What is the Value of sin 45°? Explained Using Trigonometric Ratios and Pythagorean Theorem

sin 45°

To find the value of sin 45°, we use the trigonometric ratios of a right triangle

To find the value of sin 45°, we use the trigonometric ratios of a right triangle.

Let’s consider a right triangle ABC where angle BAC is 45°.

Since it is a right triangle, one of the angles is 90°, making it a special right triangle known as an isosceles right triangle. In an isosceles right triangle, the two acute angles are 45° each.

In this triangle, let’s assume that the length of the two equal sides is ‘a’ units.

Now, using the Pythagorean theorem, we can find the length of the other side. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

So, applying the Pythagorean theorem, we have:

a^2 + a^2 = c^2

2a^2 = c^2

Taking the square root on both sides, we get:

√(2a^2) = √(c^2)

√2a = c

Since √2 is an irrational number, it is often approximated as 1.41.

Therefore, c ≈ 1.41a

Now, let’s consider the ratio sinθ = opposite/hypotenuse.

In our triangle, the opposite side to angle BAC is equal to ‘a’ and the hypotenuse is equal to ‘c’.

So, sin 45° = a/c

Replacing c with its approximate value, we have:

sin 45° ≈ a/(1.41a)

The ‘a’ units cancel out, leaving us with:

sin 45° ≈ 1/1.41

Simplifying further, we get:

sin 45° ≈ 0.707

Therefore, sin 45° is approximately equal to 0.707.

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