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 how we arrive at this value, let’s first explore what π6\frac{\pi}{6}6π represents. In radians, π6\frac{\pi}{6}6π is equivalent to 30∘30^\circ30∘. The cosine of an angle in a right triangle corresponds to the ratio of the adjacent side to the hypotenuse. For common angles, such as 30∘30^\circ30∘, 45∘45^\circ45∘, and 60∘60^\circ60∘, we often use a unit circle or special right triangles to derive their exact trigonometric values.
For a 30∘30^\circ30∘ or π6\frac{\pi}{6}6π angle, we can use a special 30-60-90 triangle, where the ratios of the sides are well-known:
- The length of the hypotenuse is 1 (since the unit circle has a radius of 1).
- The side opposite the 30∘30^\circ30∘ angle is 12\frac{1}{2}21.
- The side opposite the 60∘60^\circ60∘ angle is 32\frac{\sqrt{3}}{2}23.
In a unit circle, cos(θ)\cos(\theta)cos(θ) is the x-coordinate of the point on the circle corresponding to the angle θ\thetaθ. For the angle π6\frac{\pi}{6}6π, or 30∘30^\circ30∘, the x-coordinate is 32\frac{\sqrt{3}}{2}23. Therefore, the exact value of cos(π6)\cos\left(\frac{\pi}{6}\right)cos(6π) is:cos(π6)=32\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}cos(6π)=23
This value is derived from the geometry of the 30-60-90 triangle and the unit circle.
The cosine function is one of the basic trigonometric functions, and understanding its values for common angles like 30∘30^\circ30∘ is fundamental to solving trigonometric problems.
