Understanding the Trigonometric Function: Deriving the Value of Sin 45° Using Pythagorean Theorem

sin 45°

√2/2

sin 45° is a trigonometric function that represents the ratio of the length of the side opposite to the 45° angle to the length of the hypotenuse of a right triangle. The value of sin 45° is √2/2 or approximately 0.7071. This value can be derived by using the Pythagorean theorem, which states that for any right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. In a 45-45-90 right triangle, the two legs are equal in length, so the Pythagorean theorem can be simplified to a^2 + a^2 = c^2 or 2a^2 = c^2, where a is the length of one of the legs and c is the length of the hypotenuse. Solving for the length of the hypotenuse gives c = a√2. Since sin 45° is the ratio of the length of the opposite side to the hypotenuse, sin 45° = a√2/c = √2/2.

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