What is the exact value of cos 30° ? Enter your answer, as a simplified fraction, in the box.
The Correct Answer and Explanation is:
Correct Answer: cos30∘=32\cos 30^\circ = \frac{\sqrt{3}}{2}cos30∘=23
Explanation:
To find the exact value of cos30∘\cos 30^\circcos30∘, it’s helpful to recall the special angles in trigonometry and their corresponding values on the unit circle or within special right triangles.
One common way to find this value is by using the 30°-60°-90° special right triangle. This type of triangle has angles of 30°, 60°, and 90°, and side lengths in a consistent ratio:
- The side opposite the 30° angle is 1.
- The side opposite the 60° angle is 3\sqrt{3}3.
- The hypotenuse is 2.
This triangle can be derived by cutting an equilateral triangle (where all sides are equal and all angles are 60°) in half. When cut down the middle, two 30°-60°-90° triangles are formed.
In trigonometry, cosine of an angle in a right triangle is defined as the adjacent side divided by the hypotenuse. Applying this definition to the 30° angle in the triangle:
- The side adjacent to the 30° angle is 3\sqrt{3}3.
- The hypotenuse is 2.
Thus, cos30∘=adjacenthypotenuse=32\cos 30^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}cos30∘=hypotenuseadjacent=23
This value is exact and commonly used in both pure and applied mathematics. It appears often in geometry, physics, engineering, and trigonometric identities.
It’s also confirmed using the unit circle, where the coordinates of a point corresponding to an angle represent (cosθ,sinθ)(\cos \theta, \sin \theta)(cosθ,sinθ). For 30∘30^\circ30∘ or π6\frac{\pi}{6}6π radians, the coordinates are (32,12)\left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right)(23,21), so: cos30∘=32\cos 30^\circ = \frac{\sqrt{3}}{2}cos30∘=23
