Trigonometry Basics

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Questions and Answers

What is the ratio represented by the sine function in a right triangle?

  • Ratio of the opposite side to the hypotenuse (correct)
  • Ratio of the adjacent side to the hypotenuse
  • Ratio of the opposite side to the adjacent side
  • Ratio of the hypotenuse to the adjacent side

Which of the following statements about the Pythagorean Identity is true?

  • sin²(θ) + cos²(θ) = 1 (correct)
  • sin²(θ) - cos²(θ) = 1
  • sin²(θ) + tan²(θ) = 1
  • sin²(θ) + cos²(θ) = tan²(θ)

How is the tangent function defined in terms of sine and cosine?

  • tan(θ) = sin(θ)/cos(θ) (correct)
  • tan(θ) = cos(θ)/sin(θ)
  • tan(θ) = 1/sin(θ)
  • tan(θ) = sin²(θ)/cos²(θ)

What does the Law of Sines relate?

<p>Ratios of the lengths of sides to the angles of a triangle (B)</p> Signup and view all the answers

Which angle has a tangent value of undefined in trigonometry?

<p>90° (A)</p> Signup and view all the answers

In a unit circle, which coordinate corresponds to the cosine value of an angle?

<p>x-coordinate (A)</p> Signup and view all the answers

What is the period of the sine and cosine functions?

<p>2Ï€ (A)</p> Signup and view all the answers

What is the formula for the double angle of sine?

<p>sin(2θ) = 2sin(θ)cos(θ) (C)</p> Signup and view all the answers

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Study Notes

Trigonometry

  • Definition: Trigonometry is a branch of mathematics that studies relationships between the angles and sides of triangles, particularly right-angled triangles.

  • Key Functions:

    • Sine (sin): Ratio of the opposite side to the hypotenuse.
    • Cosine (cos): Ratio of the adjacent side to the hypotenuse.
    • Tangent (tan): Ratio of the opposite side to the adjacent side.
  • Reciprocal Functions:

    • Cosecant (csc): Reciprocal of sine (1/sin).
    • Secant (sec): Reciprocal of cosine (1/cos).
    • Cotangent (cot): Reciprocal of tangent (1/tan).
  • Pythagorean Identity: In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

    • Formula: sin²(θ) + cos²(θ) = 1
  • Unit Circle:

    • A circle with a radius of 1 centered at the origin of a coordinate plane.
    • Points on the unit circle represent the values of sin(θ) and cos(θ):
      • sin(θ) = y-coordinate
      • cos(θ) = x-coordinate
  • Common Angles:

    • 0°: sin(0) = 0, cos(0) = 1, tan(0) = 0
    • 30°: sin(30) = 1/2, cos(30) = √3/2, tan(30) = 1/√3
    • 45°: sin(45) = √2/2, cos(45) = √2/2, tan(45) = 1
    • 60°: sin(60) = √3/2, cos(60) = 1/2, tan(60) = √3
    • 90°: sin(90) = 1, cos(90) = 0, tan(90) = undefined
  • Law of Sines:

    • Relates the ratios of the lengths of sides of a triangle to the sines of its angles.
    • Formula: (a/sin(A)) = (b/sin(B)) = (c/sin(C))
  • Law of Cosines:

    • Relates the lengths of the sides of a triangle to the cosine of one of its angles.
    • Formula: c² = a² + b² - 2ab * cos(C)
  • Applications:

    • Problem-solving in geometry, physics (e.g., waves, oscillations), engineering, and navigation.
  • Trigonometric Identities:

    • Angle Sum/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(2θ) = 2sin(θ)cos(θ)
      • cos(2θ) = cos²(θ) - sin²(θ)
  • Graphs of Trigonometric Functions:

    • Sine and cosine functions have a periodic wave shape with a period of 2Ï€.
    • Tangent function has a periodic shape with asymptotes and a period of Ï€.

These notes cover fundamental concepts in trigonometry, foundational for advanced studies in mathematics and related fields.

Trigonometry Definition

  • The branch of mathematics that explores the relationship between angles and sides of triangles, especially right-angled triangles.

Key Trigonometric Functions

  • Sine (sin): The ratio of the opposite side to the hypotenuse of a right triangle.
  • Cosine (cos): The ratio of the adjacent side to the hypotenuse of a right triangle.
  • Tangent (tan): The ratio of the opposite side to the adjacent side of a right triangle.

Reciprocal Trigonometric Functions

  • Cosecant (csc): The reciprocal of sine (1/sin).
  • Secant (sec): The reciprocal of cosine (1/cos).
  • Cotangent (cot): The reciprocal of tangent (1/tan).

Pythagorean Identity

  • In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
  • Formula: sin²(θ) + cos²(θ) = 1

Unit Circle

  • A circle with a radius of 1 unit centered at the origin of a coordinate plane.
  • Points on the unit circle represent the values of sin(θ) and cos(θ):
    • sin(θ) = y-coordinate
    • cos(θ) = x-coordinate

Common Angles:

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

Law of Sines

  • Relates the ratios of the lengths of sides of a triangle to the sines of its angles.
  • Formula: (a/sin(A)) = (b/sin(B)) = (c/sin(C))

Law of Cosines

  • Relates the lengths of the sides of a triangle to the cosine of one of its angles.
  • Formula: c² = a² + b² - 2ab * cos(C)

Applications of Trigonometry

  • Solving problems in geometry, physics (e.g., waves, oscillations), engineering, and navigation.

Trigonometric Identities

  • Angle Sum/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(2θ) = 2sin(θ)cos(θ)
    • cos(2θ) = cos²(θ) - sin²(θ)

Graphs of Trigonometric Functions

  • Sine and cosine functions have a periodic wave shape with a period of 2Ï€.
  • Tangent function has a periodic shape with asymptotes and a period of Ï€.

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