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Find all angles, 0^(@) <= theta < 360^(@), that satisfy the equation below, to the nearest tenth of a degree. cos^(2)theta-1=0 Answer: theta=

Find all angles, 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.\newlinecos⁡2θ−1=0 \cos ^{2} \theta-1=0 \newlineAnswer: θ= \theta=

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Csc, sec, and cot of special anglesright chevron icon

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Q. Find all angles, 0∘≤θ<360∘ 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.\newlinecos⁡2θ−1=0 \cos ^{2} \theta-1=0 \newlineAnswer: θ= \theta=
  1. Recognize Pythagorean Identity: To solve the equation cos⁡2(θ)−1=0\cos^2(\theta) - 1 = 0, we first need to recognize that this is a Pythagorean identity. The identity is cos⁡2(θ)=1−sin⁡2(θ)\cos^2(\theta) = 1 - \sin^2(\theta), but since we have cos⁡2(θ)−1=0\cos^2(\theta) - 1 = 0, we can rewrite it as cos⁡2(θ)=1\cos^2(\theta) = 1.
  2. Take Square Root: Next, we take the square root of both sides of the equation to solve for cos⁡(θ)\cos(\theta). This gives us two possible equations: cos⁡(θ)=1\cos(\theta) = 1 and cos⁡(θ)=−1\cos(\theta) = -1, because the square root of 11 is both 11 and −1-1.
  3. Find Cosine of 11: We now find the angles where the cosine is equal to 11. The cosine of an angle is equal to 11 at 00 degrees (or 00 radians). This is the only angle between 00 degrees and 360360 degrees where the cosine is 11.
  4. Find Cosine of −1-1: Next, we find the angles where the cosine is equal to −1-1. The cosine of an angle is equal to −1-1 at 180180 degrees (or π\pi radians). This is the only angle between 00 degrees and 360360 degrees where the cosine is −1-1.
  5. Identify Satisfying Angles: We have found all the angles that satisfy the equation cos⁡2(θ)−1=0\cos^2(\theta) - 1 = 0. These angles are 00 degrees and 180180 degrees. There are no other angles in the range from 00 degrees to 360360 degrees that satisfy the equation.

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