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Trigonometric Functions on the Unit Circle quiz

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  • What do the x and y coordinates of a point on the unit circle represent in terms of trigonometric functions?
    The x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine of the angle.
  • How is the tangent of an angle calculated using the unit circle coordinates?
    Tangent is calculated by dividing the y-coordinate by the x-coordinate (tan = y/x).
  • What happens to the tangent function at 90 and 270 degrees on the unit circle?
    The tangent is undefined at 90 and 270 degrees because the x-coordinate is zero, resulting in division by zero.
  • How can you visualize any angle on the unit circle as a right triangle?
    Each angle forms a right triangle with the radius as the hypotenuse, the x-coordinate as the base, and the y-coordinate as the height.
  • Why are negative trigonometric values valid on the unit circle?
    Negative values occur depending on the quadrant, reflecting the signs of the x and y coordinates in that quadrant.
  • What is the cosine and sine of 0 degrees on the unit circle?
    Cosine of 0 degrees is 1, and sine of 0 degrees is 0.
  • What is the tangent of 0 degrees on the unit circle?
    The tangent of 0 degrees is 0 because y is 0 and x is 1.
  • What is the tangent of 180 degrees on the unit circle?
    The tangent of 180 degrees is 0 because y is 0 and x is -1.
  • How do you find the tangent when both sine and cosine are fractions with the same denominator?
    You can divide the numerators directly since the denominators cancel out.
  • What is the sine, cosine, and tangent of 217 degrees given the coordinates (-4/5, -3/5)?
    Sine is -3/5, cosine is -4/5, and tangent is 3/4.
  • How do you simplify dividing two fractions when finding tangent?
    Multiply by the reciprocal of the denominator fraction, which cancels the denominators if they are the same.
  • What is the cosine, sine, and tangent of 60 degrees given the coordinates (1/2, √3/2)?
    Cosine is 1/2, sine is √3/2, and tangent is √3.
  • How can you remember which coordinate corresponds to cosine and which to sine?
    Cosine corresponds to the x-coordinate and sine to the y-coordinate, following the alphabetical order (c before s).
  • Why is the tangent of 90 degrees undefined?
    Because the x-coordinate is 0, dividing by zero makes the tangent undefined.
  • What is the relationship between the unit circle and right triangles in trigonometry?
    Every angle on the unit circle can be visualized as forming a right triangle, making trigonometric values equal to the coordinates of the corresponding point.