Solve the equation cos2θ = 1/2sinθ 0º≤θ≤90º. (Hint: Graph each side of the equation and determine where the graphs intersect.)
cos2θ = 1 - 2sin^2 θ
so,
1 - 2sin^2 θ = 1/2 sinθ
4 sin^2θ + sinθ - 2 = 0
Now just use the quadratic formula to solve for sinθ, then take the arcsin. To check your answer, see
http://www.wolframalpha.com/input/?i=cos2%CE%B8+%3D+1%2F2+sin%CE%B8
To solve the equation cos(2θ) = 1/2sin(θ), we can follow the hint and graph each side of the equation to determine where they intersect. Let's break it down step by step:
1. Start by graphing cos(2θ) and 1/2sin(θ) separately on a coordinate plane. For simplicity, we will focus on the given range of θ: 0º ≤ θ ≤ 90º.
2. To graph cos(2θ), first note that cos(2θ) can be rewritten as cos(θ + θ). The cosine function repeats every 360º, so we can focus on the range 0º ≤ θ ≤ 180º. Plot points for θ = 0º, 30º, 60º, 90º, 120º, 150º, and 180º. Connect these points smoothly to obtain the graph of cos(2θ).
3. Now, let's graph 1/2sin(θ). The sine function also repeats every 360º, so within the given range, we will plot points for θ = 0º, 30º, 60º, and 90º. Connect these points smoothly to obtain the graph of 1/2sin(θ).
4. Once both graphs are drawn, analyze where they intersect. These intersection points represent the solutions to the equation cos(2θ) = 1/2sin(θ).
5. You can either observe the points of intersection directly from the graph, or you can determine the values of θ algebraically by setting up the equation cos(2θ) - 1/2sin(θ) = 0 and solving for θ.
By following these steps, you can visually identify the values of θ where the two graphs intersect and find the solutions to the equation.