Mastering Unit Circle Values: A Trigonometry Quiz

The sine function of ?/6 evaluates to 1/2 for the angle of 30 degrees, which is a commonly known value in trigonometry.

The cosine value of pi/4 is equivalent to sqrt(2)/2 as it represents the cosine of 45 degrees which is a common trigonometric value.

When finding the sine value of an angle in radians, the formula to use is sin(angle) = opposite/hypotenuse in a right triangle. For an angle of ?/4, the opposite side is ?2 and the hypotenuse is 2, resulting in a sine value of ?2/2.

The correct value of cos π/3 is 1/2, which is equivalent to cos 60 degrees. It represents the ratio of the adjacent side to the hypotenuse in a right triangle with a 60-degree angle.

The correct answer for sin ?/3 is actually ?3/2, as it represents the exact value of sine of the specific angle.

The sine function of any multiple of π/2 is either 1, 0, or -1, depending on the specific angle. In this case, sin(π/2) is equal to 1.

To evaluate cos(3π/4), we need to determine the cosine value at the angle 3π/4 in the unit circle. The cosine value at 3π/4 is -√2/2, which corresponds to the point (-√2/2, √2/2) on the unit circle. This value can be verified through trigonometric identities and understanding of the trigonometric unit circle.

The sine of 3pi/4 is equal to sqrt(2)/2, which can be found using the unit circle or trigonometric identities.

To find the sin of 150 degrees, we convert it to radians (150 * π / 180) which equals 5π/6. The sine of 5π/6 is 1/2.

To find the cosine of 7?/6, we use the reference angle of π/6 and determine that the cosine value is -√3/2 in the third quadrant.

In the unit circle, at 7?/6, the sine value corresponds to the y-coordinate of the point on the circle. Since the point is in the third quadrant where the y-coordinate is negative, the correct answer is -1/2.

To find the value of sin(5π/4), we need to reference the unit circle. At 5π/4, the point lies in the third quadrant where sin is negative and equal to -√2/2.

The cosine function is periodic with a period of 2π, so cos(3π/2) can be rewritten as cos(π/2) which equals 0.

In the unit circle, 3π/2 corresponds to the point (0, -1), which is the sine value of -1.

To find the sine of 5?/3, we reference the unit circle. At 5?/3, the reference angle is ?/3 in the fourth quadrant, where sine is negative and cosine is positive. Therefore, the correct answer is -√3/2.

The sine function calculates the y-coordinate of a point on the unit circle. In the case of 7π/4 radians, the point falls in the third quadrant where the y-coordinate is negative and equal to -√2/2.

The correct answer can be determined using the special right triangle with angles 30°, 60°, and 90°. By looking at the cosine ratio for 30° in this triangle, which is √3/2, we can determine that the cosine of π/6 is √3/2.

The value of cos(π/2) is 0 because at π/2 radians, the cosine function reaches its minimum value of 0. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.

The sine of 0 degrees is 0 because it represents the angle when the terminal side coincides with the x-axis in the unit circle, resulting in a sine value of 0.

To find the cosine of 5?/4, we need to determine the cosine of the angle in radians. The correct answer is -√2/2, which can be derived using the unit circle or trigonometric identities.

The cosine of 4π/3 is equal to -1/2. This can be determined by recognizing the reference angle of π/3 in Quadrant III, where cosine is negative and the value is -1/2.

When evaluating sin(4π/3), we are looking at the sin value at the angle of 4π/3, which is in the third quadrant. In the third quadrant, sin values are negative and the exact value at 4π/3 is -√3/2.

To find the cosine of an angle, we look at the x-coordinate of the point on the unit circle corresponding to that angle. In this case, 5π/3 is in the fourth quadrant, where the x-coordinate is positive and the y-coordinate is negative. Hence, the cosine is positive, and the correct answer is 1/2.

The correct answer is -1/2 because in the unit circle, at an angle of 2?/3 in the second quadrant, the cosine value is -1/2.

The sine of 2π/3 is equal to √3/2. This can be derived from the unit circle where at 2π/3 angle, the y-coordinate is equal to √3/2.

The correct answer is -√3/2 because at 5π/6, cosine function is negative and equals -√3/2.

The sine of 11π/6 is equivalent to the sine of π/6 (since 11π/6 = π/6 + 2π), which is -1/2. The incorrect answers are not the correct value for the sine of 11π/6.

The cosine function returns -1 when the input angle is 180 degrees, which is a common trigonometric property.

The cosine function is negative in the second and third quadrants. For 11π/6, the reference angle is π/6 which corresponds to a 30-degree angle. Since cos(30 degrees) = √3/2, the cosine of 11π/6 is the same but negative, resulting in -√3/2.

When evaluating cosine at 7π/4, it corresponds to the third quadrant where cosine is negative and the reference angle is π/4. Therefore, the correct answer is -√2/2.