Finding the Value of sin(x) at x = π/4 Using the Unit Circle and Trigonometric Identity

sinx pi/4

To find the value of sin(x) when x is equal to π/4, we can use the unit circle or the trigonometric identity

To find the value of sin(x) when x is equal to π/4, we can use the unit circle or the trigonometric identity.

Using the unit circle:

– Draw a circle centered at the origin with a radius of 1.
– Label the x-axis and y-axis as reference lines.
– The point on the unit circle where x intersects the circle is at an angle of π/4.
– The y-coordinate of this point is equal to sin(π/4).

Since the point lies on the unit circle and forms a right-angled triangle with the x-axis, we can use the Pythagorean theorem to find the y-coordinate.

In this case, the x-coordinate and y-coordinate are both 1/√2 (since π/4 is a 45-degree angle).

Using the Pythagorean theorem:

According to the Pythagorean theorem, the sum of the squares of the two sides of a right-angled triangle is equal to the square of the hypotenuse.

In this case, the hypotenuse is the radius of the unit circle, which is 1.

Let’s denote the sides of the triangle as a and b, where a is the x-coordinate and b is the y-coordinate.

a^2 + b^2 = hypotenuse^2
(1/√2)^2 + (1/√2)^2 = 1^2
1/2 + 1/2 = 1
1 = 1

Therefore, the y-coordinate (or sin(π/4)) is 1/√2.

In simplified form, sin(π/4) = √2/2.

So, sin(π/4) = √2/2 or approximately 0.707.

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