Cos(x) +8sin(x/2)-7=0

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To solve the equation, we will first simplify it by combining like terms:

cos(x) + 8sin(x/2) - 7 = 0

Next, we will use the trigonometric identity:

sin^2(x) + cos^2(x) = 1

Since we have both cosine and sine terms in our equation, we will convert them into a single trigonometric function using the identity:

cos(x) = sqrt(1 - sin^2(x))

Substitute this into the equation:

sqrt(1 - sin^2(x)) + 8sin(x/2) - 7 = 0

Now, we can make a substitution to simplify the equation further by letting y = sin(x/2), so:

sqrt(1 - y^2) + 8y - 7 = 0

This is now a quadratic equation in the form of:

ay^2 + by + c = 0

Where a = -1, b = 8, and c = -7.

Solve this equation for y using the quadratic formula:

y = [-b ± sqrt(b^2 - 4ac)] / 2a

y = [-(8) ± sqrt((8)^2 - 4(-1)(-7))] / 2(-1)
y = [-8 ± sqrt(64 - 28)] / -2
y = [-8 ± sqrt(36)] / -2
y = [-8 ± 6] / -2

Therefore, y = -14/(-2) or y = -2/(-2)

Therefore, y = 7 or y = 1

Now, substitute back sin(x/2) = y:

sin(x/2) = 7 or sin(x/2) = 1

Since the sine function has a range of -1 to 1, the first solution sin(x/2) = 7 is not valid.

So, we solve for x when sin(x/2) = 1:

sin(x/2) = 1
x/2 = π/2
x = π