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Precalculus: Trigonometric Functions and Angles

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이 집합의 용어 (20)

  • What defines an angle in standard position?
    An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis.
  • How is a positive angle measured?
    A positive angle is measured by counterclockwise rotation from the initial side.
  • What is a quadrantal angle?
    An angle whose terminal side lies on the x-axis or y-axis is called a quadrantal angle.
  • How many degrees are in one full revolution?
    One full revolution is 360 degrees.
  • What is the measure of a right angle in degrees and revolutions?
    A right angle measures 90 degrees or one-quarter of a revolution.
  • Define radian measure of an angle.
    One radian is the measure of a central angle that subtends an arc equal in length to the radius of the circle.
  • How do you find the length of an arc (s) given radius (r) and central angle (θ in radians)?
    Arc length is given by \(s = r\theta\).
  • Convert 45 degrees to radians.
    45 degrees equals \(\frac{\pi}{4}\) radians.
  • Convert 2.5 radians to degrees.
    2.5 radians equals approximately 143.24 degrees.
  • What is the formula for the area of a sector with radius r and central angle θ (in radians)?
    Area of a sector is \(A = \frac{1}{2} r^2 \theta\).
  • Define linear speed (v) of an object moving in circular motion.
    Linear speed is the distance traveled divided by time, \(v = \frac{s}{t}\).
  • Define angular speed (ω).
    Angular speed is the angle swept out per unit time, \(\omega = \frac{\theta}{t}\), measured in radians per unit time.
  • What are the six trigonometric functions of an acute angle?
    Sine, cosine, tangent, cosecant, secant, and cotangent.
  • How is sine of an angle θ defined in a right triangle?
    Sine is the ratio of the length of the opposite side to the hypotenuse.
  • How is cosine of an angle θ defined in a right triangle?
    Cosine is the ratio of the length of the adjacent side to the hypotenuse.
  • How is tangent of an angle θ defined in a right triangle?
    Tangent is the ratio of the length of the opposite side to the adjacent side.
  • State the Pythagorean identity involving sine and cosine.
    \(\sin^2 \theta + \cos^2 \theta = 1\).
  • What are reciprocal identities for sine and cosecant?
    \(\csc \theta = \frac{1}{\sin \theta}\).
  • What is the Complementary Angle Theorem?
    Cofunctions of complementary angles are equal, e.g., \(\sin \theta = \cos (90^\circ - \theta)\).
  • How can you find all trigonometric functions if one function value is known for an acute angle?
    Use right triangle definitions or fundamental identities and the Pythagorean theorem to find the remaining functions.