1. Definition of Trigonometric Functions

For an angle θ in a right-angled triangle:

• sin(θ) = Opposite side / Hypotenuse
• cos(θ) = Adjacent side / Hypotenuse
• tan(θ) = Opposite side / Adjacent side

On the unit circle:

• sin(θ) = y-coordinate
• cos(θ) = x-coordinate
• tan(θ) = y / x (where x ≠ 0)

2. Trigonometric Identities

Basic Identities

• Pythagorean Identity: sin2(θ) + cos2(θ) = 1
• Reciprocal Identities:
  • csc(θ) = 1 / sin(θ)
  • sec(θ) = 1 / cos(θ)
  • cot(θ) = 1 / tan(θ)
• Quotient Identities:
  • tan(θ) = sin(θ) / cos(θ)
  • cot(θ) = cos(θ) / sin(θ)

Co-Function Identities

• sin(90° - θ) = cos(θ)
• cos(90° - θ) = sin(θ)
• tan(90° - θ) = cot(θ)
• cot(90° - θ) = tan(θ)
• sec(90° - θ) = csc(θ)
• csc(90° - θ) = sec(θ)

Even-Odd Identities

• Even Functions:
  • cos(-θ) = cos(θ)
  • sec(-θ) = sec(θ)
• Odd Functions:
  • sin(-θ) = -sin(θ)
  • tan(-θ) = -tan(θ)
  • cot(-θ) = -cot(θ)
  • csc(-θ) = -csc(θ)

Sum and Difference Identities

• Sum Formulas:
  • sin(α + β) = sin(α)cos(β) + cos(α)sin(β)
  • cos(α + β) = cos(α)cos(β) - sin(α)sin(β)
  • tan(α + β) = (tan(α) + tan(β)) / (1 - tan(α)tan(β))
• Difference Formulas:
  • sin(α - β) = sin(α)cos(β) - cos(α)sin(β)
  • cos(α - β) = cos(α)cos(β) + sin(α)sin(β)
  • tan(α - β) = (tan(α) - tan(β)) / (1 + tan(α)tan(β))

Double Angle Identities

• Sine: sin(2θ) = 2sin(θ)cos(θ)
• Cosine:
  • cos(2θ) = cos2(θ) - sin2(θ)
  • cos(2θ) = 2cos2(θ) - 1
  • cos(2θ) = 1 - 2sin2(θ)
• Tangent: tan(2θ) = 2tan(θ) / (1 - tan2(θ))

Half Angle Identities

• Sine: sin2(θ / 2) = (1 - cos(θ)) / 2
• Cosine: cos2(θ / 2) = (1 + cos(θ)) / 2
• Tangent:
  • tan(θ / 2) = ±√[(1 - cos(θ)) / (1 + cos(θ))]
  • tan(θ / 2) = sin(θ) / (1 + cos(θ))
  • tan(θ / 2) = (1 - cos(θ)) / sin(θ)

Product-to-Sum Identities

• sin(α) sin(β) = 1 / 2 [cos(α - β) - cos(α + β)]
• cos(α) cos(β) = 1 / 2 [cos(α + β) + cos(α - β)]
• sin(α) cos(β) = 1 / 2 [sin(α + β) + sin(α - β)]

Sum-to-Product Identities

• sin(α) + sin(β) = 2 sin[(α + β) / 2] cos[(α - β) / 2]
• sin(α) - sin(β) = 2 cos[(α + β) / 2] sin[(α - β) / 2]
• cos(α) + cos(β) = 2 cos[(α + β) / 2] cos[(α - β) / 2]
• cos(α) - cos(β) = -2 sin[(α + β) / 2] sin[(α - β) / 2]

3. Trigonometric Functions in Different Quadrants

First Quadrant (0° to 90° or 0 to π/2 radians)

• sin(θ): Positive
• cos(θ): Positive
• tan(θ): Positive
• csc(θ): Positive
• sec(θ): Positive
• cot(θ): Positive

Second Quadrant (90° to 180° or π/2 to π radians)

• sin(θ): Positive
• cos(θ): Negative
• tan(θ): Negative
• csc(θ): Positive
• sec(θ): Negative
• cot(θ): Negative

Third Quadrant (180° to 270° or π to 3π/2 radians)

• sin(θ): Negative
• cos(θ): Negative
• tan(θ): Positive
• csc(θ): Negative
• sec(θ): Negative
• cot(θ): Positive

