Figure 10.1 should remind you of trigonometric functions you've seen before. Which of the following functions is most directly represented by the coordinates of a point on the unit circle at an angle $\theta$ from the positive x-axis?

• A. The secant and cosecant functions, since the coordinates are $\left(\sec \right(\theta ),\csc (\theta \left)\right)$
• B. The cotangent function, since the x-coordinate is $\cot \left(\theta \right)$
• C. The tangent function, since the y-coordinate is $\tan \left(\theta \right)$
• D. The sine and cosine functions, since the coordinates are $\left(\cos \right(\theta ),\sin (\theta \left)\right)$

Answer: (D)

Verified step by step guidance

1. Recall the definition of the unit circle: it is a circle with radius 1 centered at the origin of the coordinate plane.
2. Understand that any point on the unit circle corresponding to an angle \(\theta\) from the positive x-axis has coordinates given by \((x, y)\).
3. Recognize that by definition, the x-coordinate of this point is \(\cos(\theta)\) and the y-coordinate is \(\sin(\theta)\).
4. Note that other trigonometric functions like tangent, cotangent, secant, and cosecant are related to sine and cosine but are not directly the coordinates of the point on the unit circle.
5. Therefore, the coordinates of the point on the unit circle at angle \(\theta\) are most directly represented by the sine and cosine functions as \((\cos(\theta), \sin(\theta))\).