Trigonometric Ratios: Sine, Cosine, Tangent

• Trigonometry explores the correlation between triangle side lengths and angles
• Fields such as surveying, navigation, physics and engineering rely on trigonometry

Basic Trigonometric Ratios

• Trigonometric ratios involve the sides of a right-angled triangle

Sine (sin)

• Sine of an angle is the ratio of the opposite side to the hypotenuse
• sin(θ) = Opposite / Hypotenuse

Cosine (cos)

• Cosine of an angle is the ratio of the adjacent side to the hypotenuse
• cos(θ) = Adjacent / Hypotenuse

Tangent (tan)

• Tangent of an angle is the ratio of the opposite side to the adjacent side
• tan(θ) = Opposite / Adjacent
• tan(θ) = sin(θ) / cos(θ)

Cosecant (csc or cosec)

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

Secant (sec)

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

Cotangent (cot)

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

Trigonometric Values for Standard Angles

• Certain angles like 0°, 30°, 45°, 60°, and 90° are common in trigonometry

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

45° (π/4 radians)

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

60° (π/3 radians)

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

90° (π/2 radians)

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

Trigonometric Identities

• Trigonometric identities hold true for all variable values

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)

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

Sine Rule

• Relates the lengths of the sides of a triangle to the sines of its angles
• a / sin(A) = b / sin(B) = c / sin(C)
• Where a, b, and c are the side lengths, and A, B, and C are the opposite angles

Cosine Rule

• 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)
• Can be rearranged: cos(A) = (b² + c² - a²) / (2bc), to find an angle if all three sides are known

Graphs of Trigonometric Functions

• Trigonometric functions' periodic nature can be represented graphically

Sine Graph

• y = sin(x)
• Period: 2π
• Amplitude: 1
• Range: [-1, 1]

Cosine Graph

• y = cos(x)
• Period: 2π
• Amplitude: 1
• Range: [-1, 1]

Tangent Graph

• y = tan(x)
• Period: π
• Asymptotes at x = (2n+1)π/2, where n is an integer
• Range: (-∞, ∞)

Inverse Trigonometric Functions

• Inverse trigonometric functions reverse the trigonometric functions
• They return the angle using sine, cosine, or tangent of a given value

Arcsine (sin⁻¹ or arcsin)

• Returns the angle whose sine is a given number
• arcsin(x) represents angle θ where sin(θ) = x
• Domain: [-1, 1]
• Range: [-π/2, π/2]

Arccosine (cos⁻¹ or arccos)

• Returns the angle whose cosine is a given number
• arccos(x) represents angle θ where cos(θ) = x
• Domain: [-1, 1]
• Range: [0, π]

Arctangent (tan⁻¹ or arctan)

• Returns the angle whose tangent is a given number
• arctan(x) represents angle θ where tan(θ) = x
• Domain: (-∞, ∞)
• Range: (-π/2, π/2)

Applications of Trigonometry

• Trigonometry applies to numerous fields

Navigation

• Used to calculate distances and directions

Surveying

• Used to measure heights and distances on land

Physics

• Used in wave mechanics, optics, and mechanics

Engineering

• Used in structural analysis, signal processing and more