Find all angles, 0^(@)

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Find all angles,  0^(@) <= theta < 360^(@), that satisfy the equation below, to the nearest tenth of a degree.  -6cos^(2)theta-5cos theta=1 Answer:  theta=

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Rewrite Equation: Let's first rewrite the equation in a more familiar quadratic form by substituting cos(theta) with a variable, let's say 'x'. So the equation becomes: -6x^2 - 5x = 1 Now, we need to solve for 'x' before we can find the corresponding angles for theta.Solve Quadratic Equation: To solve the quadratic equation, we first move all terms to one side to set the equation to zero: -6x^2 - 5x - 1 = 0 Now, we can use the quadratic formula to find the solutions for 'x'. The quadratic formula is x = (-b ± sqrt(b^2 - 4ac)) / (2a), where a = -6, b = -5, and c = -1.Calculate Discriminant: Let's calculate the discriminant (b^2 - 4ac) first: Discriminant = (-5)^2 - 4(-6)(-1) = 25 - 24 = 1 Since the discriminant is positive, we will have two real solutions for 'x'.Find Solutions for x: Now, we can find the two solutions for 'x' using the quadratic formula: x = (-(-5) ± sqrt(1)) / (2 * -6) x = (5 ± 1) / -12 This gives us two solutions for 'x': x1 = (5 + 1) / -12 = 6 / -12 = -0.5 x2 = (5 - 1) / -12 = 4 / -12 = -1/3Find Angles for x1: Now we need to find the angles theta that correspond to the cosine values of x1 and x2. Since we are looking for angles between 0 and 360 degrees, we will use the inverse cosine function and also consider the symmetry of the cosine function. For x1 = -0.5, we find the related angle: theta1 = cos^(-1)(-0.5)Calculate theta1 and theta2: Using a calculator, we find: theta1 = cos^(-1)(-0.5) ≈ 120 degrees However, since cosine is positive in the fourth quadrant as well, we also have another angle: theta2 = 360 - theta1 ≈ 360 - 120 = 240 degreesFind Angles for x2: For x2 = -1/3, we find the related angle: theta3 = cos^(-1)(-1/3) Using a calculator, we find: theta3 ≈ cos^(-1)(-1/3) ≈ 109.5 degrees (to the nearest tenth) Again, considering the symmetry of the cosine function, we find another angle: theta4 = 360 - theta3 ≈ 360 - 109.5 = 250.5 degrees (to the nearest tenth)

Find all angles, 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree. $\newline$ $-6 \cos ^{2} \theta-5 \cos \theta=1$ $\newline$ Answer: $\theta=$

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Find all angles, 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree. $\newline$ $-6 \cos ^{2} \theta-5 \cos \theta=1$ $\newline$ Answer: $\theta=$