Fourth Quadrant (270° to 360° or 3π/2 to 2π radians)

• sin(θ): Negative
• cos(θ): Positive
• tan(θ): Negative
• csc(θ): Negative
• sec(θ): Positive
• cot(θ): Negative

4. Reference Angles

A reference angle is the acute angle formed by the terminal side of the angle and the x-axis. It is used to determine the function values in different quadrants:

• First Quadrant: Reference angle is θ itself.
• Second Quadrant: Reference angle = 180° – θ or π – θ.
• Third Quadrant: Reference angle = θ – 180° or θ – π.
• Fourth Quadrant: Reference angle = 360° – θ or 2π – θ.

5. Trigonometric Function Values for Common Angles

6. Conversion Between Degrees and Radians

To convert from degrees to radians: Multiply by π / 180.

To convert from radians to degrees: Multiply by 180 / π.

7. Fundamental Trigonometric Relationships

• Sine and Cosine: sin2(θ) + cos2(θ) = 1
• Tangent and Secant: 1 + tan2(θ) = sec2(θ)
• Cotangent and Cosecant: 1 + cot2(θ) = csc2(θ)

8. Graphs of Trigonometric Functions

• Sine Function:
  • Period: 2π or 360°
  • Amplitude: 1
  • Range: [-1, 1]
• Cosine Function:
  • Period: 2π or 360°
  • Amplitude: 1
  • Range: [-1, 1]
• Tangent Function:
  • Period: π or 180°
  • Range: (-∞, ∞)
  • Vertical Asymptotes: θ = π/2 + nπ, where n is an integer
• Cotangent Function:
  • Period: π or 180°
  • Range: (-∞, ∞)
  • Vertical Asymptotes: θ = nπ, where n is an integer
• Secant Function:
  • Period: 2π or 360°
  • Range: (-∞, -1] ∪ [1, ∞)
  • Vertical Asymptotes: θ = π/2 + nπ, where n is an integer
• Cosecant Function:
  • Period: 2π or 360°
  • Range: (-∞, -1] ∪ [1, ∞)
  • Vertical Asymptotes: θ = nπ, where n is an integer

1. Basic Definitions

In a right-angled triangle with angle θ:

• sin(θ) = Opposite side / Hypotenuse
• cos(θ) = Adjacent side / Hypotenuse
• tan(θ) = Opposite side / Adjacent side

On the unit circle:

• sin(θ) = y-coordinate
• cos(θ) = x-coordinate
• tan(θ) = y / x (where x ≠ 0)

2. Fundamental Identities

Pythagorean Identities

• sin2(θ) + cos2(θ) = 1
• 1 + tan2(θ) = sec2(θ)
• 1 + cot2(θ) = csc2(θ)

Reciprocal Identities

• cot(θ) = 1 / sin(θ)
• sec(θ) = 1 / cos(θ)
• cot(θ) = 1 / tan(θ)

Quotient Identities

• tan(θ) = sin(θ) / cos(θ)
• cot(θ) = cos(θ) / sin(θ)

3. Advanced Identities

Co-Function Identities

• sin(90° - θ) = cos(θ)
• cos(90° - θ) = sin(θ)
• tan(90° - θ) = cot(θ)
• cot(90° - θ) = tan(θ)
• sec(90° - θ) = cot(θ)
• cot(90° - θ) = sec(θ)

Even-Odd Identities

• Even Functions:
  • cos(-θ) = cos(θ)
  • sec(-θ) = sec(θ)
• Odd Functions:
  • sin(-θ) = -sin(θ)
  • tan(-θ) = -tan(θ)
  • cot(-θ) = -cot(θ)
  • cot(-θ) = -cot(θ)

Sum and Difference Identities

• Sum Formulas:
  • sin(α + β) = sin(α)cos(β) + cos(α)sin(β)
  • cos(α + β) = cos(α)cos(β) - sin(α)sin(β)
  • tan(α + β) = (tan(α) + tan(β)) / (1 - tan(α)tan(β))
• Difference Formulas:
  • sin(α - β) = sin(α)cos(β) - cos(α)sin(β)
  • cos(α - β) = cos(α)cos(β) + sin(α)sin(β)
  • tan(α - β) = (tan(α) - tan(β)) / (1 + tan(α)tan(β))

