Trigonometry Basics
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Questions and Answers

What is the sine of an angle in a right triangle, and how is it calculated?

The sine of an angle is the ratio of the length of the opposite side to the hypotenuse, given by the formula sin(θ) = Opposite / Hypotenuse.

Explain the relationship described by the Pythagorean identity in trigonometry.

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1, reflecting the relationship between the sine and cosine of the angle.

How does the tangent of an angle relate to sine and cosine?

The tangent of an angle is defined as the ratio of the opposite side to the adjacent side, which can also be expressed as tan(θ) = sin(θ) / cos(θ).

Describe the significance of key angles on the unit circle and their corresponding sine and cosine values.

<p>Key angles such as 0°, 30°, 45°, 60°, and 90° provide specific sine and cosine values, which are essential for solving trigonometric problems and understanding the unit circle's geometry.</p> Signup and view all the answers

What are the double angle formulas, and why are they useful in trigonometry?

<p>The double angle formulas express sine, cosine, and tangent of double angles in terms of single angles, such as sin(2θ) = 2sin(θ)cos(θ), making calculations simpler in various applications.</p> Signup and view all the answers

What is the Law of Sines, and how is it applied in solving triangles?

<p>The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C) and is used to find unknown angles or sides in non-right triangles.</p> Signup and view all the answers

How do the graphs of sine and cosine functions differ in their periodicity and amplitude?

<p>Both sine and cosine functions have a period of 2π and an amplitude of 1, meaning they repeat their values every 2π units and oscillate between -1 and 1.</p> Signup and view all the answers

What is the role of inverse trigonometric functions, and how do they differ from regular trigonometric functions?

<p>Inverse trigonometric functions, like sin⁻¹(x), cos⁻¹(x), and tan⁻¹(x), are used to determine the angles corresponding to given ratios, providing a reverse operation compared to regular trigonometric functions.</p> Signup and view all the answers

Study Notes

Trigonometry

  • Definition: A branch of mathematics dealing with the relationships between the angles and sides of triangles, primarily right-angled triangles.

  • Basic Functions:

    • Sine (sin): Ratio of the opposite side to the hypotenuse.
      • sin(θ) = Opposite / Hypotenuse
    • Cosine (cos): Ratio of the adjacent side to the hypotenuse.
      • cos(θ) = Adjacent / Hypotenuse
    • Tangent (tan): Ratio of the opposite side to the adjacent side.
      • tan(θ) = Opposite / Adjacent = sin(θ) / cos(θ)

  • Reciprocal Functions:

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

  • Pythagorean Identity:

    • sin²(θ) + cos²(θ) = 1

  • Angle Sum and Difference Identities:

    • 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) = 2cos²(θ) - 1 = 1 - 2sin²(θ)
    • tan(2θ) = 2tan(θ) / (1 - tan²(θ))

  • Unit Circle:

    • A circle with a radius of 1 centered at the origin.
    • Key angles: 0°, 30°, 45°, 60°, 90° (and their radian equivalents).
    • Coordinates on the unit circle represent (cos(θ), sin(θ)).

  • Applications:

    • Used in physics (wave motion, mechanics), engineering, architecture, and computer graphics.
    • Essential for solving triangles in geometry.

  • Inverse Trigonometric Functions:

    • sin⁻¹(x), cos⁻¹(x), tan⁻¹(x) to find angles from given ratios.

  • Graphs of Trigonometric Functions:

    • Periodic functions with specific amplitudes and periods:
      • sin(x): Period = 2π, Amplitude = 1
      • cos(x): Period = 2π, Amplitude = 1
      • tan(x): Period = π, Undefined at (π/2 + nπ)

  • Common Trigonometric Values:

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

  • Law of Sines and Law of Cosines:

    • Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
    • Law of Cosines: c² = a² + b² - 2ab*cos(C)

These notes provide a foundational overview of trigonometry, covering essential concepts and formulas crucial for understanding and applying trigonometric principles.

Trigonometry Overview

  • A mathematical branch focusing on the relationships between angles and sides of triangles, especially right triangles.

Basic Functions

  • Sine (sin): Opposite side over hypotenuse.
  • Cosine (cos): Adjacent side over hypotenuse.
  • Tangent (tan): Opposite side over adjacent side, also expressed as sin(θ) / cos(θ).

Reciprocal Functions

  • Cosecant (csc): 1 divided by sine (csc(θ) = 1/sin(θ)).
  • Secant (sec): 1 divided by cosine (sec(θ) = 1/cos(θ)).
  • Cotangent (cot): 1 divided by tangent, also expressed as cos(θ) / sin(θ).

Pythagorean Identity

  • Fundamental identity: sin²(θ) + cos²(θ) = 1.

Angle Sum and Difference Identities

  • Sin: sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B).
  • Cos: cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B).
  • Tan: tan(A ± B) = (tan(A) ± tan(B)) / (1 ∓ tan(A)tan(B)).

Double Angle Formulas

  • Sin: sin(2θ) = 2sin(θ)cos(θ).
  • Cos: cos(2θ) can be expressed in three ways:
    • cos²(θ) - sin²(θ)
    • 2cos²(θ) - 1
    • 1 - 2sin²(θ).
  • Tan: tan(2θ) = 2tan(θ) / (1 - tan²(θ)).

Unit Circle

  • Defined as a circle with a radius of 1, centered at the origin.
  • Key angles include 0°, 30°, 45°, 60°, and 90° along with their radian equivalents.
  • Points on the unit circle represent coordinates as (cos(θ), sin(θ)).

Applications

  • Utilized in fields like physics (wave motion, mechanics), engineering, architecture, and computer graphics.
  • Important for solving various triangle-related problems in geometry.

Inverse Trigonometric Functions

  • Functions include sin⁻¹(x), cos⁻¹(x), and tan⁻¹(x) for determining angles from given ratios.

Graphs of Trigonometric Functions

  • Functions exhibit periodic behavior with defined amplitudes and periods:
    • Sin(x): Period = 2π, Amplitude = 1.
    • Cos(x): Period = 2π, Amplitude = 1.
    • Tan(x): Period = π, undefined at odd multiples of π/2.

Common Trigonometric Values

  • Fundamental values include:
    • sin(0) = 0, cos(0) = 1, tan(0) = 0.
    • sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = √3/3.
    • sin(45°) = √2/2, cos(45°) = √2/2, tan(45°) = 1.
    • sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3.
    • sin(90°) = 1, cos(90°) = 0, tan(90°) = Undefined.

Law of Sines and Law of Cosines

  • Law of Sines states that the ratio of a side length to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C).
  • Law of Cosines relates the lengths of sides to the cosine of one angle: c² = a² + b² - 2ab*cos(C).

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Test your knowledge of trigonometry with this quiz covering the fundamental concepts, including basic and reciprocal functions. Learn about sine, cosine, tangent, and their identities to strengthen your understanding of triangles and their angles.

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