cosine 30 degrees exact value
Understand the Problem
The question is asking for the exact value of the cosine of 30 degrees. We can solve this by recalling the trigonometric values for common angles.
Answer
$\frac{\sqrt{3}}{2}$
Answer for screen readers
The exact value of the cosine of 30 degrees is $\frac{\sqrt{3}}{2}$.
Steps to Solve
-
Recall the known trigonometric values The cosine of 30 degrees is a commonly known trigonometric value. It can also be related to a special triangle.
-
Use the 30-60-90 triangle The special triangle corresponding to 30 degrees is the 30-60-90 triangle, where:
- The length of the side opposite 30 degrees is 1.
- The length of the hypotenuse is 2.
- The length of the side opposite 60 degrees is $\sqrt{3}$.
From these side lengths, we can find the cosine of 30 degrees.
- Calculate the cosine of 30 degrees The cosine function is defined as the adjacent side over the hypotenuse. For the 30-60-90 triangle, the adjacent side to the 30-degree angle is $\sqrt{3}$ and the hypotenuse is 2. Thus, we calculate:
$$ \cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2} $$
The exact value of the cosine of 30 degrees is $\frac{\sqrt{3}}{2}$.
More Information
The cosine of 30 degrees is a fundamental value in trigonometry, often used in various applications including physics and engineering. Its value, $\frac{\sqrt{3}}{2}$, also appears in calculations involving the unit circle and right triangles.
Tips
- Confusing cosine with sine: Remember, cosine corresponds to the adjacent side, while sine corresponds to the opposite side.
- Incorrectly recalling the triangle sides: Be sure to memorize the side lengths of special triangles for accurate calculations.
