Given that sin x =0.5 find the values of x between 90degrees and 360degrees?

x is a 30º reference angle (from the x-axis)

sine is positive in quadrants I and II

To find the values of x between 90 degrees and 360 degrees such that sin x = 0.5, you can use the inverse sine function or the unit circle.

1. Using the inverse sine function:
The inverse sine function, also known as arcsin or sin^(-1), gives the angle whose sine is a given value. In this case, we want to find x such that sin x = 0.5.

The inverse sine of 0.5 is written as sin^(-1)(0.5) or arcsin(0.5). Evaluate this expression using a calculator or by referring to a trigonometric table to find one angle.

arcsin(0.5) ≈ 30°

Since sine is periodic, you need to consider all possible angles that have the same value of sin x = 0.5.

In the first quadrant (0 to 90 degrees), sin x is positive. So, x = 30 degrees is a solution.

In the second quadrant (90 to 180 degrees), sin x is also positive, but as per the given range, we will exclude this quadrant.

In the third quadrant (180 to 270 degrees), sin x is negative. So, the angle that yields sin x = 0.5 is 180 degrees minus the angle found in the first quadrant.

180 - 30 = 150 degrees

In the fourth quadrant (270 to 360 degrees), sin x is again negative, and we find the angle by subtracting the angle found in the first quadrant from 360 degrees.

360 - 30 = 330 degrees

Therefore, the values of x between 90 degrees and 360 degrees such that sin x = 0.5 are 30 degrees, 150 degrees, and 330 degrees.

2. Using the unit circle:
Draw a unit circle, where the radius is 1 unit. The points on the unit circle correspond to the values of the sine and cosine functions.

Locate the point on the unit circle where the sine value is 0.5, at approximately the 30-degree angle.

Starting from this point, you can reflect it across the x-axis to find the corresponding angle in the third quadrant (180 degrees - 30 degrees = 150 degrees). Then, reflecting it across the x-axis once more gives the angle in the fourth quadrant (360 degrees - 30 degrees = 330 degrees).

Therefore, using either method, the values of x between 90 degrees and 360 degrees such that sin x = 0.5 are 30 degrees, 150 degrees, and 330 degrees.