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Chapter 1: Problem 12

Use a unit circle to compute the following trigonometric functions \(\sin 6 \pi\)

Short Answer

Expert verified
\(\sin 6 \pi = 0\)

Step by step solution

01

- Understand the Unit Circle

The unit circle is a circle with radius 1 centered at the origin of the coordinate system. It helps in understanding the sine and cosine values of angles.
02

- Convert the Angle

Since the given angle is in radians, we first need to determine its position on the unit circle. Angle \(6 \pi \) indicates six full cycles around the unit circle because \(2\pi \) represents one full cycle.
03

- Simplify the Angle

Simplify \(6 \pi \) by noting that \(6 \pi = 3 \(2 \pi \)\). Since \(2 \pi \) is one full circle, after three cycles, we end up in the same position as if starting from 0 radians: \(6 \pi \equiv 0 \).
04

- Determine the \(\sin\) Value at the Angle

At \(0 \) radians, the point on the unit circle is \((1, 0)\). The y-coordinate represents the value of \(\sin \) function at that angle. Therefore, \(\sin 0 = 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sine Function
The sine function is one of the primary functions in trigonometry. It relates an angle in a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse. In the context of the unit circle, the sine of an angle corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle. For example, at 0 radians, the unit circle point is (1, 0). This means the sine of 0 is 0. Remember, the sine function is periodic, with a period of \( 2 \pi \) radians, meaning it repeats every full cycle around the circle.
Radian Measure
Radians provide a natural way of measuring angles based on the unit circle. One radian is the angle formed when the arc length is equal to the radius of the circle. Since the circumference of a unit circle is \( 2 \pi \) radians, we can relate full cycles around the circle to radian measures directly. For example, an angle of \( 2 \pi \) radians represents a complete circle, while an angle of \( 6 \pi \) radians equates to three full cycles around the circle. Converting angles into radians is essential for using the unit circle effectively in trigonometry.
Trigonometric Cycle
The trigonometric cycle denotes the repetitive nature of trigonometric functions. Because the unit circle is circular, angles beyond \( 2 \pi \) radians or less than 0 radians can be normalized by subtracting or adding \( 2 \pi \) until they fall within one full cycle (0 to \( 2 \pi \) radians). This property allows us to simplify the evaluation of trigonometric functions for any angle. For example, evaluating \( \sin 6 \pi \) simplifies to evaluating \( \sin 0 \) because \( 6 \pi \) represents three full cycles, bringing us back to 0 radians on the unit circle.

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Most popular questions from this chapter

Find all solutions of the given equation. $$ \sin t=-1 $$

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