Answer :

To find sin(2x) when cot(x) = 13/15, we use the Pythagorean theorem to find all sides of the triangle, figure out sin(x) and cos(x), and then apply the double-angle formula for sine to get sin(2x) = 390/361.

To find sin(2x) when given cot(x) = 13/15 and knowing that we are in the first quadrant where all trigonometric functions are positive, we can use trigonometric identities and the Pythagorean theorem.

Firstly, we know that cot(x) is the reciprocal of tan(x), so tan(x) = 15/13. Using the definition of the tangent function, which is tan(x) = opposite/adjacent, we set the opposite side to 15 and the adjacent side to 13 in a right triangle.

By the Pythagorean theorem, we can find the hypotenuse.
hypotenuse =


[tex]sqrt(13^2 + 15^2) = sqrt(169 + 225) = sqrt(394) = 19 (since 19^2 = 361).[/tex]

Now we have all sides of the triangle: opposite = 15, adjacent = 13, and hypotenuse = 19. We can find sin(x) and cos(x) using the definitions of sine and cosine:

  • [tex]sin(x) = opposite/hypotenuse = 15/19cos(x) = adjacent/hypotenuse = 13/19[/tex]

Using the double-angle formula for sine, sin(2x) = 2sin(x)cos(x), we substitute the values found:

[tex]sin(2x) = 2 * (15/19) * (13/19) = 30 * 13 / 361 = 390/361.[/tex]

Therefore, sin(2x) is 390/361.

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Rewritten by : Barada