Trigonometric Ratios

• Trigonometry studies relationships between triangle sides and angles.
• Fields such as geometry, navigation, physics, and engineering rely on it.

Basic Trigonometric Ratios

• Trigonometric ratios connect right triangle angles to side ratios.
• Sine (sin), cosine (cos), and tangent (tan) form the primary trigonometric ratios.

Sine (sin)

• Sine represents the ratio of the opposite side's length to the hypotenuse.
• sin(θ) = Opposite / Hypotenuse

Cosine (cos)

• Cosine represents the ratio of the adjacent side's length to the hypotenuse.
• cos(θ) = Adjacent / Hypotenuse

Tangent (tan)

• Tangent represents the ratio of the opposite side's length to the adjacent side.
• tan(θ) = Opposite / Adjacent
• Tangent is also sin(θ) / cos(θ).

Reciprocal Trigonometric Ratios

• Cosecant (csc), secant (sec), and cotangent (cot) make up the reciprocal trigonometric ratios.

Cosecant (csc)

• Cosecant is the sine reciprocal.
• csc(θ) = Hypotenuse / Opposite = 1 / sin(θ)

Secant (sec)

• Secant is the cosine reciprocal.
• sec(θ) = Hypotenuse / Adjacent = 1 / cos(θ)

Cotangent (cot)

• Cotangent is the tangent reciprocal.
• cot(θ) = Adjacent / Opposite = 1 / tan(θ) = cos(θ) / sin(θ)

Trigonometric Values for Standard Angles

• 0°, 30°, 45°, 60°, and 90° are frequently used angles.
• Knowing the trigonometric values for these angles is helpful.

0°

• sin(0°) = 0
• cos(0°) = 1
• tan(0°) = 0

30° (π/6 radians)

• sin(30°) = 1/2
• cos(30°) = √3/2
• tan(30°) = 1/√3 = √3/3

45° (π/4 radians)

• sin(45°) = 1/√2 = √2/2
• cos(45°) = 1/√2 = √2/2
• tan(45°) = 1

60° (π/3 radians)

• sin(60°) = √3/2
• cos(60°) = 1/2
• tan(60°) = √3

90°

• sin(90°) = 1
• cos(90°) = 0
• tan(90°) = undefined

Trigonometric Identities

• Trigonometric identities consist of equations with trigonometric functions that hold true for all variable values where the functions are defined.

Pythagorean Identities

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

Angle Sum and Difference Identities

• 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))

Double Angle Identities

• sin(2θ) = 2sin(θ)cos(θ)
• cos(2θ) = cos²(θ) - sin²(θ) = 2cos²(θ) - 1 = 1 - 2sin²(θ)
• tan(2θ) = (2tan(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)

Law of Sines

• The Law of Sines connects triangle side lengths with the sines of their angles.
• a / sin(A) = b / sin(B) = c / sin(C), where a, b, c represent side lengths and A, B, C represent opposite angles.
• It helps solve triangles when you have two angles and a side (AAS) or two sides and an angle opposite one of them (SSA).
• The SSA case may result in ambiguous scenarios, leading to zero, one, or two possible triangles.

Law of Cosines

• The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
• a² = b² + c² - 2bc * cos(A)
• b² = a² + c² - 2ac * cos(B)
• c² = a² + b² - 2ab * cos(C)
• Useful for solving triangles when given three sides (SSS) or two sides and an included angle (SAS).

Applications of Trigonometry

• Trigonometry finds extensive use across various fields.

Navigation

• Used in determining positions and directions using angles and distances.
• Essential in GPS systems, maritime navigation, and aviation.

Physics

• Determines projectile motion, forces, and oscillations in mechanics.
• Describes wave behavior and interference patterns in optics and electromagnetism.

Engineering

• Determines loads and stresses in structures for structural engineering.
• Analyzes alternating current (AC) circuits and signal processing in electrical engineering.

Surveying

• Measures distances, heights, and angles in land surveying, which helps create accurate maps and boundaries.

Computer Graphics

• Performs object transformations (rotation, scaling, translation) in 2D and 3D spaces.
• Extensively used in game development, animation, and virtual reality.

Inverse Trigonometric Functions

• Inverse trigonometric functions are inverses of trigonometric functions.
• They are used to find the angle when a trigonometric ratio value is known.

Arcsine (sin⁻¹)

• Arcsine, written as sin⁻¹(x) or arcsin(x), inverts the sine function.
• It gives the angle with sine x.
• The range of arcsin(x) is [-π/2, π/2].

Arccosine (cos⁻¹)

• Arccosine, written as cos⁻¹(x) or arccos(x), inverts the cosine function.
• It gives the angle with cosine x.
• The range of arccos(x) is [0, π].

Arctangent (tan⁻¹)

• Arctangent, written as tan⁻¹(x) or arctan(x), inverts the tangent function.
• It gives the angle with tangent x.
• The range of arctan(x) is (-π/2, π/2).

Trigonometric Equations

• Trigonometric equations contain trigonometric functions.
• Solving them involves identifying all variable values that satisfy the equation.
• Solutions typically involve angles within a specific range, like [0, 2π) or [0°, 360°).
• General solutions consider the periodic nature of trigonometric functions.

Strategies for Solving Trigonometric Equations

• Use algebraic techniques to isolate the trigonometric function.
• Use trigonometric identities to simplify the equation.
• Find the reference angle and determine the quadrants where the solutions lie.
• Write the general solution by adding integer multiples of the period to the solutions within one period.