Introduction to Trigonometry

• Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles.
• Trigonometry is especially useful for right triangles (triangles with one angle equal to 90 degrees).
• Trigonometric functions are commonly used to relate angles of a right triangle to ratios of its sides.
• These functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

Basic Trigonometric Functions

• Sine (sin) of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse.
• Cosine (cos) of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse.
• Tangent (tan) of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
• Cosecant (csc) is the reciprocal of sine (csc θ = 1/sin θ).
• Secant (sec) is the reciprocal of cosine (sec θ = 1/cos θ).
• Cotangent (cot) is the reciprocal of tangent (cot θ = 1/tan θ).

Acronyms

• SOH CAH TOA is a mnemonic often used to remember the definitions of sine, cosine, and tangent:
  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent

Common Angles

• Trigonometric functions of certain angles are commonly used and should be memorized:
  • 0 degrees (0 radians): sin(0) = 0, cos(0) = 1, tan(0) = 0
  • 30 degrees (π/6 radians): sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3
  • 45 degrees (π/4 radians): sin(45°) = √2/2, cos(45°) = √2/2, tan(45°) = 1
  • 60 degrees (π/3 radians): sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3
  • 90 degrees (π/2 radians): sin(90) = 1, cos(90) = 0, tan(90) = undefined

Trigonometric Identities

• Trigonometric identities are equations involving trigonometric functions that are true for all values of the angles for which the functions are defined.
• Pythagorean Identity: sin²θ + cos²θ = 1.
• Other forms of the Pythagorean identity:
  • sin²θ = 1 - cos²θ
  • cos²θ = 1 - sin²θ
• Tangent and Cotangent Identities:
  • tan θ = sin θ / cos θ
  • cot θ = cos θ / sin θ
• Reciprocal Identities:
  • csc θ = 1 / sin θ
  • sec θ = 1 / cos θ
  • cot θ = 1 / tan 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 = 2 sin θ cos θ
  • cos 2θ = cos²θ - sin²θ = 2 cos²θ - 1 = 1 - 2 sin²θ
  • tan 2θ = (2 tan θ) / (1 - tan²θ)
• 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 relates the lengths of the sides of a triangle to the sines of its angles.
• Given a triangle with sides a, b, c and angles A, B, C opposite those sides respectively, the Law of Sines states: a / sin A = b / sin B = c / sin C
• The Law of Sines is useful for solving triangles when you know:
  • Two angles and one side (AAS or ASA).
  • Two sides and an angle opposite one of them (SSA, which may have zero, one, or two possible solutions).

Law of Cosines

• The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
• Given a triangle with sides a, b, c and angle C opposite side c, the Law of Cosines states: c² = a² + b² - 2ab cos C
• The Law of Cosines can be rearranged to find the cosine of an angle: cos C = (a² + b² - c²) / (2ab)
• The Law of Cosines is useful for solving triangles when you know:
  • Three sides (SSS).
  • Two sides and the included angle (SAS).

Applications of Trigonometry

• Surveying: Determining distances and heights in land surveying.
• Navigation: Calculating directions and distances in air and sea navigation.
• Engineering: Designing structures, bridges, and mechanical systems.
• Physics: Analyzing wave motion, optics, and mechanics.
• Astronomy: Measuring distances to stars and planets.
• Computer Graphics: Creating realistic images and animations.