What is cos 60 degrees in fraction form?
The Correct Answer and Explanation is:
The correct answer is 1/2
The value of cos(60∘)\cos(60^\circ)cos(60∘) in fraction form is 12\frac{1}{2}21.
Explanation:
The cosine function, cos(θ)\cos(\theta)cos(θ), is a trigonometric function that is defined on the unit circle. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. The cosine of an angle θ\thetaθ is the x-coordinate of the point where the terminal side of the angle intersects the unit circle.
To understand why cos(60∘)=12\cos(60^\circ) = \frac{1}{2}cos(60∘)=21, we need to recall the basic properties of the unit circle and specific angles. The angle 60∘60^\circ60∘ (or π3\frac{\pi}{3}3π radians) is one of the special angles in trigonometry, and we can calculate the cosine of 60∘60^\circ60∘ using the coordinates of the point on the unit circle that corresponds to this angle.
Unit Circle Reference:
At 60∘60^\circ60∘, the corresponding point on the unit circle has coordinates (cos(60∘),sin(60∘))\left( \cos(60^\circ), \sin(60^\circ) \right)(cos(60∘),sin(60∘)). This point is located on the first quadrant of the unit circle, where both x and y values are positive.
From the known values of trigonometric functions for standard angles, we have:
- cos(60∘)=12\cos(60^\circ) = \frac{1}{2}cos(60∘)=21
- sin(60∘)=32\sin(60^\circ) = \frac{\sqrt{3}}{2}sin(60∘)=23
Thus, the x-coordinate of the point on the unit circle at 60∘60^\circ60∘ is 12\frac{1}{2}21, which is the value of cos(60∘)\cos(60^\circ)cos(60∘).
Why This Is True:
In a 30-60-90 triangle, which is a right triangle with angles of 30∘30^\circ30∘, 60∘60^\circ60∘, and 90∘90^\circ90∘, the ratio of the lengths of the sides opposite these angles is well-known:
- The side opposite the 30∘30^\circ30∘ angle has a length of 12\frac{1}{2}21.
- The side opposite the 60∘60^\circ60∘ angle has a length of 32\frac{\sqrt{3}}{2}23.
- The hypotenuse of the triangle has a length of 1 (since we are working on the unit circle).
Using this triangle, we can confirm that cos(60∘)\cos(60^\circ)cos(60∘), which corresponds to the ratio of the adjacent side to the hypotenuse, is 12\frac{1}{2}21.
Thus, cos(60∘)=12\cos(60^\circ) = \frac{1}{2}cos(60∘)=21.