Trigonometry Basics
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Questions and Answers

What is the definition of sine in a right triangle?

  • The ratio of the hypotenuse to the opposite side
  • The ratio of the adjacent side to the opposite side
  • The ratio of the opposite side to the hypotenuse (correct)
  • The ratio of the adjacent side to the hypotenuse

    • What is the sum formula for sine in trigonometry?

  • sin(A + B) = sin(A)sin(B) + cos(A)cos(B)
  • sin(A + B) = sin(A)cos(B) - cos(A)sin(B)
  • sin(A + B) = sin(A)cos(B) + cos(A)sin(B) (correct)
  • sin(A + B) = sin(A)cos(B) + tan(A)tan(B)

    • What is the Pythagorean Identity in trigonometry?

  • sin(A) + cos(A) = 1
  • tan(A) + cot(A) = 1
  • sin^2(A) + cos^2(A) = 1 (correct)
  • sec(A) + csc(A) = 1

    • What is the application of trigonometry in navigation?

    <p>Calculating distances and directions</p> Signup and view all the answers

    What type of triangle is not a right triangle?

    <p>Oblique triangle</p> Signup and view all the answers

    What is the unit of measurement for angles in radians?

    <p>Radians</p> Signup and view all the answers

    What is the trigonometric ratio of the adjacent side to the hypotenuse in a right triangle?

    <p>Cosine</p> Signup and view all the answers

    What is the application of trigonometry in modeling periodic phenomena?

    <p>Wave motion</p> Signup and view all the answers

    What is the reciprocal of the sine function?

    <p>cosecant</p> Signup and view all the answers

    What is the period of the tangent function?

    <p>π</p> Signup and view all the answers

    What is the formula for the cosine of a double angle?

    <p>cos^2(A) - sin^2(A)</p> Signup and view all the answers

    What is the identity for sin(A + B) in terms of sine and cosine of A and B?

    <p>sin(A)cos(B) + cos(A)sin(B)</p> Signup and view all the answers

    What is the trigonometric function that models wave motion?

    <p>sine</p> Signup and view all the answers

    What is the trigonometric identity for sin(-A)?

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

    What is the type of triangle used to define trigonometric ratios?

    <p>right triangle</p> Signup and view all the answers

    What is the value of sin(45°) in a unit circle?

    <p>1/√2</p> Signup and view all the answers

    What is the formula for the tangent of a double angle?

    <p>2tan(A) / (1 - tan^2(A))</p> Signup and view all the answers

    What is the amplitude of a trigonometric function?

    <p>the maximum value of the function</p> Signup and view all the answers

    Study Notes

    Trigonometry

    Definition

    • Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.

    Key Concepts

    Angles

    • Degrees: A unit of measurement for angles, with 360 degrees in a full circle.
    • Radians: A unit of measurement for angles, with 2π radians in a full circle.

    Triangles

    • Right Triangle: A triangle with one right angle (90 degrees).
    • Oblique Triangle: A triangle that is not a right triangle.

    Trigonometric Ratios

    • Sine (sin): The ratio of the opposite side to the hypotenuse in a right triangle.
    • Cosine (cos): The ratio of the adjacent side to the hypotenuse in a right triangle.
    • Tangent (tan): The ratio of the opposite side to the adjacent side in a right triangle.
    • Cotangent (cot): The ratio of the adjacent side to the opposite side in a right triangle.
    • Secant (sec): The ratio of the hypotenuse to the adjacent side in a right triangle.
    • Cosecant (csc): The ratio of the hypotenuse to the opposite side in a right triangle.

    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)
      • sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
      • cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
      • cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

    Applications

    • Triangulation: Solving triangles using trigonometric ratios and identities.
    • Wave Motion: Modeling periodic phenomena, such as sound and light waves, using trigonometric functions.
    • Navigation: Calculating distances and directions using trigonometric principles.

    Trigonometry

    Definition

    • Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.

    Angles

    • A full circle has 360 degrees or 2π radians.
    • Degrees and radians are units of measurement for angles.

    Triangles

    • A right triangle has one right angle (90 degrees).
    • An oblique triangle is a triangle that is not a right triangle.

    Trigonometric Ratios

    • Sine (sin) is the ratio of the opposite side to the hypotenuse in a right triangle.
    • Cosine (cos) is the ratio of the adjacent side to the hypotenuse in a right triangle.
    • Tangent (tan) is the ratio of the opposite side to the adjacent side in a right triangle.
    • Cotangent (cot) is the ratio of the adjacent side to the opposite side in a right triangle.
    • Secant (sec) is the ratio of the hypotenuse to the adjacent side in a right triangle.
    • Cosecant (csc) is the ratio of the hypotenuse to the opposite side in a right triangle.

    Identities

    • The Pythagorean Identity states that sin^2(A) + cos^2(A) = 1.
    • Sum and Difference Formulas include:
      • sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
      • sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
      • cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
      • cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

    Applications

    • Triangulation involves solving triangles using trigonometric ratios and identities.
    • Wave motion can be modeled using trigonometric functions to represent periodic phenomena, such as sound and light waves.
    • Trigonometry has navigation applications, including calculating distances and directions.

    Trigonometry

    Definitions and Concepts

    • An angle is a measure of rotation between two lines or planes
    • A triangle is a polygon with three sides and three angles
    • A right triangle is a triangle with one right angle (90 degrees)
    • The hypotenuse is the side opposite the right angle in a right triangle
    • Trigonometric ratios are relationships between the sides and angles of a right triangle

    Trigonometric Ratios

    • Sine (sin) is the ratio of the opposite side to the hypotenuse
    • Cosine (cos) is the ratio of the adjacent side to the hypotenuse
    • Tangent (tan) is the ratio of the opposite side to the adjacent side
    • Cotangent (cot) is the ratio of the adjacent side to the opposite side
    • Secant (sec) is the ratio of the hypotenuse to the opposite side
    • Cosecant (csc) is the ratio of the hypotenuse to the adjacent side

    Identities and Formulas

    • The Pythagorean Identity states that sin^2(A) + cos^2(A) = 1
    • The Sum and Difference Formulas are:
      • 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 are:
      • sin(2A) = 2sin(A)cos(A)
      • cos(2A) = cos^2(A) - sin^2(A)
      • tan(2A) = 2tan(A) / (1 - tan^2(A))

    Applications and Graphs

    • Right triangle problems involve solving for sides and angles using trigonometric ratios
    • Wave motion can be modeled using sine and cosine functions
    • Graphs of trigonometric functions have:
      • A period of 2π for sine and cosine
      • A period of π for tangent
      • Amplitude and phase shifts can be applied to trigonometric functions

    Important Values and Identities

    • The unit circle is a circle with a radius of 1, used to define trigonometric ratios
    • Special angles are 30°, 45°, 60°, and their corresponding trigonometric ratios
    • Trigonometric identities are equations that are true for all angles, such as sin(-A) = -sin(A)

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    Quiz Team

    Description

    Learn about the fundamental concepts of trigonometry, including angles, triangles, and their relationships. Covers degrees, radians, right triangles, and oblique triangles.

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