If θ∈[−2π,2π], then the number of solutions of 22cos2θ+(2−6)cosθ−3=0 is equal to:
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Step-by-Step Solution
Step 1: Substitute and Simplify
To simplify the given trigonometric equation, we can make a substitution. Let x=cosθ. This transforms the equation into a quadratic equation in terms of x.
Step 2: Solve the Quadratic Equation
We use the quadratic formula x=2a−b±b2−4ac to find the values of x. Here, a=22, b=2−6, and c=−3.
Step 3: Calculate Discriminant and Roots
First, we calculate the discriminant b2−4ac. Then, we simplify the square root term. Notice that 10+46 can be written as (4+6)2. Finally, we find the two possible values for x.
Step 4: Find Solutions for cosθ=23
For cosθ=23, the principal value is 6π. Since the cosine function is positive in the first and fourth quadrants, and the interval is [−2π,2π], the solutions are 6π, −6π, 611π (which is 2π−6π), and −611π (which is −2π+6π). This gives 4 solutions.
Step 5: Find Solutions for cosθ=−21
For cosθ=−21, the principal value is 43π. Since the cosine function is negative in the second and third quadrants, and the interval is [−2π,2π], the solutions are 43π, −43π, 45π (which is 2π−43π is incorrect, it should be 2π−43π is not in the range, it should be π+4π), and −45π. This gives 4 solutions.
Step 6: Total Number of Solutions
Combining the solutions from both cases, cosθ=23 and cosθ=−21, we get a total of 4+4=8 distinct solutions in the interval [−2π,2π].