Trigonometric Ratios Overview

Trigonometric Ratios

• Definition: Trigonometric ratios relate the angles of a right triangle to the ratios of its sides.

• Basic Ratios:

  1. Sine (sin):
     • sin(θ) = Opposite side / Hypotenuse
  2. Cosine (cos):
     • cos(θ) = Adjacent side / Hypotenuse
  3. Tangent (tan):
     • tan(θ) = Opposite side / Adjacent side

• Reciprocal Ratios:

  1. Cosecant (csc):
     • csc(θ) = 1/sin(θ) = Hypotenuse / Opposite side
  2. Secant (sec):
     • sec(θ) = 1/cos(θ) = Hypotenuse / Adjacent side
  3. Cotangent (cot):
     • cot(θ) = 1/tan(θ) = Adjacent side / Opposite side

• Important Angles:

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

• Identities:

  • Pythagorean Identity:
    • sin²(θ) + cos²(θ) = 1
  • Sum and Difference Formulas:
    • sin(A ± B) = sinA cosB ± cosA sinB
    • cos(A ± B) = cosA cosB ∓ sinA sinB
    • tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)

• Applications:

  • Used in solving triangles (finding sides and angles).
  • Essential in physics, engineering, and various fields of science.

• Visualization:

  • Right triangle representation helps in understanding the ratios.
  • Unit circle can also be utilized for understanding angles beyond 0° to 360°.

• Calculator Use:

  • Ensure calculator is in the correct mode (degrees or radians) based on the angle measurement needed.

Trigonometric Ratios Overview

• Trigonometric ratios are vital in connecting the angles of a right triangle to their side lengths.

Basic Ratios

• Sine (sin): Represents the ratio of the length of the opposite side to the hypotenuse.
• Cosine (cos): Represents the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (tan): Represents the ratio of the length of the opposite side to the adjacent side.

Reciprocal Ratios

• Cosecant (csc): The reciprocal of sine; calculated as hypotenuse divided by the opposite side.
• Secant (sec): The reciprocal of cosine; calculated as hypotenuse divided by the adjacent side.
• Cotangent (cot): The reciprocal of tangent; calculated as adjacent side divided by the opposite side.

Important Angles

• Key values of sine, cosine, and tangent for standard angles:
  • Sine:
    • sin(0°) = 0
    • sin(30°) = 1/2
    • sin(45°) = √2/2
    • sin(60°) = √3/2
    • sin(90°) = 1
  • Cosine:
    • cos(0°) = 1
    • cos(30°) = √3/2
    • cos(45°) = √2/2
    • cos(60°) = 1/2
    • cos(90°) = 0
  • Tangent:
    • tan(0°) = 0
    • tan(30°) = 1/√3
    • tan(45°) = 1
    • tan(60°) = √3
    • tan(90°) is undefined.

Identities

• Pythagorean Identity: Fundamental identity of trigonometry, stating that the squares of sine and cosine of an angle sum to one: sin²(θ) + cos²(θ) = 1.
• Sum and Difference Formulas:
  • sin(A ± B) = sinA cosB ± cosA sinB
  • cos(A ± B) = cosA cosB ∓ sinA sinB
  • tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB).

Applications

• Essential for resolving triangles to find unknown sides and angles.
• Widely used in fields such as physics and engineering, highlighting their importance across the sciences.

Visualization

• Right triangles provide a clear representation for understanding trigonometric ratios.
• The unit circle serves as a tool to grasp angles outside the range of 0° to 360°.

Calculator Use

• Ensure that the calculator is set to the appropriate mode (degrees or radians) to match the angle measurement required for calculations.

This quiz covers the fundamental trigonometric ratios, including sine, cosine, and tangent, as well as their reciprocal counterparts. It also explores important angles and their corresponding ratio values, providing a solid foundation for understanding how these concepts apply in right triangles.