This question tests understanding of special right triangles and how their ratios determine exact trigonometric values on the unit circle. A 30-60-90 triangle has side ratios of 1 : √3 : 2 (short leg : long leg : hypotenuse), and when scaled to the unit circle (hypotenuse = 1), the short leg is 1/2 and the long leg is √3/2. In a 30-60-90 triangle with hypotenuse 1, the side opposite the 30° angle (π/6 radians) has length 1/2, and the side opposite the 60° angle (π/3 radians) has length √3/2. Therefore, sin(π/6) = 1/2, cos(π/6) = √3/2, while sin(π/3) = √3/2, cos(π/3) = 1/2. Choice C is correct because for θ=π/3 (60°), the adjacent side to the angle is the shorter leg 1/2, matching cos(π/3) from the scaled 30-60-90 ratios. Choice A reverses sine and cosine, giving the y-coordinate value (sine) when the question asks for the x-coordinate value (cosine). Remember that on the unit circle, sin(π/6) and cos(π/3) are equal (both = 1/2), and sin(π/3) and cos(π/6) are equal (both = √3/2), because these are complementary angles in a 30-60-90 triangle. To remember which value goes with which angle: the smaller angle (30° or π/6) has the smaller sine value (1/2), and the larger angle (60° or π/3) has the larger sine value (√3/2).

This question tests understanding of special right triangles and how their ratios determine exact trigonometric values on the unit circle. A 30-60-90 triangle has side ratios of 1 : √3 : 2 (short leg : long leg : hypotenuse), and when scaled to the unit circle (hypotenuse = 1), the short leg is 1/2 and the long leg is √3/2. In a 30-60-90 triangle with hypotenuse 1, the side opposite the 30° angle (π/6 radians) has length 1/2, and the side opposite the 60° angle (π/3 radians) has length √3/2. Therefore, sin(π/6) = 1/2, cos(π/6) = √3/2, while sin(π/3) = √3/2, cos(π/3) = 1/2. Choice C is correct because cos(π/3) equals the adjacent side to the 60° angle divided by the hypotenuse, which is the short leg (1/2) in the 30-60-90 triangle scaled to unit circle. Choice A confuses sine and cosine values, giving sin(π/3) = √3/2 when the question asks for cosine. Remember that on the unit circle, sin(π/6) and cos(π/3) are equal (both = 1/2), and sin(π/3) and cos(π/6) are equal (both = √3/2), because these are complementary angles in a 30-60-90 triangle.

This question tests understanding of special right triangles and how their ratios determine exact trigonometric values on the unit circle. Special triangles allow us to find exact trigonometric values without a calculator: the 30-60-90 triangle gives values for angles of π/6 and π/3, while the 45-45-90 triangle gives values for angles of π/4. For angle 5π/6, we find the reference angle by subtracting from π: π - 5π/6 = π/6, so we use a 30-60-90 triangle with reference angle π/6 in the second quadrant where sine is positive and cosine is negative. Choice C is correct because sin(5π/6) = sin(π/6) = 1/2, since 5π/6 is in the second quadrant where sine values are positive and the reference angle π/6 has sine value 1/2. Choice B incorrectly applies a negative sign to the sine value, when sine is positive in the second quadrant where 5π/6 is located. To remember which value goes with which angle: the smaller angle (30° or π/6) has the smaller sine value (1/2), and the larger angle (60° or π/3) has the larger sine value (√3/2).

This question tests understanding of special right triangles and how their ratios determine exact trigonometric values on the unit circle. A 45-45-90 triangle has side ratios of 1 : 1 : √2 (leg : leg : hypotenuse), and when scaled to fit the unit circle (hypotenuse = 1), each leg has length √2/2, giving sin(π/4) = cos(π/4) = √2/2. For angle 5π/4, we are in the third quadrant where the reference angle is 5π/4 - π = π/4, so we use a 45-45-90 triangle where both cosine (x-coordinate) and sine (y-coordinate) are negative. Choice A is correct because cos(5π/4) = -cos(π/4) = -√2/2, since 5π/4 is in the third quadrant where cosine values are negative and the reference angle π/4 has cosine value √2/2. Choice B gives the positive value √2/2, failing to account for the negative sign required in the third quadrant where 5π/4 is located. For 45-45-90 triangles, the key insight is that the two legs are equal, so sin(π/4) = cos(π/4) = √2/2, and this same value appears at all 45° angles around the unit circle (with appropriate signs).

This question tests understanding of special right triangles and how their ratios determine exact trigonometric values on the unit circle. A 45-45-90 triangle has side ratios of 1 : 1 : √2 (leg : leg : hypotenuse), and when scaled to fit the unit circle (hypotenuse = 1), each leg has length √2/2, giving sin(π/4) = cos(π/4) = √2/2. For angle 3π/4, we are in the second quadrant where the reference angle is π - 3π/4 = π/4, so we use a 45-45-90 triangle where cosine is negative (x-coordinate) and sine is positive (y-coordinate). Choice B is correct because at 3π/4 in the second quadrant, cos(3π/4) = -√2/2 (negative x-coordinate) and sin(3π/4) = √2/2 (positive y-coordinate), using the 45-45-90 triangle values with appropriate signs. Choice A fails to account for the quadrant, giving positive values for both coordinates when cosine should be negative in the second quadrant. For 45-45-90 triangles, the key insight is that the two legs are equal, so sin(π/4) = cos(π/4) = √2/2, and this same value appears at all 45° angles around the unit circle (with appropriate signs).

