Trigonometry Basics: Angles and Triangles

Angles and Triangles

• Trigonometry is the study of relationships between the sides and angles of triangles.
• Angles can be measured in degrees, radians, or gradians.
• A triangle has three angles, which always add up to 180 degrees.

Trigonometric Ratios

• The three basic trigonometric ratios are:
  1. Sine (sin): opposite side / hypotenuse
  2. Cosine (cos): adjacent side / hypotenuse
  3. Tangent (tan): opposite side / adjacent side
• These ratios can be used to find the lengths of sides and angles in a right-angled triangle.

Identities and Formulas

• 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) - tan(A)tan(B)
• Double and half-angle formulas:
  • sin(2A) = 2sin(A)cos(A)
  • cos(2A) = cos^2(A) - sin^2(A)
  • tan(2A) = 2tan(A) / (1 - tan^2(A))

Graphs and Properties

• Sine, cosine, and tangent functions have periodic graphs with different amplitudes and periods.
• The sine function has a maximum value of 1 and a minimum value of -1.
• The cosine function has a maximum value of 1 and a minimum value of -1.
• The tangent function has no maximum or minimum value.

Applications

• Trigonometry is used in:
  • Navigation and surveying
  • Physics and engineering
  • Computer graphics and game development
  • Medical imaging and analysis
  • Music and acoustics

Angles and Triangles

• Trigonometry studies relationships between sides and angles of triangles.
• Angles can be measured in degrees, radians, or gradians.
• A triangle has three angles, adding up to 180 degrees.

Trigonometric Ratios

• Sine (sin) is the ratio of opposite side to hypotenuse.
• Cosine (cos) is the ratio of adjacent side to hypotenuse.
• Tangent (tan) is the ratio of opposite side to adjacent side.
• Trigonometric ratios help find lengths of sides and angles in right-angled triangles.

Identities and Formulas

• 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), and tan(A + B) = tan(A) + tan(B) - tan(A)tan(B).
• Double and half-angle formulas: sin(2A) = 2sin(A)cos(A), cos(2A) = cos^2(A) - sin^2(A), and tan(2A) = 2tan(A) / (1 - tan^2(A)).

Graphs and Properties

• Sine, cosine, and tangent functions have periodic graphs with different amplitudes and periods.
• Sine function has a maximum value of 1 and a minimum value of -1.
• Cosine function has a maximum value of 1 and a minimum value of -1.
• Tangent function has no maximum or minimum value.

Applications

• Trigonometry is used in navigation and surveying.
• Trigonometry is used in physics and engineering.
• Trigonometry is used in computer graphics and game development.
• Trigonometry is used in medical imaging and analysis.
• Trigonometry is used in music and acoustics.

Learn about the basics of trigonometry, including measuring angles and understanding trigonometric ratios such as sine, cosine, and tangent.