cos OPA = (x^2-8x+40)/(sqrt(x^2-16x+80)sqrt(x^2+100)) Find the positive value of x such that OPA = 60 degrees

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember that `cos 60^o = 1/2` , hence you need to substitute `1/2`  for cos OPA in equation such that:

`1/2 = (x^2-8x+40)/(sqrt((x^2-16x+80)(x^2+100)))`

`2(x^2-8x+40) = sqrt((x^2-16x+80)(x^2+100))`

You need to raise to square to remove the square root such that:

`4(x^2-8x+40)^2 = (x^2-16x+80)(x^2+100)`

You need to expand binomial to the left such that:

`4x^4 - 256x^2 + 6400 - 64x^3 + 320x^2 - 2560 = x^4 + 100x^2 - 16x^3 - 1600x + 80x^2 + 8000`

You need to move all terms to the left side and you need to collect like terms:

`3x^4 -48 x^3 -116x^2 + 1600x - 4160 = 0`

You need to write the factored form of polynomial such that:

`3(x + 6.642)(x - 16.701)(x^2 - 5.941x + 12.5) = 0`

Hence, evaluating the positive real roots to equation above, for `cos OPA = 1/2` , yields `x = 16.701` .

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