Trigonometric Identities and Triangles

Trigonometric Identities

• Pythagorean Identity: sin²(A) + cos²(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) = cos²(A) - sin²(A)
  • tan(2A) = (2tan(A)) / (1 - tan²(A))

Triangles And Angles

• Right Triangle: one angle is 90° (π/2 radians)
• Angles in Standard Position:
  • Initial side: x-axis (positive)
  • Terminal side: where the angle ends (can be any quadrant)
• Reference Angle:
  • The acute angle between the terminal side and the x-axis
  • Used to find trig values for non-acute angles

Sine Cosine And Tangent

• Sine (sin):
  • Opposite side / Hypotenuse
  • sin(A) = opposite side / hypotenuse
• Cosine (cos):
  • Adjacent side / Hypotenuse
  • cos(A) = adjacent side / hypotenuse
• Tangent (tan):
  • Opposite side / Adjacent side
  • tan(A) = opposite side / adjacent side

Trigonometric Identities

• The Pythagorean Identity is a fundamental concept in trigonometry, stating that sin²(A) + cos²(A) = 1.
• The Sum and Difference Formulas describe the trigonometric relationships between angles A and B:
  • 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 provide trigonometric values for double angles:
  • sin(2A) = 2sin(A)cos(A)
  • cos(2A) = cos²(A) - sin²(A)
  • tan(2A) = (2tan(A)) / (1 - tan²(A))

Triangles and Angles

• A right triangle has one angle measuring 90° (π/2 radians).
• Angles in standard position have their initial side on the x-axis (positive) and their terminal side in any quadrant.
• The reference angle is the acute angle between the terminal side and the x-axis, used to find trigonometric values for non-acute angles.

Sine, Cosine, and Tangent

• Sine (sin) is defined as the opposite side divided by the hypotenuse: sin(A) = opposite side / hypotenuse.
• Cosine (cos) is defined as the adjacent side divided by the hypotenuse: cos(A) = adjacent side / hypotenuse.
• Tangent (tan) is defined as the opposite side divided by the adjacent side: tan(A) = opposite side / adjacent side.