This question tests understanding of special right triangles and how their ratios determine exact trigonometric values on the unit circle. A 30-60-90 triangle has side ratios of 1 : √3 : 2 (short leg : long leg : hypotenuse), and when scaled to the unit circle (hypotenuse = 1), the short leg is 1/2 and the long leg is √3/2. Comparing sin(π/6) and sin(π/3), we see that sin(π/6) = 1/2 (opposite the 30° angle) and sin(π/3) = √3/2 (opposite the 60° angle), and since 1/2 < √3/2, we have sin(π/6) < sin(π/3). Choice C is correct because sin(π/6) = 1/2 and sin(π/3) = √3/2, and since 1/2 ≈ 0.5 and √3/2 ≈ 0.866, we clearly have sin(π/6) < sin(π/3). Choice A incorrectly states these values are equal, confusing the fact that sin(π/6) = cos(π/3) (both equal 1/2) with a comparison of the two sine values. To remember which value goes with which angle: the smaller angle (30° or π/6) has the smaller sine value (1/2), and the larger angle (60° or π/3) has the larger sine value (√3/2).

This question tests understanding of special right triangles and how their ratios determine exact trigonometric values on the unit circle. A 45-45-90 triangle has side ratios of 1 : 1 : √2 (leg : leg : hypotenuse), and when scaled to fit the unit circle (hypotenuse = 1), each leg has length √2/2, giving sin(π/4) = cos(π/4) = √2/2. In a 45-45-90 triangle on the unit circle, both legs are equal and have length √2/2 (since the legs² + legs² = 1²), which means at angle π/4, the coordinates are (√2/2, √2/2), so both sine and cosine equal √2/2. Choice B is correct because the 45-45-90 triangle's leg ratio, scaled by dividing by √2, gives the opposite side (sin) as √2/2 for θ=π/4. Choice C confuses the 30-60-90 triangle with the 45-45-90 triangle, using √3/2 when the correct ratio yields √2/2. For 45-45-90 triangles, the key insight is that the two legs are equal, so sin(π/4) = cos(π/4) = √2/2, and this same value appears at all 45° angles around the unit circle (with appropriate signs). Always express special angle values exactly using radicals (1/2, √2/2, √3/2) rather than decimal approximations—this is both more precise and expected in mathematics.

This question tests understanding of special right triangles and how their ratios determine exact trigonometric values on the unit circle. A 30-60-90 triangle has side ratios of 1 : √3 : 2 (short leg : long leg : hypotenuse), and when scaled to the unit circle (hypotenuse = 1), the short leg is 1/2 and the long leg is √3/2. In a 30-60-90 triangle with hypotenuse 1, the side opposite the 30° angle (π/6 radians) has length 1/2, and the side opposite the 60° angle (π/3 radians) has length √3/2. Therefore, sin(π/6) = 1/2, cos(π/6) = √3/2, while sin(π/3) = √3/2, cos(π/3) = 1/2. Choice C is correct because for θ=π/6, the triangle places the x-coordinate (cos) as the longer adjacent side √3/2 and the y-coordinate (sin) as the shorter opposite side 1/2. Choice B confuses the values for π/6 and π/3, swapping sine and cosine by using sin(π/6) = √3/2 when actually sin(π/6) = 1/2. Key to special angle problems: memorize the two special triangles (1:1:√2 for 45-45-90, and 1:√3:2 for 30-60-90), then remember that sin uses the opposite side and cos uses the adjacent side when the angle is at the origin. To remember which value goes with which angle: the smaller angle (30° or π/6) has the smaller sine value (1/2), and the larger angle (60° or π/3) has the larger sine value (√3/2).

This question tests understanding of special right triangles and how their ratios determine exact trigonometric values on the unit circle. A 45-45-90 triangle has side ratios of 1 : 1 : √2 (leg : leg : hypotenuse), and when scaled to fit the unit circle (hypotenuse = 1), each leg has length √2/2, giving sin(π/4) = cos(π/4) = √2/2. In a 45-45-90 triangle on the unit circle, both legs are equal and have length √2/2 (since the legs² + legs² = 1²), which means at angle π/4, the coordinates are (√2/2, √2/2), so both sine and cosine equal √2/2. Choice A is correct because for θ=3π/4 in quadrant II, the reference 45-45-90 triangle gives absolute values √2/2, with cos negative and sin positive. Choice C confuses the 30-60-90 triangle with the 45-45-90 triangle, using √3/2 and 1/2 when the correct ratios are both √2/2. For 45-45-90 triangles, the key insight is that the two legs are equal, so sin(π/4) = cos(π/4) = √2/2, and this same value appears at all 45° angles around the unit circle (with appropriate signs). Always express special angle values exactly using radicals (1/2, √2/2, √3/2) rather than decimal approximations—this is both more precise and expected in mathematics.

This question tests understanding of special right triangles and how their ratios determine exact trigonometric values on the unit circle. A 30-60-90 triangle has side ratios of 1 : √3 : 2 (short leg : long leg : hypotenuse), and when scaled to the unit circle (hypotenuse = 1), the short leg is 1/2 and the long leg is √3/2. For angle π/6, we use the 30-60-90 triangle; tan(θ) = sin(θ)/cos(θ), with opposite side 1/2 and adjacent side √3/2, giving tan(π/6) = (1/2)/(√3/2) = 1/√3 = √3/3. Choice B is correct because the scaled ratios yield sin(π/6) = 1/2 and cos(π/6) = √3/2, so tan = (1/2)/(√3/2) = √3/3. Choice A swaps the values for 30° and 60°, using tan(π/3) = √3 when actually tan(π/6) = √3/3. Key to special angle problems: memorize the two special triangles (1:1:√2 for 45-45-90, and 1:√3:2 for 30-60-90), then remember that sin uses the opposite side and cos uses the adjacent side when the angle is at the origin. To remember which value goes with which angle: the smaller angle (30° or π/6) has the smaller sine value (1/2), and the larger angle (60° or π/3) has the larger sine value (√3/2).