Trigonometry Basics

Angles and Measurement

• Angles can be measured in degrees, radians, or gradians.
• A full circle is 360 degrees, 2π radians, or 400 gradians.
• 1 radian is approximately equal to 57.3 degrees.

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: sin^2(A) + cos^2(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))
• Double Angle Formulas:
  • sin(2A) = 2sin(A)cos(A)
  • cos(2A) = cos^2(A) - sin^2(A)
  • tan(2A) = 2tan(A) / (1 - tan^2(A))

Solving Triangles

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

Graphs of Trigonometric Functions

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

Angles and Measurement

• A full circle measures 360 degrees, 2π radians, or 400 gradians.
• 1 radian is approximately equal to 57.3 degrees.
• Angles can be measured in three different units: degrees, radians, and gradians.

Trigonometric Ratios

• The sine (sin) of an angle is the ratio of the opposite side to the hypotenuse.
• The cosine (cos) of an angle is the ratio of the adjacent side to the hypotenuse.
• The tangent (tan) of an angle is the ratio of the opposite side to the adjacent side.
• The cotangent (cot) of an angle is the ratio of the adjacent side to the opposite side.
• The secant (sec) of an angle is the ratio of the hypotenuse to the opposite side.
• The cosecant (csc) of an angle is the ratio of the hypotenuse to the adjacent side.

Trigonometric Identities

• The Pythagorean Identity states that sin^2(A) + cos^2(A) = 1.
• The Sum and Difference Formulas are used to find the sine, cosine, and tangent of the sum or difference of two angles.
• The Double Angle Formulas are used to find the sine, cosine, and tangent of an angle that is twice another angle.

Solving Triangles

• Right Triangles can be solved using trigonometric ratios and the Pythagorean Theorem.
• The Pythagorean Theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
• Oblique Triangles can be solved using the Law of Sines and the Law of Cosines.
• The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is the same for all three sides.
• The Law of Cosines states that the square of the length of a side is equal to the sum of the squares of the lengths of the other two sides, minus twice the product of the lengths of the other two sides and the cosine of the angle between them.

Graphs of Trigonometric Functions

• The sine and cosine functions have a period of 2π and an amplitude of 1.
• The tangent function has a period of π and asymptotes at x = π/2 + kπ, where k is an integer.
• The graphs of trigonometric functions can be used to model periodic phenomena, such as sound waves and light waves.