Understanding Sin(60°) Using the Unit Circle and Trigonometric Ratios | Step-by-Step Explanation and Calculations

sin(60°)

To find the value of sin(60°), we can use the unit circle or trigonometric ratios

To find the value of sin(60°), we can use the unit circle or trigonometric ratios.

Using the unit circle:
In the unit circle, we represent angles by measuring the rotation from the positive x-axis in a counter-clockwise direction. At 60°, we have an angle in the first quadrant. We can draw a right triangle within the unit circle to find the values of sin(60°), cos(60°), and tan(60°).

To construct this triangle, we draw a line from the origin (center of the circle) to the point on the unit circle corresponding to 60°. This line represents the hypotenuse of the triangle. The x-coordinate of the point is equal to cos(60°), and the y-coordinate is equal to sin(60°). The remaining side of the triangle is the opposite side with respect to the angle of 60°.

For a 30°-60°-90° triangle, the sides have a ratio of 1:√3:2. Therefore, we know that the hypotenuse is 2 (as it corresponds to the radius of the unit circle), the opposite side (sin(60°)) is √3, and the adjacent side (cos(60°)) is 1.

So, sin(60°) = √3/2.

Using trigonometric ratios:
We can also use the trigonometric ratio of a right triangle to find sin(60°).

For a right triangle, sin(x) is defined as the ratio of the length of the side opposite to the angle x to the length of the hypotenuse. In this case, the opposite side (sin(60°)) is √3 (as we found earlier using the unit circle), and the hypotenuse is 2 (radius of the unit circle).

So, sin(60°) = (√3)/2.

Therefore, the value of sin(60°) is (√3)/2 or approximately 0.866.

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