sin 45 in fraction
Understand the Problem
The question asks for the sine of 45 degrees in fraction form. This involves understanding the trigonometric value of sine at that angle.
Answer
The sine of 45 degrees is $\frac{\sqrt{2}}{2}$.
Answer for screen readers
The sine of 45 degrees in fraction form is $\frac{\sqrt{2}}{2}$.
Steps to Solve
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Identify the Angle The angle given is 45 degrees. We need to find the sine value of this angle.
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Use the Special Triangle For an angle of 45 degrees, we can use the properties of a special right triangle, known as the 45-45-90 triangle. In this triangle, the two legs are of equal length.
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Calculate the Hypotenuse If we take each leg of the triangle to be of length 1, we can find the length of the hypotenuse using the Pythagorean theorem: $$ \text{Hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2} $$
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Find the Sine Value The sine of an angle in a right triangle is defined as the opposite side over the hypotenuse. For 45 degrees: $$ \sin(45^\circ) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}} $$
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Rationalize the Denominator To express the sine in a more standard fraction form, we usually rationalize the denominator: $$ \sin(45^\circ) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} $$
The sine of 45 degrees in fraction form is $\frac{\sqrt{2}}{2}$.
More Information
The value of $\frac{\sqrt{2}}{2}$ is significant in trigonometry and is often encountered in problems involving 45-degree angles. It is approximately equal to 0.707, which is useful in various mathematical and engineering applications.
Tips
- A common mistake is to forget to rationalize the denominator. Always express trigonometric values in standard fraction form with a rational denominator.
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