Question

Solve on the interval (0, 2pi) 1+cos theta=sqrt3 +2/2

Answer 1

The solution to the equation
`\(1+\cos\theta = (√(3) + 2)/(2)\)`on the interval
`\((0, 2\pi)\) is \(\theta = (\pi)/(3)\).`

Step-by-step explanation:

To find the solution, first, isolate the cosine term:

`\[ \cos\theta = (√(3))/(2) \]`

Next, identify the angle whose cosine is
`\((√(3))/(2)\)` within the given interval. In the interval
`\((0, 2\pi)\),` this corresponds to
`\(\theta = (\pi)/(3)\)`. Thus, \(\theta =
`(\pi)/(3)\)` is the solution to the equation.

This can be further confirmed by substituting
`\(\theta = (\pi)/(3)\)` back into the original equation:

`\[ 1 + \cos(\pi)/(3) = 1 + (√(3))/(2) = (√(3) + 2)/(2) \]`

Therefore,
`\(\theta = (\pi)/(3)\)`satisfies the equation on the specified interval.

Answer 2

Theta = π/6, 11π/6

Step-by-step explanation:

To solve the equation \(1+\cos(\theta)=\frac{\sqrt{3}+2}{2}\) on the interval \((0, 2\pi)\), first isolate the cosine term by subtracting 1 from both sides: \(\cos(\theta) = \frac{\sqrt{3}+2}{2} - 1\). Simplifying, \(\cos(\theta) = \frac{\sqrt{3}}{2}\). On the unit circle, \(\cos(\theta) = \frac{\sqrt{3}}{2}\) at angles π/6 and 11π/6. These values fall within the interval \((0, 2\pi)\), so the solutions for θ are θ = π/6 and 11π/6.

The equation involves solving for the angle θ such that the cosine of that angle equals the given value within the specified interval (0, 2π). By recognizing the cosine values corresponding to specific angles on the unit circle, the solutions for θ are determined.

Understanding trigonometric functions, the unit circle, and their relationship to angles within specific intervals helps in solving equations involving trigonometric identities and finding solutions within given ranges.