Trigonometry Basics

Angles and Measurement

• Angles can be measured in degrees, radians, or gradients.
• 1 radian = 180/π degrees (approximately 57.3°)
• 1 degree = π/180 radians (approximately 0.0175 rad)

Trigonometric Ratios

• Sine (sin): opposite side / hypotenuse
• Cosine (cos): adjacent side / hypotenuse
• Tangent (tan): opposite side / adjacent side
• Cotangent (cot): adjacent side / opposite side
• Secant (sec): hypotenuse / opposite side
• Cosecant (csc): hypotenuse / adjacent side

Trigonometric Identities

• Pythagorean Identity: sin²(A) + cos²(A) = 1
• Sum and Difference Formulas:
  • 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))

Solving Triangles

• Right Triangles:
  • Use trigonometric ratios to find unknown sides or angles.
  • Apply Pythagorean theorem: a² + b² = c² (where c is the hypotenuse)
• Oblique Triangles:
  • Use law of sines: a / sin(A) = b / sin(B) = c / sin(C)
  • Use law of cosines: c² = a² + b² - 2ab * cos(C)

Graphs of Trigonometric Functions

• Sine and cosine functions:
  • Period: 2π
  • Amplitude: 1
  • Range: [-1, 1]
• Tangent function:
  • Period: π
  • Asymptotes: x = π/2 + kπ (where k is an integer)
  • Range: all real numbers

Applications of Trigonometry

• Triangulation: used in navigation, surveying, and physics.
• Wave motion: used to model sound and light waves.
• Analytic geometry: used to solve problems involving right triangles and trigonometric identities.

Angles and Measurement

• Angles can be measured in degrees, radians, or gradients.
• 1 radian is equal to 180/π degrees, approximately 57.3°.
• 1 degree is equal to π/180 radians, approximately 0.0175 rad.

Trigonometric Ratios

• Sine (sin) is the ratio of the opposite side to the hypotenuse.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan) is the ratio of the opposite side to the adjacent side.
• Cotangent (cot) is the ratio of the adjacent side to the opposite side.
• Secant (sec) is the ratio of the hypotenuse to the opposite side.
• Cosecant (csc) is the ratio of the hypotenuse to the adjacent side.

Trigonometric Identities

• The Pythagorean Identity states that sin²(A) + cos²(A) = 1.
• The Sum and Difference Formulas are:
  • 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))

Solving Triangles

• In right triangles, trigonometric ratios can be used to find unknown sides or angles.
• The Pythagorean theorem is a² + b² = c², where c is the hypotenuse.
• In oblique triangles, the law of sines is a / sin(A) = b / sin(B) = c / sin(C).
• The law of cosines is c² = a² + b² - 2ab * cos(C).

Graphs of Trigonometric Functions

• Sine and cosine functions have a period of 2π, amplitude of 1, and range [-1, 1].
• Tangent functions have a period of π, asymptotes at x = π/2 + kπ, and a range of all real numbers.

Applications of Trigonometry

• Triangulation is used in navigation, surveying, and physics.
• Trigonometry is used to model wave motion, such as sound and light waves.
• Analytic geometry uses trigonometry to solve problems involving right triangles and trigonometric identities.

Learn about angles and measurement, trigonometric ratios, and more. Understand degrees, radians, and gradients, and how to calculate sine, cosine, tangent, and other trigonometric functions.