Answer:
x = pi/3
Explanation:
To solve this equation, we can start by using the double angle formula for cosine, which states that cos(2x) = 2cos^2(x) - 1. This formula allows us to rewrite the left-hand side of the equation as 2cos^2(x) + 7sinx = 2(cos^2(x) + (1/2)sin(2x)) + 7sinx.
Next, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to rewrite the term cos^2(x) as 1 - sin^2(x). Substituting this into our equation, we get 2(1 - sin^2(x) + (1/2)sin(2x)) + 7sinx = 5. We can now combine like terms to get 2 - 2sin^2(x) + sin(2x) + 7sinx = 5.
Next, we can use the double angle formula for sine, which states that sin(2x) = 2sinxcosx. Substituting this into our equation, we get 2 - 2sin^2(x) + 2sinxcosx + 7sinx = 5. We can now combine like terms to get 2 - sin^2(x) + 3sinxcosx = 5.
Finally, we can use the Pythagorean identity once again to rewrite the term sin^2(x) as 1 - cos^2(x). Substituting this into our equation, we get 2 - (1 - cos^2(x)) + 3sinxcosx = 5. We can now combine like terms to get cos^2(x) + 3sinxcosx = 3.
To solve this equation, we can start by squaring both sides, which gives us cos^4(x) + 6cos^2(x)sin^2(x) + 9sin^2(x)cos^2(x) = 9. We can then use the identity sin^2(x) = 1 - cos^2(x) to rewrite the term sin^2(x)cos^2(x) as (1 - cos^2(x))cos^2(x). Substituting this into our equation, we get cos^4(x) + 6cos^2(x)(1 - cos^2(x)) + 9(1 - cos^2(x))cos^2(x) = 9. We can now combine like terms to get cos^4(x) - 3cos^4(x) + 9cos^2(x) - 9cos^4(x) = 9. Solving for cos^4(x), we get cos^4(x) = 3/8.
Since the cosine is always between -1 and 1, the only possible value of cos^2(x) is 1/2. Therefore, the only solution to the original equation is x = pi/3, which is the only value of x that satisfies the equation cos^2(x) = 1/2.
Therefore, the solution to the equation 2cos^2(x) + 7sinx = 5 is x = pi/3.