Mastering Trigonometric Functions and Unit Circle Quiz

The cosine function returns the ratio of the adjacent side to the hypotenuse in a right triangle. When evaluating cos(2π), we are looking at the cosine of a full circle, which is equivalent to 1.

The tangent function has a period of π, which means tan(2π) is equivalent to tan(0), resulting in a value of 0. The incorrect answers may result from misunderstanding the properties of the tangent function and periodicity.

The sine of 0 degrees is 0. In trigonometry, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle.

When the angle is 5?/4, sin(5?/4) equals -1/?2 because the sine function evaluates to the y-coordinate of the point on the unit circle corresponding to the angle.

The sine of 0 degrees is equal to 0. In trigonometry, the sine function represents the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle. When the angle is 0 degrees, the opposite side has a length of 0, resulting in a sine value of 0.

To find the cotangent of an angle, use the formula cot(angle) = 1/tan(angle). In this case, tan(11π/6) = -√3. Hence, cot(11π/6) = 1/(-√3) = -√3.

The sine of 2θ is equal to 0 when 2θ is a multiple of π. This occurs when 2θ = 0, π, 2π, 3π, and so on, resulting in a value of 0 for sin(2θ).

The cosecant function, csc(x), is undefined when x is equal to 0 or any multiple of π; therefore, csc(2θ) is undefined.

The secant of an angle is the reciprocal of the cosine of that angle. Since the cosine of 2 radians is 0.416, the secant of 2 radians is 1.

The cotangent of an angle where the sine is 0 (such as 2π) is undefined because it results in division by 0.

To find the value of sin(?/3), we can use the fact that sin(π/3) = √3/2, sin(π/2) = 1, sin(2π/3) = √3/2, and sin(π) = 0. Given that sin(?/3) = √3/2, we can see that ? = π. Therefore, sin(π/3) = √3/2.

The correct formula for the sine of an angle in radians is opposite/hypotenuse, which in this case simplifies to 1/?2.

The value of sin(?/2) is 1 because sin(π/2) or sin(90°) is equal to 1. In trigonometry, the sine function represents the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.

In trigonometry, sin(3π/4) represents the sin value of the angle 3π/4 radians. The correct answer is 1/√2 because sin(3π/4) is equivalent to 1/√2.

To find the value of sin(5π/6), we can refer to the unit circle where 5π/6 corresponds to the point with coordinates (-1/2, √3/2). The sine value is the y-coordinate of this point, which is 1/2.

In trigonometry, Sin(4?/3) evaluates to -√3/2 as it corresponds to the sin of an angle in radians equal to 4?/3. This value can be determined by referencing the unit circle or using trigonometric identity formulas.

In the unit circle, the sine of 3π/2 is equal to -1 as it corresponds to the point on the unit circle with coordinates (0, -1). This position indicates the lowest point on the circle in the downward direction.

To find sin(5?/3), we convert 5?/3 to radians. 5?/3 * π = 5π/3. The sine of 5π/3 is -√3/2. Therefore, the correct answer is -√3/2.

In the fourth quadrant, sin is negative and cos is positive. When evaluating sin(7π/4), the reference angle is π/4 (45 degrees), so sin(π/4) = √2/2. However, since we are in the fourth quadrant, where sin is negative, the correct answer is -√2/2 or -1/√2.

The correct formula for cos(x/4) is 1/?2. Remember that the cosine function is periodic, so it repeats every 2?.

The cosine function of any multiple of π/2 (including 0) is always equal to 0 because the cosine of any multiple of π/2 is zero, as Cos(0) = 1 and Cos(π/2) = 0.

To find the cosine of 3π/4, you need to consider the unit circle. At 3π/4, the angle falls in the second quadrant where the x-coordinate is negative and the y-coordinate is positive. The cosine value in this quadrant is represented as -1/√2.

The correct answer can be determined by knowing the exact values of cosine function at common angles. In this case, the cosine of 5π/6 is equal to -√3/2 which corresponds to the cosine value at this angle in standard trigonometric functions.