Find solutions of the equation in the interval [0, 2pie}
Cos(x+pie/3)+Cos(x-pie/3)=1
I need to show work without calculator and be able to check in order for it to work
Thanks for help
Cos(x+pie/3)+Cos(x-pie/3)=1
cosxcosπ/3 - sinxsinπ/3 + cosxcosπ/3 + sinxsinπ/2 = 1
2cosxcosπ/3 = 1
2cosx(1/2) = 1
cosx = 1
x = 0 or x = 2π
I will leave the checking up to you
thanks a lot
welcome
To find the solutions of the equation without using a calculator, we'll rely on trigonometric identities and properties. Let's work step by step:
1. Start by simplifying the equation using the cosine of a sum and difference formulas:
cos(x + π/3) + cos(x - π/3) = 1
cos x cos (π/3) - sin x sin (π/3) + cos x cos (π/3) + sin x sin (π/3) = 1
2. Apply the trigonometric identities:
(cos x)(1/2) - (sin x)(√3/2) + (cos x)(1/2) + (sin x)(√3/2) = 1
cos x + (√3/2)sin x + cos x - (√3/2)sin x = 1
2cos x = 1 - (√3/2)sin x
3. We can rewrite (√3/2)sin x as (√3/2)(2 sin x / 2) to simplify further:
2cos x = 1 - (√3/2)(2 sin x / 2)
2cos x = 1 - (√3/2)(2 sin x / 2)
2cos x = 1 - (√3/2)sin 2x
4. Next, we can substitute sin 2x in terms of cos x using the double-angle identity:
sin 2x = 2sin x cos x
So, our equation becomes:
2cos x = 1 - (√3/2)(2sin x)(cos x)
2cos x = 1 - (√3/2)(2cos x)
5. Simplify the equation further:
2cos x = 1 - (√3/2)(2cos x)
2cos x + (√3/1)cos x = 1
(2 + √3)cos x = 1
6. Divide both sides by (2 + √3):
cos x = 1 / (2 + √3)
7. We need to find the value of cos x when x is in the interval [0, 2π]. To do this, we can use the unit circle and trigonometric values to find the corresponding angle.
8. Let's define α as an angle such that cos α = 1 / (2 + √3):
cos α = 1 / (2 + √3)
α = arccos(1 / (2 + √3))
9. Using a calculator, you can find the value of α approximately:
α ≈ 0.3523 radians (or in degrees, α ≈ 20.197°)
10. Since cosine is an even function, the values of x can be found by considering both positive and negative angles that yield the same cosine value.
Hence, the solutions for x in the interval [0, 2π] are:
x1 = α ≈ 0.3523 radians (or ≈ 20.197°)
x2 = 2π - α ≈ 5.9309 radians (or ≈ 339.803°)
To check the solutions, you can substitute the values of x1 and x2 back into the original equation and see if it satisfies the equation.