How Do You Find Cos 60 Degrees?

1. Understanding the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. Each point on the unit circle corresponds to an angle, and the coordinates of that point represent the cosine and sine of the angle.

• The x-coordinate of a point on the unit circle represents cos⁡θ\cos \theta.
• The y-coordinate of the same point represents sin⁡θ\sin \theta.

2. 60∘60^\circ on the Unit Circle

Now, let’s focus on the specific angle 60∘60^\circ, which is a common angle in trigonometry. The angle 60∘60^\circ is in the first quadrant of the unit circle.

If you draw a line from the center of the circle to the point on the unit circle at an angle of 60∘60^\circ, this forms a right triangle. The hypotenuse of this triangle is the radius of the unit circle, which is 1.

3. Using the 30-60-90 Triangle

The triangle formed by the radius and the line making a 60∘60^\circ angle is a special type of right triangle known as a 30-60-90 triangle. The key property of a 30-60-90 triangle is that the sides have specific ratios:

• The side opposite the 30∘30^\circ angle has length 12\frac{1}{2} (since the hypotenuse is 1).
• The side opposite the 60∘60^\circ angle has length 32\frac{\sqrt{3}}{2}.

Since the radius of the unit circle is 1, the length of the side adjacent to the 60∘60^\circ angle (which corresponds to the cosine of the angle) is 12\frac{1}{2}.

4. Cosine Definition

The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. For the angle 60∘60^\circ, the adjacent side has length 12\frac{1}{2}, and the hypotenuse has length 1. Therefore:

    cos60∘=HypotenuseAdjacent Side​=21​

5. Final Answer

So, based on this reasoning and the unit circle properties, we can conclude that:
   cos60∘=21​

This value is widely recognized in trigonometry and comes from both geometric reasoning (using the 30-60-90 triangle) and the unit circle.