Trigonometry Basics

Definition and Basics

• Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles.
• It involves the use of trigonometric functions such as sine, cosine, and tangent to describe these relationships.

Trigonometric Functions

• Sine (sin): The ratio of the opposite side to the hypotenuse of a right-angled triangle.
  • sin(A) = opposite side / hypotenuse
• Cosine (cos): The ratio of the adjacent side to the hypotenuse of a right-angled triangle.
  • cos(A) = adjacent side / hypotenuse
• Tangent (tan): The ratio of the opposite side to the adjacent side of a right-angled triangle.
  • tan(A) = opposite side / adjacent side
• Cosecant (csc): The reciprocal of sine.
  • csc(A) = 1 / sin(A)
• Secant (sec): The reciprocal of cosine.
  • sec(A) = 1 / cos(A)
• Cotangent (cot): The reciprocal of tangent.
  • cot(A) = 1 / tan(A)

Trigonometric 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)

Graphs of Trigonometric Functions

• Sine and Cosine Graphs:
  • Period: 2π
  • Amplitude: 1
• Tangent Graph:
  • Period: π
  • Asymptotes: x = π/2 + kπ, where k is an integer

Solving Triangles

• Right Triangles:
  • Use trigonometric functions to find missing sides and angles.
• Oblique Triangles:
  • Use the Law of Sines or the Law of Cosines to find missing sides and angles.
  • Law of Sines: a / sin(A) = b / sin(B) = c / sin(C)
  • Law of Cosines: c^2 = a^2 + b^2 - 2ab * cos(C)

Definition and Basics

• Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, involving trigonometric functions such as sine, cosine, and tangent.

Trigonometric Functions

• Sine (sin) is the ratio of the opposite side to the hypotenuse of a right-angled triangle, calculated as sin(A) = opposite side / hypotenuse.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse of a right-angled triangle, calculated as cos(A) = adjacent side / hypotenuse.
• Tangent (tan) is the ratio of the opposite side to the adjacent side of a right-angled triangle, calculated as tan(A) = opposite side / adjacent side.
• Cosecant (csc) is the reciprocal of sine, calculated as csc(A) = 1 / sin(A).
• Secant (sec) is the reciprocal of cosine, calculated as sec(A) = 1 / cos(A).
• Cotangent (cot) is the reciprocal of tangent, calculated as cot(A) = 1 / tan(A).

Trigonometric 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)

Graphs of Trigonometric Functions

• Sine and Cosine Graphs:
  • Have a period of 2π
  • Have an amplitude of 1
• Tangent Graph:
  • Has a period of π
  • Has asymptotes at x = π/2 + kπ, where k is an integer

Solving Triangles

• Right Triangles:
  • Use trigonometric functions to find missing sides and angles
• Oblique Triangles:
  • Use the Law of Sines or the Law of Cosines to find missing sides and angles
  • Law of Sines: a / sin(A) = b / sin(B) = c / sin(C)
  • Law of Cosines: c^2 = a^2 + b^2 - 2ab * cos(C)