1.
What is the formula for a unit circle?
Correct Answer
B. X^{2}+y^{2}=1
Explanation
The formula for a unit circle is x2+y2=1. This equation represents all the points on a circle with a radius of 1, centered at the origin (0,0) in a coordinate plane. It states that the square of the x-coordinate plus the square of the y-coordinate equals 1. This equation is derived from the Pythagorean theorem and is fundamental in trigonometry and geometry.
2.
What is the radius of a unit circle?
Correct Answer
A. 1
Explanation
The radius of a unit circle is always 1. A unit circle is defined as a circle with a radius of 1 unit. It is a special circle used in mathematics to simplify calculations and understand geometric properties. The radius is the distance from the center of the circle to any point on its circumference, and in the case of a unit circle, this distance is always 1 unit.
3.
What are the properties of a unit circle?
Correct Answer
D. All of the above
Explanation
The properties of a unit circle include having its center at the origin, having a radius of one, and having a circumference of 2π. Therefore, the correct answer is "All of the above."
4.
What is the value of sin of 1 unit circle? (in rad)
Correct Answer
B. 0.8414709848
Explanation
The value of sin(1) on the unit circle is approximately 0.8414709848. This can be determined by calculating the y-coordinate of the point on the unit circle that corresponds to an angle of 1 radian. Since the unit circle has a radius of 1, the y-coordinate represents the sine of the angle.
5.
In Calculus, almost all the references to the trigonometric functions are based on the unit circle.
Correct Answer
A. True
Explanation
In Calculus, the unit circle is commonly used to define and analyze trigonometric functions. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is useful because it allows us to relate the angles formed by the circle to the values of the trigonometric functions. By using the unit circle, we can easily determine the values of sine, cosine, and other trigonometric functions for any given angle. Therefore, it is accurate to say that almost all references to trigonometric functions in Calculus are based on the unit circle.
6.
What is 2 radians on a unit circle?
Correct Answer
D. 360 degrees
Explanation
2 radians on a unit circle is equivalent to 360 degrees. A unit circle has a radius of 1, and a full rotation around the circle is equal to 2π radians or 360 degrees. Since 2 radians is the same as a full rotation, the answer is 360 degrees.
7.
What is the area of a unit circle?
Correct Answer
B. A = πr^{2}
Explanation
The area of a unit circle is given by the formula A = πr^2, where r is the radius of the circle. In this case, the radius is 1 (since it's a unit circle), so the area is A = π(1)^2 = π. Therefore, the correct answer is A = πr^2.
8.
The Unit Circle is a circle with its center at
Correct Answer
A. Origin (0,0)
Explanation
The Unit Circle is a circle with its center at the origin (0,0) because in the coordinate plane, the origin represents the point where the x-axis and y-axis intersect. Since the Unit Circle is used to represent angles and distances in trigonometry, it makes sense for it to be centered at the origin. This allows for easy calculations and comparisons of trigonometric functions for various angles.
9.
What is the value of π/3?
Correct Answer
B. 60 degrees
Explanation
The value of π/3 is equal to 60 degrees. This is because π radians is equivalent to 180 degrees, so to find the value of π/3 in degrees, we divide 180 by 3, resulting in 60 degrees.
10.
What is the value of 2π/3?
Correct Answer
C. 120 degrees
Explanation
The value of 2π/3 is equal to 120 degrees. This is because a full circle is equal to 360 degrees, and 2π radians is equal to 360 degrees. Therefore, to find the value in degrees, we can set up a proportion: 2π/3 = 360/x. Solving for x, we find that x is equal to 120 degrees.