Question

Circle O below has radius 1. Eight segment lengths are labeled with lowercase letters. Six of these equal a trigonometric function of theta. Your answer to this problem should be a six letter sequence whose letters represent the segment lengths that equal the following functions (in the correct order):

sin(theta),
cos(theta),
tan(theta),
csc(theta),
sec(theta),
cot(theta).
So, for example, you would answer a,k,h,c,b,d if you thought
sin(theta) = a,
cos(theta) = k,
tan(theta) = h,
csc(theta) = c,
sec(theta) = b,
cot(theta) = d.
I was able to come up with:
sin(theta) = d,
cos(theta) = a,
tan(theta) = h,
csc(theta) = f,
sec(theta) = g,
cot(theta) = h.

Answer 1

In a unit circle, the trigonometric functions correspond to specific segment lengths with sin(theta) equaling the y-coordinate, cos(theta) the x-coordinate, and so on. The correct sequence of segment lengths for sin(theta), cos(theta), tan(theta), csc(theta), sec(theta), and cot(theta) are d, a, e, b, c, f respectively.

Step-by-step explanation:

In a circle with a radius of 1, we can derive the segment lengths that correspond to the six trigonometric functions of a given angle θ (theta). These functions are defined as follows for a right triangle within the circle:

• sin(θ) is the ratio of the opposite side to the hypotenuse.
• cos(θ) is the ratio of the adjacent side to the hypotenuse.
• tan(θ) is the ratio of the opposite side to the adjacent side.
• csc(θ) (cosecant) is the reciprocal of sin(θ).
• sec(θ) (secant) is the reciprocal of cos(θ).
• cot(θ) (cotangent) is the reciprocal of tan(θ).

Given these definitions, the correct sequence of segment lengths representing these trigonometric functions, in a circle of radius 1, would be d, a, e, b, c, and f, respectively.

Since the radius is 1, sin(θ) will be equal to the length of the segment from the center of the circle to the point on the circumference opposite of the angle, which is generally depicted as the y-coordinate in the unit circle. Therefore, sin(θ) would correspond to d. Similarly, cos(θ) corresponds to the x-coordinate, or a, in a unit circle. For tan(θ), it is the ratio of the lengths of the opposite side to the adjacent side, here represented as e. The cosecant, csc(θ), is the reciprocal of d, so it's represented by the segment b. For the secant, sec(θ), taking the reciprocal of a gives us c. Finally, cot(θ) would be the reciprocal of e, represented as f.