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How does the unit circle connect the right triangle definitions of sine, cosine, and tangent to coordinate...

How does the unit circle connect the right triangle definitions of sine, cosine, and tangent to coordinate points?
Why does the equation x^2 + y^2 = 1 make sense for every point on the unit circle?

Answer

  1. The unit circle connects the right triangle definitions of sine, cosine, and tangent to coordinate points by using the coordinates of a point (x, y) on the circle. The x-coordinate gives the cosine value, and the y-coordinate gives the sine value. Tangent is the ratio of sine to cosine, or y/x, as long as x ≠ 0. This allows the trigonometric functions to be expressed in terms of the coordinates of points on the unit circle.

  2. The equation x^2 + y^2 = 1 makes sense for every point on the unit circle because the unit circle is defined as having a radius of 1. According to the Pythagorean Theorem, for any point (x, y) on the circle, the sum of the squares of the x and y coordinates must equal the square of the radius, which is 1. Therefore, x^2 + y^2 = 1 for all points on the unit circle.

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