## sin 45°

### √2/ 2

sin 45° is an exact value that can be calculated without using a calculator.

To find sin 45°, we start by drawing a right triangle with one angle measuring 45°. We know that in a right triangle, the sine of an angle is defined as the length of the opposite side divided by the length of the hypotenuse.

Next, we label the sides of the triangle. Let’s call the hypotenuse h and the legs a and b. Since the triangle has a 45° angle, we know that the two legs are congruent and the hypotenuse is √2 times longer than each leg.

Using the Pythagorean Theorem, we can find the length of the legs as follows:

a^2 + b^2 = h^2

a^2 + a^2 = h^2 (since the legs are congruent)

2a^2 = h^2

a = h/√2

We can substitute this value of a into the sine formula:

sin 45° = opposite/hypotenuse

sin 45° = a/h

sin 45° = (h/√2) / h

sin 45° = √2/2

Therefore, the exact value of sin 45° is √2/2.

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