The Pythagorean identity states that for any angle x x x:
sin2x+cos2x=1 \sin^2 x + \cos^2 x = 1 sin2x+cos2x=1
Given that sinx=2425 \sin x = \frac{24}{25} sinx=2524, we can substitute this value into the identity.
Substitute sinx=2425 \sin x = \frac{24}{25} sinx=2524 into the Pythagorean identity:
(2425)2+cos2x=1 \left(\frac{24}{25}\right)^2 + \cos^2 x = 1 (2524)2+cos2x=1
Calculate (2425)2\left(\frac{24}{25}\right)^2(2524)2:
(2425)2=576625 \left(\frac{24}{25}\right)^2 = \frac{576}{625} (2524)2=625576
Substitute back into the equation:
576625+cos2x=1 \frac{576}{625} + \cos^2 x = 1 625576+cos2x=1
Subtract 576625\frac{576}{625}625576 from both sides:
cos2x=1−576625=625625−576625=49625 \cos^2 x = 1 - \frac{576}{625} = \frac{625}{625} - \frac{576}{625} = \frac{49}{625} cos2x=1−625576=625625−625576=62549
Take the square root of both sides to find cosx\cos xcosx:
cosx=±49625=±725 \cos x = \pm \sqrt{\frac{49}{625}} = \pm \frac{7}{25} cosx=±62549=±257
Since the problem does not specify the quadrant, both positive and negative values are possible. Therefore, the solutions are:
cosx=725 or cosx=−725 \boxed{\cos x = \frac{7}{25} \text{ or } \cos x = -\frac{7}{25}} cosx=257 or cosx=−257
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