Precalculus: Trigonometric Functions Quiz

Explanation
The sine of pi/4 is equal to 1/root2. This can be determined by using the unit circle, where pi/4 corresponds to a 45-degree angle. At this angle, the y-coordinate on the unit circle is equal to 1/root2, which represents the sine value. Therefore, the correct answer is 1/root2.

Explanation
The cosine of 120 degrees is equal to -1/2. In a unit circle, at an angle of 120 degrees, the x-coordinate is -1/2. Therefore, the correct answer is -1/2.

Explanation
The question asks for the value of the tangent of 60 degrees. In trigonometry, the tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. For a 60-degree angle in a right triangle, the opposite side length is equal to the length of the side opposite the 30-degree angle, which is equal to the length of the adjacent side. Therefore, the tangent of 60 degrees is equal to 1, which is equivalent to root3 divided by root3. Hence, the correct answer is root3.

Explanation
The question asks for the value of cos(5pi/6). The correct answer is -root3/2. This can be determined by using the unit circle or the special triangles in trigonometry. In the unit circle, 5pi/6 corresponds to an angle of 150 degrees in the second quadrant. The x-coordinate of the point on the unit circle corresponding to this angle is -root3/2, which is the value of cos(5pi/6).

Explanation
The tangent function of 225 degrees is equal to 1. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. In this case, the angle 225 degrees falls in the third quadrant of the unit circle, where the tangent function is positive. Therefore, the correct answer is 1.

Explanation
The sine function of an angle measures the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. In this case, the angle is 5pi/3, which corresponds to a reference angle of pi/3 in the fourth quadrant. In the fourth quadrant, the sine function is negative. The value of sin(pi/3) is root3/2, so the value of sin(5pi/3) would be the negative of that, which is -root3/2.

Explanation
The correct answer is 1/root2 because when we evaluate cos(7pi/4), we can convert it to a reference angle by subtracting 2pi from it, which gives us -pi/4. Cosine of -pi/4 is equal to 1/root2.

Explanation
The sine function relates the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle. In this case, the angle is 210 degrees. Since the sine function is negative in the third quadrant, the value of sin(210 deg) is negative. The exact value can be determined by finding the reference angle, which is 30 degrees in this case. The sine of 30 degrees is 1/2, so the sine of 210 degrees is -1/2. Therefore, the correct answer is -1/2.

Explanation
The tangent function is undefined at certain angles, including 3π/2. This is because the tangent function is defined as the ratio of the sine and cosine functions, and at 3π/2, the cosine function is equal to zero. Since division by zero is undefined, the value of tan(3π/2) is undefined.

Explanation
The correct answer is -1/root2. When we evaluate cos(135 deg), we can use the unit circle to find the corresponding angle. In the unit circle, the angle 135 degrees is in the third quadrant. The cosine function is negative in the third quadrant, so the value of cos(135 deg) is negative. Additionally, the cosine of 45 degrees is 1/root2, so the cosine of 135 degrees is the negative of that, which is -1/root2.