solve---**Question 12** **Question Stem:** By considering a unit circle, state whether the following are true or false. **Sub-parts / Statements:** a. sin 10° < cos 10° b. sin 50° < tan 50° c. cos 80° > sin 80° d. cos 90° = sin 0° e. tan 180° = sin 180° f. cos 170° > sin 170° g. sin 120° > tan 120° h. sin 90° = cos 180° i. tan 230° < cos 230° j. cos 350° < sin 85° k. sin 260° < cos 110° l. tan 270° = cos 180°

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The unit circle is our fundamental tool for evaluating trigonometric functions. Any point on the unit circle with radius 1 has coordinates cosine theta and sine theta, where theta is the angle from the positive x-axis. The sine function gives us the y-coordinate, cosine gives the x-coordinate, and tangent equals sine divided by cosine, which represents the slope of the radius line. We divide the circle into four quadrants to analyze different angle ranges. Let's analyze the first quadrant statements using our unit circle. For statement a, sin 10 degrees versus cos 10 degrees: at 10 degrees, the y-coordinate is smaller than the x-coordinate, so sin 10 is less than cos 10. This is true. For statement b, sin 50 degrees versus tan 50 degrees: since tangent equals sin divided by cos, and cos 50 is less than 1, tan 50 is greater than sin 50. This is true. For statement d, cos 90 degrees equals sin 0 degrees: both equal zero, so this is true. Now let's examine the second quadrant where cosine becomes negative while sine remains positive. For statement c, cos 80 degrees versus sin 80 degrees: 80 degrees is very close to 90 degrees, so sin 80 is much larger than the small positive cos 80. This statement is false. For statement f, cos 170 degrees versus sin 170 degrees: cos 170 is negative while sin 170 is positive, so cos 170 cannot be greater than sin 170. This is false. For statement h, sin 90 degrees equals cos 180 degrees: sin 90 equals 1 while cos 180 equals negative 1. This is false. Let's complete our analysis with quadrants three and four. For statement e, tan 180 degrees equals sin 180 degrees: both equal zero, so this is true. For statement g, sin 120 degrees versus tan 120 degrees: sin 120 is positive while tan 120 is negative, so this is true. For statement i, tan 230 degrees versus cos 230 degrees: both are negative in quadrant three, but tan 230 has larger magnitude, so this is false. For statement j, cos 350 degrees versus sin 85 degrees: cos 350 is positive while sin 85 is close to 1, so this is true. For statement k, sin 260 degrees versus cos 110 degrees: both are negative, but sin 260 has larger magnitude, so this is true. For statement l, tan 270 degrees equals cos 180 degrees: tan 270 is undefined while cos 180 equals negative 1, so this is false. Let's summarize our complete solution. We analyzed 12 trigonometric statements using the unit circle approach. Seven statements are true: a, b, d, e, g, j, and k. Five statements are false: c, f, h, i, and l. Our systematic approach involved four key steps: first, identify which quadrant the angle belongs to; second, determine the signs of sine, cosine, and tangent in that quadrant; third, use reference angles when helpful; and fourth, compare the actual magnitudes. Remember the sign patterns: quadrant one has all positive, quadrant two has sine positive but cosine and tangent negative, quadrant three has tangent positive but sine and cosine negative, and quadrant four has cosine positive but sine and tangent negative. Always watch for undefined values like tangent at 90 and 270 degrees. The unit circle remains our most powerful tool for trigonometric analysis.