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Trigonometric Identities and Equations Using a Pythagorean identity to find solutions in an interval for a trigonometri... Find all solutions of the equation in the interval [0,2pi). 4cos x=-sin^(2)x+4 Write your answer in radians in terms of pi. If there is more than one solution, separate them with commas.

Trigonometric Identities and Equations\newlineUsing a Pythagorean identity to find solutions in an interval for a trigonometri...\newlineFind all solutions of the equation in the interval [0,2π) [0,2 \pi) .\newline4cos⁡x=−sin⁡2x+4 4 \cos x=-\sin ^{2} x+4 \newlineWrite your answer in radians in terms of π \pi .\newlineIf there is more than one solution, separate them with commas.

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Q. Trigonometric Identities and Equations\newlineUsing a Pythagorean identity to find solutions in an interval for a trigonometri...\newlineFind all solutions of the equation in the interval [0,2π) [0,2 \pi) .\newline4cos⁡x=−sin⁡2x+4 4 \cos x=-\sin ^{2} x+4 \newlineWrite your answer in radians in terms of π \pi .\newlineIf there is more than one solution, separate them with commas.
  1. Rewrite using Pythagorean identity: Rewrite the equation using a Pythagorean identity.\newlineWe know that sin⁡2(x)+cos⁡2(x)=1\sin^2(x) + \cos^2(x) = 1, so we can rewrite sin⁡2(x)\sin^2(x) as 1−cos⁡2(x)1 - \cos^2(x).\newlineThe equation becomes 4cos⁡(x)=−(1−cos⁡2(x))+44\cos(x) = -(1 - \cos^2(x)) + 4.
  2. Simplify the equation: Simplify the equation.\newlineDistribute the negative sign on the right side of the equation to get 4cos⁡(x)=−1+cos⁡2(x)+44\cos(x) = -1 + \cos^2(x) + 4.\newlineCombine like terms to get 4cos⁡(x)=cos⁡2(x)+34\cos(x) = \cos^2(x) + 3.
  3. Rearrange for quadratic equation: Rearrange the equation to form a quadratic equation in terms of cos⁡(x)\cos(x). Subtract 4cos⁡(x)4\cos(x) from both sides to get 0=cos⁡2(x)−4cos⁡(x)+30 = \cos^2(x) - 4\cos(x) + 3.
  4. Factor the quadratic equation: Factor the quadratic equation.\newlineWe are looking for factors of 33 that add up to −4-4. The factors are −1-1 and −3-3.\newlineSo, the factored form is 0=(cos⁡(x)−1)(cos⁡(x)−3)0 = (\cos(x) - 1)(\cos(x) - 3).
  5. Solve for cos⁡(x)\cos(x): Solve for cos⁡(x)\cos(x).\newlineSet each factor equal to zero: cos⁡(x)−1=0\cos(x) - 1 = 0 and cos⁡(x)−3=0\cos(x) - 3 = 0.\newlineSolving for cos⁡(x)\cos(x) gives us cos⁡(x)=1\cos(x) = 1 and cos⁡(x)=3\cos(x) = 3.\newlineHowever, the cosine of an angle cannot be greater than 11, so cos⁡(x)=3\cos(x) = 3 has no solution.
  6. Find angles for cos⁡(x)\cos(x): Find the angles for cos⁡(x)=1\cos(x) = 1. The only angle in the interval [0,2π)[0, 2\pi) where cos⁡(x)=1\cos(x) = 1 is x=0x = 0.

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