Trigonometric Functions Quiz
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Questions and Answers

What is the value of $ an rac{ heta}{4}$ if $ heta = 0°$?

  • 3
  • 0 (correct)
  • Undefined
  • 1

    • Which of the following statements regarding sine and cosine functions is correct?

  • cos(-x) = -cos(x)
  • sin(-x) = sin(x)
  • cos(-x) = cos(x) (correct)
  • sin(x) is always positive

    • What is the range of the cosine function?

  • 0 to 1
  • 0 to $ rac{ heta}{2}$
  • -1 to 0
  • -1 to 1 (correct)

    • What is the reciprocal of sin x?

    <p>cosec x</p> Signup and view all the answers

    In which quadrant are both sine and cosine values negative?

    <p>Third quadrant</p> Signup and view all the answers

    What is the radius of the circle when a central angle of 60° intercepts an arc of length 37.4 cm?

    <p>35.7 cm</p> Signup and view all the answers

    How far does the tip of the minute hand of a watch move in 40 minutes if the length of the minute hand is 1.5 cm?

    <p>6.28 cm</p> Signup and view all the answers

    What is the relationship between the radii of two circles if the arcs of the same lengths subtend angles of 65° and 110° respectively?

    <p>22 : 13</p> Signup and view all the answers

    Convert 25° to radians. What is the correct measure?

    <p>$\frac{5\pi}{36}$</p> Signup and view all the answers

    What is the radian measure of –47°30′?

    <p>$-\frac{47.5\pi}{180}$</p> Signup and view all the answers

    What is the radian conversion of 240°?

    <p>$\frac{4\pi}{3}$</p> Signup and view all the answers

    In how many minutes does the minute hand make a complete revolution?

    <p>60 minutes</p> Signup and view all the answers

    What would the length of an arc be in a circle with a radius of 3 cm and an angle of 90°?

    <p>1.5π cm</p> Signup and view all the answers

    What is the value of cos(π)?

    <p>-1</p> Signup and view all the answers

    Which of the following statements is true regarding the sine function?

    <p>sin x = 0 when x = nπ.</p> Signup and view all the answers

    For which values of x does cos(x) equal zero?

    <p>x = (2n + 1)π/2</p> Signup and view all the answers

    What relationship holds true for all real x concerning sine and cosine?

    <p>sin² x + cos² x = 1</p> Signup and view all the answers

    When is sec x defined?

    <p>When cos x = 0</p> Signup and view all the answers

    What is the value of sin(3π)?

    <p>0</p> Signup and view all the answers

    What is the value of tan(π/4)?

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

    For what integer multiples does tan(x) become undefined?

    <p>(2n + 1)π/2</p> Signup and view all the answers

    What does the term 'trigonometry' literally mean?

    <p>Measuring the sides of a triangle</p> Signup and view all the answers

    Which profession is NOT mentioned as having historically used trigonometry?

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

    In which of the following applications is trigonometry used?

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

    What does a positive angle represent?

    <p>Counterclockwise rotation</p> Signup and view all the answers

    What is the initial side of an angle?

    <p>The starting point of the ray</p> Signup and view all the answers

    What is the vertex in the context of an angle?

    <p>The point of rotation of the ray</p> Signup and view all the answers

    Which of the following units are commonly used for measuring angles?

    <p>Degrees and radians</p> Signup and view all the answers

    What is indicated by the amount of rotation in angle measurement?

    <p>The measure of an angle</p> Signup and view all the answers

    What is the radian measure of an angle measuring 60°?

    <p>$\frac{\pi}{3}$</p> Signup and view all the answers

    Which formula correctly relates degree and radian measures?

    <p>Degree measure = $\frac{180}{\pi} \times$ Radian measure</p> Signup and view all the answers

    How many degrees are in $ rac{3 heta}{ heta}$ radians?

    <p>$\frac{180 \cdot 3\theta}{\pi \cdot \theta}$</p> Signup and view all the answers

    What is the degree equivalent of $ rac{7 heta}{6}$ radians?

    <p>$210°$</p> Signup and view all the answers

    What is the radian measure of an angle that is 270°?

    <p>$\frac{3\pi}{2}$</p> Signup and view all the answers

    If 40° 20′ is converted into radians, what is its equivalent?

    <p>$\frac{121\pi}{540}$</p> Signup and view all the answers

    Which of the following radian measures corresponds to 180°?

    <p>$\pi$</p> Signup and view all the answers

    What is the degree measure corresponding to 6 radians?

    <p>$343°$</p> Signup and view all the answers

    Study Notes

    Trigonometric Functions

    • Trigonometry is the study of the relationships between the sides and angles of triangles.
    • Angles are measured in degrees or radians.
    • Degree measure divides a circle into 360 degrees (360°).
    • Radian measure defines an angle as the ratio of the arc length to the radius of a circle.
    • Conversion formulas:
      • Radian measure = (180/π) × Degree measure.
      • Degree measure = (π/180) × Radian measure.
    • Quadrantal angles are angles whose terminal side coincides with one of the axes. Examples include 0°, 90°, 180°, and 270°.
    • Trigonometric functions are functions that relate an angle to the ratios of the sides of a right-angled triangle.
    • Basic trigonometric functions:
      • Sine (sin): Opposite side / Hypotenuse
      • Cosine (cos): Adjacent side / Hypotenuse
      • Tangent (tan): Opposite side / Adjacent side
      • Cosecant (csc): 1 / Sin
      • Secant (sec): 1 / Cos
      • Cotangent (cot): 1 / Tan

    Trigonometric Identities

    • Fundamental trigonometric identity: sin²x + cos²x = 1
    • Other identities:
      • 1 + tan²x = sec²x
      • 1 + cot²x = csc²x

    Values of Trigonometric Functions

    • The values of trigonometric functions for certain common angles, such as 0°, 30°, 45°, 60°, and 90°, can be determined using the unit circle.

    Sign of Trigonometric Functions

    • The sign of a trigonometric function depends on the quadrant in which the angle lies.
    • First Quadrant (0° < x < 90°): All trigonometric functions are positive.
    • Second Quadrant (90° < x < 180°): Sine and cosecant are positive; others are negative.
    • Third Quadrant (180° < x < 270°): Tangent and cotangent are positive; others are negative.
    • Fourth Quadrant (270° < x < 360°): Cosine and secant are positive; others are negative.
    • Key formulas:
      • cos (-x) = cos x
      • sin (-x) = -sin x
      • -1 ≤ cos x ≤ 1 and -1 ≤ sin x ≤ 1 for all x.

    Periodicity of Trigonometric Functions

    • The values of sine and cosine functions repeat after a full cycle of 2π (360°).
    • sin(2nπ + x) = sin x, n ∈ Z
    • cos(2nπ + x) = cos x, n ∈ Z.

    Special Cases

    • sin x = 0 implies x = nπ, where n is any integer.
    • cos x = 0 implies x = (2n + 1)(π/2), where n is any integer.

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    Trigonometric Functions PDF

    Description

    Test your knowledge of trigonometric functions and their relationships to angles and triangles. This quiz covers concepts like degree and radian measures, as well as the basic trigonometric functions such as sine, cosine, and tangent.

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