value of sin 135′ in simplest form with a rational Find the exact denominator

value of sin 135′ in simplest form with a rational Find the exact denominator:

The Correct Answer and Explanation is:

The exact value of sin 135° in simplest form with a rational denominator is:sin⁡135∘=22\sin 135^\circ = \frac{\sqrt{2}}{2}sin135∘=22​​

Explanation:

To find the exact value of sin 135°, we can use our knowledge of reference angles and the unit circle.

First, note that 135° lies in the second quadrant. Angles in the second quadrant range from 90° to 180°, and for these angles, the sine function is positive. This is important because the sign of the trigonometric function depends on the quadrant in which the angle lies.

To simplify sin 135°, we find its reference angle. The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. For 135°, the reference angle is:180∘−135∘=45∘180^\circ – 135^\circ = 45^\circ180∘−135∘=45∘

So, sin 135° has the same absolute value as sin 45°, but since it lies in the second quadrant and sine is positive there, we have:sin⁡135∘=sin⁡45∘\sin 135^\circ = \sin 45^\circsin135∘=sin45∘

Now we need to recall the exact value of sin 45°. This is a common angle in trigonometry, and its sine is:sin⁡45∘=22\sin 45^\circ = \frac{\sqrt{2}}{2}sin45∘=22​​

Therefore:sin⁡135∘=22\sin 135^\circ = \frac{\sqrt{2}}{2}sin135∘=22​​

This value is already in its simplest form and has a rationalized denominator. A rational denominator means that the denominator is a rational number. In this case, 2 is a rational number, and the irrational part (√2) is in the numerator. Thus, the answer is both exact and properly simplified.

This value is used often in trigonometry, geometry, and physics. It is one of the standard values students are encouraged to memorize because it helps solve a wide variety of problems involving angles, especially those related to 45°, 135°, and their equivalents in radians.