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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-cos theta=0
Answer: 
theta=

Find all angles, 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.\newlinecos2θcosθ=0 \cos ^{2} \theta-\cos \theta=0 \newlineAnswer: θ= \theta=

Full solution

Q. Find all angles, 0θ<360 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.\newlinecos2θcosθ=0 \cos ^{2} \theta-\cos \theta=0 \newlineAnswer: θ= \theta=
  1. Identify Trigonometric Equation: First, let's identify the trigonometric equation we need to solve: cos2(θ)cos(θ)=0\cos^2(\theta) - \cos(\theta) = 0. We can factor this equation to find the values of θ\theta that satisfy it. Factor out cos(θ)\cos(\theta): cos(θ)(cos(θ)1)=0\cos(\theta)(\cos(\theta) - 1) = 0. Now we have two factors that can be individually set to zero to find the solutions for θ\theta.
  2. Factor and Solve: Set the first factor equal to zero: cos(θ)=0\cos(\theta) = 0. To find the angles where the cosine of θ\theta is zero, we look at the unit circle. Cosine is zero at 9090 degrees (π/2\pi/2 radians) and 270270 degrees (3π/23\pi/2 radians).
  3. Find Cosine Values: Set the second factor equal to zero: cos(θ)1=0\cos(\theta) - 1 = 0. Solve for cos(θ)\cos(\theta): cos(θ)=1\cos(\theta) = 1. The angle where the cosine of θ\theta is one is at 00 degrees (00 radians).
  4. Solve for Theta: Now we have all the angles that satisfy the equation within the range of 00 to 360360 degrees.\newlineThe angles are 00 degrees, 9090 degrees, and 270270 degrees.\newlineConvert these angles to the nearest tenth of a degree, which in this case does not change the values since they are already whole numbers.

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