the sin of theta =1/2

how do i figure out what theta is without using the calculator?

To find the value of theta without using a calculator, you can use the inverse sine function or the concept of special triangles.

Inverse Sine Function Method:
1. Start with the equation sin(theta) = 1/2.
2. Apply the inverse sine function (also known as arcsin) to both sides of the equation: arcsin(sin(theta)) = arcsin(1/2).
3. Since arcsin and sin are inverse functions, they cancel each other out: theta = arcsin(1/2).

Now, to evaluate arcsin(1/2), you need to remember the special right triangle ratios or the unit circle:

Special Triangle Method:
1. Recall that for a 30-60-90 right triangle, the side lengths are in the ratio 1 : sqrt(3) : 2, where the opposite side to the 30-degree angle is half the hypotenuse.
2. In this case, the opposite side is 1, so the hypotenuse is 2. Thus, sin(30 degrees) = 1/2.
3. Since you're looking for theta, which is the angle whose sine is 1/2, you can conclude that theta equals 30 degrees.

Therefore, without a calculator, the value of theta in sin(theta) = 1/2 is 30 degrees.