Trigonometric Functions and Unit Circle Concepts

Trigonometric Functions

• Trigonometry studies relationships between triangle sides and angles.
• Trigonometric functions relate triangle angles to side ratios.
• Essential for solving problems related to triangles and wave phenomena.
• Primary trigonometric functions: sine, cosine, and tangent.
• Typically defined for right-angled triangles.
• Sine (sin θ) is the ratio of the opposite side to the hypotenuse.
• Cosine (cos θ) is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan θ) is the ratio of the opposite side to the adjacent side.
• Reciprocal trigonometric functions: cosecant, secant, and cotangent.
• Cosecant (csc θ) is the reciprocal of sine, csc θ = 1/sin θ.
• Secant (sec θ) is the reciprocal of cosine, sec θ = 1/cos θ.
• Cotangent (cot θ) is the reciprocal of tangent, cot θ = 1/tan θ.

Unit Circle

• Circle with a radius of 1, centered at the origin.
• Useful for understanding trigonometric functions for all real numbers.
• Coordinates of a point on the unit circle: (cos θ, sin θ), where θ is the angle from the positive x-axis.
• Helps visualize the signs and values of trigonometric functions in different quadrants.
• First quadrant (0 < θ < π/2): cosine and sine are positive.
• Second quadrant (π/2 < θ < π): sine is positive, cosine is negative.
• Third quadrant (π < θ < 3π/2): sine and cosine are negative.
• Fourth quadrant (3π/2 < θ < 2π): cosine is positive, sine is negative.

Trigonometric Identities

• Equations involving trigonometric functions, true for all defined variable values.
• Useful for simplifying expressions, solving equations, and proving other identities.
• Pythagorean identities: sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, 1 + cot² θ = csc² θ.
• 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.
• Even-odd identities: sin(-θ) = -sin θ, cos(-θ) = cos θ, tan(-θ) = -tan θ.
• Angle sum and difference identities:
  • sin(A ± B) = sin A cos B ± cos 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)
• Double-angle identities:
  • sin(2θ) = 2 sin θ cos θ
  • cos(2θ) = cos² θ - sin² θ = 2 cos² θ - 1 = 1 - 2 sin² θ
  • tan(2θ) = (2 tan θ) / (1 - tan² θ)
• Half-angle identities:
  • sin(θ/2) = ±√((1 - cos θ)/2)
  • cos(θ/2) = ±√((1 + cos θ)/2)
  • tan(θ/2) = ±√((1 - cos θ)/(1 + cos θ)) = (sin θ) / (1 + cos θ) = (1 - cos θ) / (sin θ)

Inverse Trigonometric Functions

• Inverse operations of trigonometric functions, used to find angles.
• Arcsine (sin⁻¹ x or asin x): angle whose sine is x.
• Arccosine (cos⁻¹ x or acos x): angle whose cosine is x.
• Arctangent (tan⁻¹ x or atan x): angle whose tangent is x.
• Domains and ranges are restricted to ensure single-valued functions.
• arcsin x: domain [-1, 1], range [-π/2, π/2].
• arccos x: domain [-1, 1], range [0, π].
• arctan x: domain (-∞, ∞), range (-π/2, π/2).

Trigonometric Equations

• Equations involving trigonometric functions.
• Solving involves finding variable values that make the equation true.
• Infinitely many solutions due to the periodic nature of trigonometric functions.
• To find all solutions, find solutions within a specific interval and use periodicity.
• Example: if sin x = 0.5, then x = π/6 + 2πk or x = 5π/6 + 2πk, where k is an integer.

Applications of Trigonometry

• Navigation: Used to determine distances and directions.
• Physics: Used to analyze wave motion, projectile motion, and oscillations.
• Engineering: Used in structural design, surveying, and electrical engineering.
• Computer Graphics: Used for rotations, scaling, and transformations of objects.
• Astronomy: Used to measure distances to stars and planets.

Laws of Sines and Cosines

• Law of Sines: a/sin A = b/sin B = c/sin C (relates sides and sines of opposite angles).
• Law of Cosines: c² = a² + b² - 2ab cos C (relates sides and cosine of one angle).
• Used to solve triangles when given certain side and angle information.
• Useful for non-right triangles.