cos(π/4) in fraction
Understand the Problem
The question is asking for the value of cos(π/4) expressed in fraction form. This involves recalling the cosine value of a common angle in trigonometry.
Answer
The value of $\cos\left(\frac{\pi}{4}\right)$ is $\frac{\sqrt{2}}{2}$.
Answer for screen readers
The value of $\cos\left(\frac{\pi}{4}\right)$ expressed in fraction form is $\frac{\sqrt{2}}{2}$.
Steps to Solve
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Recall the cosine value for the angle
We need to recall the value of $\cos\left(\frac{\pi}{4}\right)$. This is a common angle in trigonometry. -
Recognize the special triangle
For the angle $\frac{\pi}{4}$, we can use the special 45-45-90 triangle. In this triangle, both legs are of equal length, and the length of each leg can be considered as 1. The hypotenuse can be calculated as: $$ \text{Hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2} $$ -
Calculate cosine
The cosine of an angle in a right triangle is given by the ratio of the adjacent side to the hypotenuse. For $\frac{\pi}{4}$: $$ \cos\left(\frac{\pi}{4}\right) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}} $$ -
Rationalize the denominator
To express this in a more standard form, we multiply the numerator and the denominator by $\sqrt{2}$: $$ \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} $$
The value of $\cos\left(\frac{\pi}{4}\right)$ expressed in fraction form is $\frac{\sqrt{2}}{2}$.
More Information
The cosine value of $\frac{\pi}{4}$ is commonly used in trigonometry, particularly in problems involving 45-degree angles. This angle is significant because it has equal leg lengths in the special triangle, making calculations straightforward.
Tips
- Forgetting to rationalize the denominator when expressing $\frac{1}{\sqrt{2}}$.
- Mixing up the values of sine and cosine for the same angle.
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