how do I find this without a calculator? sin 90 degrees?

Sketch the triangle on scrap paper.

You will see that as the angle approaches 90, the opposite approaches the hypotenuse
so
sin 90 = 1

jess fus ro dah

To find the value of sin 90 degrees without a calculator, you can rely on your knowledge of trigonometric values for common angles.

The value of sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse, in a right triangle. For a right triangle with a 90-degree angle, the hypotenuse is the longest side while the opposite side is the side which is opposite to the angle.

In the case of a 90-degree angle, the opposite side is the side of the triangle that is not the hypotenuse. Since there are no other angles in a 90-degree triangle apart from the right angle, this side has a length of 1 (assuming we are working with a unit circle).

Therefore, in a right triangle with a 90-degree angle, the value of sin 90 degrees is 1.

To summarize, if you have memorized the trigonometric values for common angles, you can recall that sin 90 degrees equals 1 without using a calculator.