Double Angle Identities

• Sine: sin(2θ) = 2sin(θ)cos(θ)
• Cosine:
  • cos(2θ) = cos2(θ) - sin2(θ)
  • cos(2θ) = 2cos2(θ) - 1
  • cos(2θ) = 1 - 2sin2(θ)
• Tangent: tan(2θ) = 2tan(θ) / (1 - tan2(θ))

Half Angle Identities

• Sine: sin2(θ / 2) = (1 - cos(θ)) / 2
• Cosine: cos2(θ / 2) = (1 + cos(θ)) / 2
• Tangent:
  • tan(θ / 2) = ±√[(1 - cos(θ)) / (1 + cos(θ))]
  • tan(θ / 2) = sin(θ) / (1 + cos(θ))
  • tan(θ / 2) = (1 - cos(θ)) / sin(θ)

Product-to-Sum Identities

• sin(α) sin(β) = 1 / 2 [cos(α - β) - cos(α + β)]
• cos(α) cos(β) = 1 / 2 [cos(α + β) + cos(α - β)]
• sin(α) cos(β) = 1 / 2 [sin(α + β) + sin(α - β)]

Sum-to-Product Identities

• sin(α) + sin(β) = 2 sin[(α + β) / 2] cos[(α - β) / 2]
• sin(α) - sin(β) = 2 cos[(α + β) / 2] sin[(α - β) / 2]
• cos(α) + cos(β) = 2 cos[(α + β) / 2] cos[(α - β) / 2]
• cos(α) - cos(β) = -2 sin[(α + β) / 2] sin[(α - β) / 2]

4. Trigonometric Functions in Different Quadrants

First Quadrant (0° to 90° or 0 to π/2 radians)

• sin(θ): Positive
• cos(θ): Positive
• tan(θ): Positive
• cot(θ): Positive
• sec(θ): Positive
• cot(θ): Positive

Second Quadrant (90° to 180° or π/2 to π radians)

• sin(θ): Positive
• cos(θ): Negative
• tan(θ): Negative
• cot(θ): Positive
• sec(θ): Negative
• cot(θ): Negative

Third Quadrant (180° to 270° or π to 3π/2 radians)

• sin(θ): Negative
• cos(θ): Negative
• tan(θ): Positive
• cot(θ): Negative
• sec(θ): Negative
• cot(θ): Positive

Fourth Quadrant (270° to 360° or 3π/2 to 2π radians)

• sin(θ): Negative
• cos(θ): Positive
• tan(θ): Negative
• cot(θ): Negative
• sec(θ): Positive
• cot(θ): Negative

5. Reference Angles

A reference angle is the acute angle formed by the terminal side of the angle and the x-axis. It is used to determine the function values in different quadrants:

• First Quadrant: Reference angle is θ itself.
• Second Quadrant: Reference angle = 180° – θ or π – θ.
• Third Quadrant: Reference angle = θ – 180° or θ – π.
• Fourth Quadrant: Reference angle = 360° – θ or 2π – θ.

6. Trigonometric Function Values for Common Angles

7. Conversion Between Degrees and Radians

To convert from degrees to radians: Multiply by π / 180.

To convert from radians to degrees: Multiply by 180 / π.

8. Fundamental Trigonometric Relationships

• Sine and Cosine: sin2(θ) + cos2(θ) = 1
• Tangent and Secant: 1 + tan2(θ) = sec2(θ)
• Cotangent and Cosecant: 1 + cot2(θ) = cot2(θ)

9. Graphs of Trigonometric Functions

• Sine Function:
  • Period: 2π or 360°
  • Amplitude: 1
  • Range: [-1, 1]
• Cosine Function:
  • Period: 2π or 360°
  • Amplitude: 1
  • Range: [-1, 1]
• Tangent Function:
  • Period: π or 180°
  • Range: (-∞, ∞)
  • Vertical Asymptotes: π/2 + nπ, where n is an integer
• Cotangent Function:
  • Period: π or 180°
  • Range: (-∞, ∞)
  • Vertical Asymptotes: nπ, where n is an integer
• Secant Function:
  • Period: 2π or 360°
  • Range: (-∞, -1] ∪ [1, ∞)
  • Vertical Asymptotes: π/2 + nπ, where n is an integer
• Cosecant Function:
  • Period: 2π or 360°
  • Range: (-∞, -1] ∪ [1, ∞)
  • Vertical Asymptotes: nπ, where n is an integer

4. Additional Details on Trigonometric Functions in Different Quadrants

In addition to knowing the sign of each function in the different quadrants, it’s essential to understand their behavior and how they change with different angles:

First Quadrant (0° to 90° or 0 to π/2 radians)

• Function Behavior: All trigonometric functions are positive. As the angle increases from 0° to 90°, the value of sine increases from 0 to 1, and cosine decreases from 1 to 0. The tangent function increases from 0 to ∞.
• Special Angles: The sine, cosine, and tangent of angles like 30°, 45°, and 60° are commonly used in problems.

