What is [tex]$\cos 60^{\circ} ?$[/tex]
A. [tex]$\frac{\sqrt{3}}{2}$[/tex]
B. [tex]$\frac{1}{\sqrt{2}}$[/tex]
C. 1
D. [tex]$\frac{1}{2}$[/tex]
E. [tex]$\sqrt{3}$[/tex]
F. [tex]$\frac{1}{\sqrt{3}}$[/tex]
Answer (1)
Recall the definition of cosine in a right triangle as the ratio of the adjacent side to the hypotenuse.
Recognize the side ratios in a 30-60-90 triangle are 1 : 3 : 2 .
Determine that cos 6 0 ∘ is the ratio of the adjacent side (1) to the hypotenuse (2).
Conclude that cos 6 0 ∘ = 2 1 .
Explanation
Problem Analysis The problem asks for the value of cos 6 0 ∘ . To find this value, we can recall the unit circle or the properties of a 30-60-90 triangle.
30-60-90 Triangle Consider a 30-60-90 triangle. In such a triangle, the side lengths are in the ratio 1 : 3 : 2 , where 1 is the length of the side opposite the 30-degree angle, 3 is the length of the side opposite the 60-degree angle, and 2 is the length of the hypotenuse.
Calculate Cosine The cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse. For a 60-degree angle, the adjacent side has length 1, and the hypotenuse has length 2. Therefore, cos 6 0 ∘ = 2 1 .
Unit Circle Alternatively, using the unit circle, the coordinates of the point on the unit circle corresponding to 6 0 ∘ are ( 2 1 , 2 3 ) . The x-coordinate represents the cosine of the angle, so cos 6 0 ∘ = 2 1 .
Final Answer The value of cos 6 0 ∘ is 2 1 .
Examples
Understanding the cosine of angles is crucial in many fields, such as physics and engineering. For example, when analyzing the trajectory of a projectile, the initial velocity can be broken down into horizontal and vertical components using cosine and sine functions. If a ball is launched at an angle of 60 degrees with an initial velocity of v , the horizontal component of the velocity is v cos 6 0 ∘ = 2 1 v . This helps determine how far the ball will travel horizontally.
