Trigonometry Basics

Trigonometry

Definition

• Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.

Key Concepts

Angles

• Degrees: A unit of measurement for angles, with 360 degrees in a full circle.
• Radians: A unit of measurement for angles, with 2π radians in a full circle.

Triangles

• Right Triangle: A triangle with one right angle (90 degrees).
• Oblique Triangle: A triangle that is not a right triangle.

Trigonometric Ratios

• Sine (sin): The ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (cos): The ratio of the adjacent side to the hypotenuse in a right triangle.
• Tangent (tan): The ratio of the opposite side to the adjacent side in a right triangle.
• Cotangent (cot): The ratio of the adjacent side to the opposite side in a right triangle.
• Secant (sec): The ratio of the hypotenuse to the adjacent side in a right triangle.
• Cosecant (csc): The ratio of the hypotenuse to the opposite side in a right triangle.

Identities

• Pythagorean Identity: sin^2(A) + cos^2(A) = 1
• Sum and Difference Formulas:
  • 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)

Applications

• Triangulation: Solving triangles using trigonometric ratios and identities.
• Wave Motion: Modeling periodic phenomena, such as sound and light waves, using trigonometric functions.
• Navigation: Calculating distances and directions using trigonometric principles.

Trigonometry

Definition

• Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.

Angles

• A full circle has 360 degrees or 2π radians.
• Degrees and radians are units of measurement for angles.

Triangles

• A right triangle has one right angle (90 degrees).
• An oblique triangle is a triangle that is not a right triangle.

Trigonometric Ratios

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

Identities

• The Pythagorean Identity states that sin^2(A) + cos^2(A) = 1.
• Sum and Difference Formulas include:
  • 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)

Applications

• Triangulation involves solving triangles using trigonometric ratios and identities.
• Wave motion can be modeled using trigonometric functions to represent periodic phenomena, such as sound and light waves.
• Trigonometry has navigation applications, including calculating distances and directions.

Trigonometry

Definitions and Concepts

• An angle is a measure of rotation between two lines or planes
• A triangle is a polygon with three sides and three angles
• A right triangle is a triangle with one right angle (90 degrees)
• The hypotenuse is the side opposite the right angle in a right triangle
• Trigonometric ratios are relationships between the sides and angles of a right triangle

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

Identities and Formulas

• The Pythagorean Identity states that sin^2(A) + cos^2(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))
• The Double Angle Formulas are:
  • sin(2A) = 2sin(A)cos(A)
  • cos(2A) = cos^2(A) - sin^2(A)
  • tan(2A) = 2tan(A) / (1 - tan^2(A))

Applications and Graphs

• Right triangle problems involve solving for sides and angles using trigonometric ratios
• Wave motion can be modeled using sine and cosine functions
• Graphs of trigonometric functions have:
  • A period of 2π for sine and cosine
  • A period of π for tangent
  • Amplitude and phase shifts can be applied to trigonometric functions

Important Values and Identities

• The unit circle is a circle with a radius of 1, used to define trigonometric ratios
• Special angles are 30°, 45°, 60°, and their corresponding trigonometric ratios
• Trigonometric identities are equations that are true for all angles, such as sin(-A) = -sin(A)