Give the exact value of each of the following. cos pi/ 6
The Correct Answer and Explanation is:
The exact value of cos(π6)\cos\left(\frac{\pi}{6}\right)cos(6π) is 32\frac{\sqrt{3}}{2}23.
Explanation:
To understand why, let’s first recall that π\piπ is approximately 3.14159, and π6\frac{\pi}{6}6π corresponds to a 30° angle in radians. We can find the cosine of a 30° angle by referring to the unit circle or by using standard trigonometric values for commonly used angles (like 30°, 45°, 60°).
Unit Circle:
On the unit circle, an angle of π6\frac{\pi}{6}6π radians (or 30°) corresponds to a point on the circle. The coordinates of this point are (cos(π6),sin(π6))(\cos(\frac{\pi}{6}), \sin(\frac{\pi}{6}))(cos(6π),sin(6π)).
For a 30° angle, the cosine value (which is the x-coordinate of the point) is 32\frac{\sqrt{3}}{2}23.
30°-60°-90° Triangle:
Another way to derive this value is by considering the 30°-60°-90° right triangle, which is a special triangle with known side ratios. In this triangle:
- The side opposite the 30° angle is 111.
- The side opposite the 60° angle is 3\sqrt{3}3.
- The hypotenuse is 222.
The cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse. For a 30° angle:cos(30∘)=adjacent sidehypotenuse=32\cos(30^\circ) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}cos(30∘)=hypotenuseadjacent side=23
Thus, cos(π6)=32\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}cos(6π)=23.
Summary:
- cos(π6)=32\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}cos(6π)=23.
- This result comes from both the unit circle and the 30°-60°-90° triangle.
This value is widely used in trigonometry and is part of the set of standard trigonometric values for key angles (30°, 45°, 60°).
