Consider the point on a circle with radius . Determine the exact values of , and .Choose the correct response from the options below.

Question

Consider the point on a circle with radius . Determine the exact values of ,  and .Choose the correct response from the options below.

BlackTom AI · Accepted Answer

cos(θ) = x/r, sin(θ) = y/r and tan(θ) = y/x.

1) Relevant knowledge: For a point P(x,y) on a circle of radius r centered at the origin, cos(θ)=x/r, sin(θ)=y/r, and tan(θ)=y/x. Signs follow quadrant: x>0, y<0 gives quadrant IV so cosine positive, sine negative, tangent negative.

2) Compute r from the coordinates to confirm the given radius: x = 3/4, y = −√55/4.
   r = sqrt(x^2 + y^2)
     = sqrt((3/4)^2 + (−√55/4)^2).

3) Intermediate arithmetic:
   (3/4)^2 = 9/16.
   (−√55/4)^2 = 55/16.
   Sum = 9/16 + 55/16 = 64/16 = 4.
   r = sqrt(4) = 2, which matches the stated radius.

4) Use definitions to find the trigonometric values:
   cos(θ) = x/r = (3/4) ÷ 2 = (3/4) * (1/2) = 3/8.
   sin(θ) = y/r = (−√55/4) ÷ 2 = (−√55/4) * (1/2) = −√55/8.
   tan(θ) = y/x = (−√55/4) ÷ (3/4) = (−√55/4) * (4/3) = −√55/3.

5) Why this is correct and why other choices are wrong:
   - The correct values use r = 2 in the denominator for sine and cosine; any choice that gives sin or cos divided by 1 (or by 2 incorrectly) is incorrect. For example, sin = −√55/2 and cos = 3/2 would be too large (cos > 1), so that option is invalid.
   - Sign errors (positive instead of negative for sin or tan) contradict the quadrant IV sign pattern.
   - The tangent value must equal y/x; options that do not simplify to −√55/3 are algebraically incorrect.

6) Conclusion: The exact values are sin(θ) = −√55/8, cos(θ) = 3/8, and tan(θ) = −√55/3, which matches the provided correct answer.

Consider the point on a circle with radius . Determine the exact values of , and .Choose the correct response from the options below.单项选择题

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