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Trigonometry: Angles and Their Properties

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Termini in questo insieme (20)

  • What is an angle in standard position?
    An angle with its vertex at the origin, initial side on the positive x-axis, and terminal side rotated from the initial side.
  • How is the measure of an angle in standard position determined?
    By the amount of rotation from the initial side to the terminal side, measured in degrees or radians.
  • What are the four types of angles based on their measure?
    Acute (less than 90°), Right (exactly 90°), Obtuse (between 90° and 180°), and Straight (exactly 180°).
  • Define complementary angles.
    Two angles whose measures add up to \(90^\circ\).
  • Define supplementary angles.
    Two angles whose measures add up to \(180^\circ\).
  • What is the complement of a 30° angle?
    60°, because 30° + 60° = 90°.
  • What is the supplement of a 110° angle?
    70°, because 110° + 70° = 180°.
  • How do you find the complement of an angle \(x\)?
    Subtract the angle from 90°, i.e., \(90^\circ - x\).
  • How do you find the supplement of an angle \(x\)?
    Subtract the angle from 180°, i.e., \(180^\circ - x\).
  • What is the sum of the interior angles of a triangle?
    Always \(180^\circ\).
  • How can complementary and supplementary angles help solve triangle problems?
    They help find unknown angles by using the relationships that angles add up to 90° or 180°.
  • What is a radian?
    A radian is a measure of angle based on the radius of a circle; one radian is the angle subtended by an arc equal in length to the radius.
  • How many radians are in a full circle?
    A full circle is \(2\pi\) radians.
  • How many degrees are in a full circle?
    A full circle is 360 degrees.
  • What is the formula to convert degrees to radians?
    Multiply degrees by \(\frac{\pi}{180}\\).
  • What is the formula to convert radians to degrees?
    Multiply radians by \(\frac{180}{\pi}\\).
  • Convert 120° to radians.
    120° × \(\frac{\pi}{180}\\) = \(\frac{2\pi}{3}\\) radians.
  • Convert \(\frac{\pi}{5}\\) radians to degrees.
    \(\frac{\pi}{5}\\) × \(\frac{180}{\pi}\\) = 36°.
  • What is the relationship between the initial side and terminal side in an angle in standard position?
    The initial side lies on the positive x-axis; the terminal side is the rotated ray that forms the angle.
  • What is the difference between acute and obtuse angles?
    Acute angles measure less than 90°, while obtuse angles measure more than 90° but less than 180°.