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In a right triangle, cos(2x)^(@)=sin(7x-7)^(@). Solve for x. Round your answer to the nearest hundredth if necessary.

In a right triangle, cos⁡(2x)∘=sin⁡(7x−7)∘ \cos (2 x)^{\circ}=\sin (7 x-7)^{\circ} . Solve for x x . Round your answer to the nearest hundredth if necessary.

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Full solution

Q. In a right triangle, cos⁡(2x)∘=sin⁡(7x−7)∘ \cos (2 x)^{\circ}=\sin (7 x-7)^{\circ} . Solve for x x . Round your answer to the nearest hundredth if necessary.
  1. Apply Pythagorean Identity: Since we have cos⁡(2x)2\cos(2x)^2 and sin⁡(7x−7)2\sin(7x-7)^2, we can use the Pythagorean identity sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1. This means that cos⁡2(θ)=1−sin⁡2(θ)\cos^2(\theta) = 1 - \sin^2(\theta). So, we can replace cos⁡(2x)2\cos(2x)^2 with 1−sin⁡2(2x)1 - \sin^2(2x).
  2. Set Equation to Zero: Now we have 1−sin⁡2(2x)=sin⁡2(7x−7)1 - \sin^2(2x) = \sin^2(7x-7). Let's move everything to one side to set the equation to zero: sin⁡2(2x)+sin⁡2(7x−7)−1=0\sin^2(2x) + \sin^2(7x-7) - 1 = 0.
  3. Use Sine Double Angle Identity: We can use the sine double angle identity sin⁡(2θ)=2sin⁡(θ)cos⁡(θ)\sin(2\theta) = 2\sin(\theta)\cos(\theta). But we need to express sin⁡2(2x)\sin^2(2x) in terms of sin⁡\sin and cos⁡\cos. So, we'll use the identity sin⁡2(θ)=1−cos⁡(2θ)2\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}.
  4. Simplify Equation: Applying the identity to sin⁡2(2x)\sin^2(2x), we get (1−cos⁡(4x))/2(1 - \cos(4x))/2. Now our equation looks like (1−cos⁡(4x))/2+sin⁡2(7x−7)−1=0(1 - \cos(4x))/2 + \sin^2(7x-7) - 1 = 0.
  5. Combine Like Terms: Simplify the equation by multiplying everything by 22 to get rid of the fraction: 1−cos⁡(4x)+2sin⁡2(7x−7)−2=01 - \cos(4x) + 2\sin^2(7x-7) - 2 = 0.
  6. Use Sine Double Angle Identity Again: Combine like terms: −cos⁡(4x)+2sin⁡2(7x−7)−1=0-\cos(4x) + 2\sin^2(7x-7) - 1 = 0.
  7. Substitute into Equation: We can use the identity sin⁡2(θ)=1−cos⁡(2θ)2\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} again for sin⁡2(7x−7)\sin^2(7x-7). This gives us 1−cos⁡(14x−14)2\frac{1 - \cos(14x-14)}{2}.
  8. Simplify Equation: Substitute this into our equation: −cos⁡(4x)+2(1−cos⁡(14x−14)2)−1=0-\cos(4x) + 2\left(\frac{1 - \cos(14x-14)}{2}\right) - 1 = 0.
  9. Combine Like Terms: Simplify the equation: −cos⁡(4x)+(1−cos⁡(14x−14))−1=0-\cos(4x) + (1 - \cos(14x-14)) - 1 = 0.
  10. Combine Like Terms: Simplify the equation: −cos⁡(4x)+(1−cos⁡(14x−14))−1=0-\cos(4x) + (1 - \cos(14x-14)) - 1 = 0. Combine like terms again: −cos⁡(4x)−cos⁡(14x−14)=0-\cos(4x) - \cos(14x-14) = 0.

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