Introduction to Trigonometry

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Questions and Answers

Which of the following equations is a correct trigonometric identity?

  • tan θ = sin θ * cos θ
  • sin(2θ) = 2sin θ cos θ (correct)
  • sin² θ + cos² θ = 2
  • tan θ + cot θ = 1

What is the value of sin 45° when expressed as a ratio?

  • $\frac{\sqrt{2}}{2}$ (correct)
  • $\frac{\sqrt{3}}{3}$
  • $\frac{1}{2}$
  • $\frac{1}{\sqrt{2}}$

Which trigonometric function helps determine the adjacent side when the angle and hypotenuse are known?

  • Cosine (correct)
  • Cosecant
  • Sine
  • Tangent

If sin θ = 0.5, which of the following angles could θ represent within the range 0° to 360°?

<p>150° (B), 30° (D)</p> Signup and view all the answers

Which of the following is NOT an application of trigonometry?

<p>Creating complex algebraic equations (D)</p> Signup and view all the answers

Which of these relationships represents the tangent function in terms of sine and cosine?

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

Which inverse trigonometric function is used to determine the angle from a known cosine value?

<p>arccos (C)</p> Signup and view all the answers

What is the graphical representation of sine and cosine functions characterized by?

<p>Cyclical patterns (D)</p> Signup and view all the answers

Flashcards

Sine (sin)

The ratio of the side opposite to the angle to the hypotenuse.

Cosine (cos)

The ratio of the side adjacent to the angle to the hypotenuse.

Tangent (tan)

The ratio of the side opposite to the angle to the side adjacent to the angle.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables involved. They are used to simplify trigonometric expressions and prove other identities.

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Trigonometric Equations

Equations that involve trigonometric functions. Solving them often requires using identities and algebraic techniques.

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Trigonometric Functions

These functions relate angles to the ratios of sides of right-angled triangles. There are three main trigonometric ratios: sine, cosine, and tangent.

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Inverse Trigonometric Functions

Functions that are defined as the inverse of the basic trigonometric functions (sine, cosine, and tangent). They are used to find the angle when the trigonometric value is known.

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Pythagorean Theorem

This theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. It is a crucial relationship used in trigonometry.

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Study Notes

Introduction to Trigonometry

  • Trigonometry studies relationships between angles and sides of triangles.
  • It is crucial in fields like navigation, surveying, and engineering.
  • Trigonometric functions link angles to ratios of sides in right-angled triangles.

Basic Trigonometric Ratios

  • Sine (sin): Opposite side / Hypotenuse.
  • Cosine (cos): Adjacent side / Hypotenuse.
  • Tangent (tan): Opposite side / Adjacent side.

Relationship between Trigonometric Ratios

  • tan θ = sin θ / cos θ.
  • sin² θ + cos² θ = 1 (Pythagorean identity).

Trigonometric Functions and Their Graphs

  • Sine, cosine, and tangent functions are cyclical.
  • They exhibit periodic behavior, repeating values at regular intervals.
  • Their graphs display these cyclical patterns and are essential for understanding their properties.

Trigonometric Identities

  • Equations true for all variable values.
  • Used to simplify trigonometric expressions.
  • Examples include:
    • sin² θ + cos² θ = 1 (Pythagorean identity).
    • tan θ = sin θ / cos θ.

Trigonometric Functions of Special Angles

  • Knowing trigonometric function values for specific angles (30°, 45°, and 60°) eases calculations.
  • These values stem from 30-60-90 and 45-45-90 triangles.
  • Memorization is beneficial.

Inverse Trigonometric Functions

  • Inverse functions reverse the original function's operation.
  • Used to find angles given trigonometric values.
  • Notation includes arcsin (sin⁻¹), arccos (cos⁻¹), and arctan (tan⁻¹).

Applications of Trigonometry

  • Navigation: Determining distances and directions using angles.
  • Surveying: Measuring land areas and distances via angles.
  • Engineering: Designing structures and planning projects using angles.
  • Astronomy: Studying celestial objects and positions with angles.
  • Physics: Describing motion and forces using angles and lengths.

Trigonometric Equations

  • Equations involving trigonometric functions.
  • Solving them often involves trigonometric identities and algebraic methods.
  • Solutions may have multiple values due to function cycles.

Unit Circle

  • A circle of radius 1 centered at the coordinate system's origin.
  • Points on the circle represent trigonometric functions for angles.
  • A powerful visual aid for understanding trigonometric functions.

Graphs of Trigonometric Functions

  • Visual representations of trigonometric functions.
  • Show periodicity and other key properties.
  • Graphs are valuable for recognizing patterns and problem-solving.

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