Trigonometry Basics

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Study Notes

Trigonometry

Definition: Trigonometry is a branch of mathematics that studies relationships between the angles and sides of triangles, particularly right-angled triangles.
Key Functions:
- Sine (sin): Ratio of the opposite side to the hypotenuse.
- Cosine (cos): Ratio of the adjacent side to the hypotenuse.
- Tangent (tan): Ratio of the opposite side to the adjacent side.
Reciprocal Functions:
- Cosecant (csc): Reciprocal of sine (1/sin).
- Secant (sec): Reciprocal of cosine (1/cos).
- Cotangent (cot): Reciprocal of tangent (1/tan).
Pythagorean Identity: In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
- Formula: sin²(θ) + cos²(θ) = 1
Unit Circle:
- A circle with a radius of 1 centered at the origin of a coordinate plane.
- Points on the unit circle represent the values of sin(θ) and cos(θ):
  - sin(θ) = y-coordinate
  - cos(θ) = x-coordinate
Common Angles:
- 0°: sin(0) = 0, cos(0) = 1, tan(0) = 0
- 30°: sin(30) = 1/2, cos(30) = √3/2, tan(30) = 1/√3
- 45°: sin(45) = √2/2, cos(45) = √2/2, tan(45) = 1
- 60°: sin(60) = √3/2, cos(60) = 1/2, tan(60) = √3
- 90°: sin(90) = 1, cos(90) = 0, tan(90) = undefined
Law of Sines:
- Relates the ratios of the lengths of sides of a triangle to the sines of its angles.
- Formula: (a/sin(A)) = (b/sin(B)) = (c/sin(C))
Law of Cosines:
- Relates the lengths of the sides of a triangle to the cosine of one of its angles.
- Formula: c² = a² + b² - 2ab * cos(C)
Applications:
- Problem-solving in geometry, physics (e.g., waves, oscillations), engineering, and navigation.
Trigonometric Identities:
- Angle Sum/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))
- Double Angle Formulas:
  - sin(2θ) = 2sin(θ)cos(θ)
  - cos(2θ) = cos²(θ) - sin²(θ)
Graphs of Trigonometric Functions:
- Sine and cosine functions have a periodic wave shape with a period of 2π.
- Tangent function has a periodic shape with asymptotes and a period of π.

These notes cover fundamental concepts in trigonometry, foundational for advanced studies in mathematics and related fields.

Trigonometry Definition

The branch of mathematics that explores the relationship between angles and sides of triangles, especially right-angled triangles.

Key Trigonometric Functions

Sine (sin): The ratio of the opposite side to the hypotenuse of a right triangle.
Cosine (cos): The ratio of the adjacent side to the hypotenuse of a right triangle.
Tangent (tan): The ratio of the opposite side to the adjacent side of a right triangle.

Reciprocal Trigonometric Functions

Cosecant (csc): The reciprocal of sine (1/sin).
Secant (sec): The reciprocal of cosine (1/cos).
Cotangent (cot): The reciprocal of tangent (1/tan).

Pythagorean Identity

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Formula: sin²(θ) + cos²(θ) = 1

Unit Circle

A circle with a radius of 1 unit centered at the origin of a coordinate plane.
Points on the unit circle represent the values of sin(θ) and cos(θ):
- sin(θ) = y-coordinate
- cos(θ) = x-coordinate

Common Angles:

0°: sin(0) = 0, cos(0) = 1, tan(0) = 0
30°: sin(30) = 1/2, cos(30) = √3/2, tan(30) = 1/√3
45°: sin(45) = √2/2, cos(45) = √2/2, tan(45) = 1
60°: sin(60) = √3/2, cos(60) = 1/2, tan(60) = √3
90°: sin(90) = 1, cos(90) = 0, tan(90) = undefined

Law of Sines

Relates the ratios of the lengths of sides of a triangle to the sines of its angles.
Formula: (a/sin(A)) = (b/sin(B)) = (c/sin(C))

Law of Cosines

Relates the lengths of the sides of a triangle to the cosine of one of its angles.
Formula: c² = a² + b² - 2ab * cos(C)

Applications of Trigonometry

Solving problems in geometry, physics (e.g., waves, oscillations), engineering, and navigation.

Trigonometric Identities

Angle Sum/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))
Double Angle Formulas:
- sin(2θ) = 2sin(θ)cos(θ)
- cos(2θ) = cos²(θ) - sin²(θ)

Graphs of Trigonometric Functions

Sine and cosine functions have a periodic wave shape with a period of 2π.
Tangent function has a periodic shape with asymptotes and a period of π.

Description

This quiz covers fundamental concepts of trigonometry, including key functions like sine, cosine, and tangent, as well as their reciprocal functions and identities. Test your understanding of these relationships and their application in right-angled triangles and the unit circle.