties

Basic Definitions

• Sine (sin): In a right triangle, sin(θ) = opposite/hypotenuse
• Cosine (cos): In a right triangle, cos(θ) = adjacent/hypotenuse
• Tangent (tan): In a right triangle, tan(θ) = opposite/adjacent = sin(θ)/cos(θ)
• Cotangent (cot): cot(θ) = 1/tan(θ) = cos(θ)/sin(θ)
• Secant (sec): sec(θ) = 1/cos(θ)
• Cosecant (csc): csc(θ) = 1/sin(θ)

Domains and Ranges

• sin(x): Domain: ℝ, Range: [-1, 1]
• cos(x): Domain: ℝ, Range: [-1, 1]

• tan(x): Domain: {x ∈ ℝ

  x ≠ π/2 + nπ}, Range: ℝ

• cot(x): Domain: {x ∈ ℝ

  x ≠ nπ}, Range: ℝ

• sec(x): Domain: {x ∈ ℝ

  x ≠ π/2 + nπ}, Range: (-∞, -1] ∪ [1, ∞)

• csc(x): Domain: {x ∈ ℝ

  x ≠ nπ}, Range: (-∞, -1] ∪ [1, ∞)

Key Function Values

Periodicity

• sin(x + 2π) = sin(x)
• cos(x + 2π) = cos(x)
• tan(x + π) = tan(x)
• cot(x + π) = cot(x)
• sec(x + 2π) = sec(x)
• csc(x + 2π) = csc(x)

Trigonometric Function Properties

Evenness and Oddness

• sin(-x) = -sin(x) (odd function)
• cos(-x) = cos(x) (even function)
• tan(-x) = -tan(x) (odd function)
• cot(-x) = -cot(x) (odd function)
• sec(-x) = sec(x) (even function)
• csc(-x) = -csc(x) (odd function)

Derivatives

• d/dx[sin(x)] = cos(x)
• d/dx[cos(x)] = -sin(x)
• d/dx[tan(x)] = sec²(x)
• d/dx[cot(x)] = -csc²(x)
• d/dx[sec(x)] = sec(x)tan(x)
• d/dx[csc(x)] = -csc(x)cot(x)

Integrals

• ∫sin(x)dx = -cos(x) + C
• ∫cos(x)dx = sin(x) + C

• ∫tan(x)dx = -ln

  cos(x)

  + C

• ∫cot(x)dx = ln

  sin(x)

  + C

• ∫sec(x)dx = ln

  sec(x) + tan(x)

  + C

• ∫csc(x)dx = ln

  csc(x) - cot(x)

  + C

Solving Trigonometric Inequalities

Basic Approach

1. Isolate the trigonometric expression
2. Find critical points by solving the corresponding equation
3. Test intervals between critical points
4. Express the solution in terms of the appropriate interval

Examples for Solving Inequalities like 2sin²x > 1

For 2sin²x > 1:

1. Divide both sides by 2: sin²x > 1/2

2. Take square root:

   sin x

   > 1/√2

3. Solve: sin x > 1/√2 or sin x < -1/√2
4. Find all solutions in [0, 2π]:
   • sin x > 1/√2 gives x ∈ (π/4, 3π/4)
   • sin x < -1/√2 gives x ∈ (5π/4, 7π/4)
5. Extend to all reals: x ∈ (π/4 + 2nπ, 3π/4 + 2nπ) ∪ (5π/4 + 2nπ, 7π/4 + 2nπ), where n ∈ ℤ

For 4sin x cos x + 1 < 0

1. Use identity: sin(2x) = 2sin x cos x
2. Rewrite: 2sin(2x) + 1 < 0
3. Solve: sin(2x) < -1/2
4. Find solutions in [0, 4π]:
   • 2x ∈ (7π/6 + 2nπ, 11π/6 + 2nπ), where n ∈ ℤ
   • x ∈ (7π/12 + nπ, 11π/12 + nπ), where n ∈ ℤ

Special Triangle Relationships

30-60-90 Triangle

• If the shortest leg length is a:
  • The hypotenuse is 2a
  • The longer leg is a√3

45-45-90 Triangle

• If the leg length is a:
  • The hypotenuse is a√2
  • Both legs are equal (a)

See

• Trigonometric Identities
• Trigonometric Equations
• Trigonometric Inequalities
• 30-60-90 Triangle
• 45-45-90 Triangle
• Derivatives - For derivatives of trigonometric functions
• Integration Techniques - For integration of trigonometric functions

Exercises

1. Solve trigonometric inequalities like $2\sin^2 x > 1$, $\cos^2 x \leq 1$, $4\sin x \cos x + 1 < 0$.