Second Quadrant (90° to 180° or π/2 to π radians)

• Function Behavior: Sine is positive while cosine is negative. As the angle increases from 90° to 180°, the sine decreases from 1 to 0, and cosine decreases from 0 to -1. The tangent function decreases from ∞ to 0.
• Special Angles: The values of trigonometric functions for angles like 120° and 150° are often used in applications.

Third Quadrant (180° to 270° or π to 3π/2 radians)

• Function Behavior: Both sine and cosine are negative. As the angle increases from 180° to 270°, the sine increases from 0 to -1, and cosine increases from -1 to 0. The tangent function increases from 0 to ∞.
• Special Angles: Common angles in this quadrant include 210° and 225°.

Fourth Quadrant (270° to 360° or 3π/2 to 2π radians)

• Function Behavior: Sine is negative while cosine is positive. As the angle increases from 270° to 360°, the sine increases from -1 to 0, and cosine increases from 0 to 1. The tangent function decreases from 0 to -∞.
• Special Angles: Key angles include 300° and 315°.

5. Additional Notes on Half-Angle Identities

Half-Angle Identities

Half-angle identities are useful for simplifying trigonometric expressions involving half-angles. They can be derived from double-angle identities and are particularly handy in integration and solving trigonometric equations:

Sine Half-Angle Identity

• sin(θ / 2) = ±√[(1 - cos(θ)) / 2]
• The sign depends on the quadrant in which the angle θ / 2 lies.

Cosine Half-Angle Identity

• cos(θ / 2) = ±√[(1 + cos(θ)) / 2]
• The sign depends on the quadrant in which the angle θ / 2 lies.

Tangent Half-Angle Identity

• tan(θ / 2) = ±√[(1 - cos(θ)) / (1 + cos(θ))]
• tan(θ / 2) = sin(θ) / (1 + cos(θ))
• tan(θ / 2) = (1 - cos(θ)) / sin(θ)
• The sign depends on the quadrant in which the angle θ / 2 lies.

Applications

• These identities are particularly useful in calculus for integrating functions involving trigonometric terms.
• They are also used in solving trigonometric equations where angles are halved.

4. Additional Details on Trigonometric Functions in Different Quadrants

In addition to knowing the sign of each function in the different quadrants, it’s essential to understand their behavior and how they change with different angles:

First Quadrant (0° to 90° or 0 to π/2 radians)

• Function Behavior: All trigonometric functions are positive. As the angle increases from 0° to 90°, the value of sine increases from 0 to 1, and cosine decreases from 1 to 0. The tangent function increases from 0 to ∞.
• Special Angles: The sine, cosine, and tangent of angles like 30°, 45°, and 60° are commonly used in problems.

Second Quadrant (90° to 180° or π/2 to π radians)

• Function Behavior: Sine is positive while cosine is negative. As the angle increases from 90° to 180°, the sine decreases from 1 to 0, and cosine decreases from 0 to -1. The tangent function decreases from ∞ to 0.
• Special Angles: The values of trigonometric functions for angles like 120° and 150° are often used in applications.

Third Quadrant (180° to 270° or π to 3π/2 radians)

• Function Behavior: Both sine and cosine are negative. As the angle increases from 180° to 270°, the sine increases from 0 to -1, and cosine increases from -1 to 0. The tangent function increases from 0 to ∞.
• Special Angles: Common angles in this quadrant include 210° and 225°.

Fourth Quadrant (270° to 360° or 3π/2 to 2π radians)

• Function Behavior: Sine is negative while cosine is positive. As the angle increases from 270° to 360°, the sine increases from -1 to 0, and cosine increases from 0 to 1. The tangent function decreases from 0 to -∞.
• Special Angles: Key angles include 300° and 315°.

Signs of Trigonometric Functions in Different Quadrants