Introduction to Trigonometry

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Questions and Answers

What is the definition of cosine in a right-angled triangle?

  • Adjacent/Hypotenuse (correct)
  • Hypotenuse/Adjacent
  • Opposite/Hypotenuse
  • Opposite/Adjacent

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

  • sin²θ + tan²θ = 1
  • tan²θ = sin²θ + cos²θ
  • sin²θ - cos²θ = 1
  • sin²θ + cos²θ = 1 (correct)

Which trigonometric ratio is the reciprocal of the sine function?

  • Secant
  • Cotangent
  • Cosecant (correct)
  • Cosine

What does the unit circle help to represent in trigonometry?

<p>Cosine as the x-coordinate and sine as the y-coordinate (B)</p> Signup and view all the answers

Which of the following identities is true for the tangent function?

<p>tan(-θ) = -tanθ (D)</p> Signup and view all the answers

In which of the following applications is trigonometry crucial?

<p>Analyzing the motion of objects in physics (D)</p> Signup and view all the answers

How can trigonometric equations be solved effectively?

<p>With algebraic manipulation and trigonometric identities (B)</p> Signup and view all the answers

What is the relationship between sine and cosine for any angle θ as defined by the unit circle?

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

Flashcards

What is Trigonometry?

Trigonometry is a branch of mathematics that studies relationships between angles and sides of triangles. It has applications in fields like navigation, surveying, engineering, and physics.

What are the fundamental trigonometric ratios?

The fundamental trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). These ratios are defined for acute angles in a right-angled triangle. - sin θ = opposite/hypotenuse - cos θ = adjacent/hypotenuse - tan θ = opposite/adjacent

What are the other trigonometric ratios?

Cosecant (csc θ) is the reciprocal of sine (csc θ = 1/sin θ). Secant (sec θ) is the reciprocal of cosine (sec θ = 1/cos θ). Cotangent (cot θ) is the reciprocal of tangent (cot θ = 1/tan θ).

What is the Pythagorean identity?

The Pythagorean identity states that sin²θ + cos²θ = 1, which is a fundamental relationship between sine and cosine.

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How are trigonometric functions extended?

Trigonometric functions can be extended to any angle (positive, negative, or zero) using the unit circle.

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

sin(-θ) = -sinθ

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What are trigonometric equations?

Trigonometric equations involve trigonometric functions and are solved using algebraic manipulation, trigonometric identities, and the unit circle.

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How is trigonometry used in real-world applications?

Trigonometry plays a crucial role in solving problems involving triangles, which are foundational for numerous applications. Examples include: - Calculating heights and distances in surveying. - Determining angles of elevation and depression in navigation. - Analyzing the motion of objects in physics. - Designing structures in engineering.

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Study Notes

Introduction to Trigonometry

  • Trigonometry studies relationships between angles and sides of triangles.
  • Applications include navigation, surveying, engineering, and physics.

Basic Trigonometric Ratios

  • Fundamental ratios are sine (sin), cosine (cos), and tangent (tan).
  • Defined for acute angles in right-angled triangles.
  • sin θ = opposite side / hypotenuse
  • cos θ = adjacent side / hypotenuse
  • tan θ = opposite side / adjacent side

Other Trigonometric Ratios

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

Relationship Between Trigonometric Ratios

  • Pythagorean identity: sin²θ + cos²θ = 1.

Trigonometric Functions for Any Angle

  • Extended to any angle using the unit circle.
  • Unit circle is a circle with radius 1 centered at the origin.
  • X-coordinate represents cosine of angle, y-coordinate represents sine.
  • Tangent of an angle is the ratio of sine to cosine (tan θ = sin θ / cos θ).

Trigonometric Identities

  • Equations true for all valid values.
  • Key identities include:
    • sin(-θ) = -sinθ
    • cos(-θ) = cosθ
    • tan(-θ) = -tanθ
  • Used to simplify expressions and solve equations.

Trigonometric Equations

  • Involve trigonometric functions, solved using manipulation, identities, and unit circle.
  • May have multiple solutions due to periodicity.

Applications of Trigonometry

  • Crucial for solving triangle-related problems.
  • Applications include:
    • Calculating heights and distances (surveying).
    • Determining angles (navigation).
    • Analyzing object motion (physics).
    • Designing structures (engineering).

Graphs of Trigonometric Functions

  • Periodic functions repeating at intervals.
  • Sine, cosine, and tangent graphs show characteristic waves.
  • Sine and cosine period is 360° (or 2π radians).
  • Tangent period is 180° (or π radians).

Inverse Trigonometric Functions

  • Used to find angles given trigonometric ratios.
  • Notations include arcsin (sin⁻¹), arccos (cos⁻¹), arctan (tan⁻¹), etc.
  • Produce angles within specific ranges.

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