(square root 5) cos (2x+0.464) = 1
0<=x<pi
Supposed to give 2 solutions:
0.322, 2.36 (2d.p.)
I can get the first solution, but not the second. Help!!
cos(2x + 0.464) = 1
2x + 0.464 = cos-1(1)
But: cos-1(1) = 0 --- and only 0, so there will be only one answer within that domain.
And: 2x + 0.464 = 0 ---> 2x = -0.464 ---> x = -0.232
To get within the domain, add 3.14159 ---> x = 2.909.
I did miss the √5 --- thanks for the correction!
When you (corectly!) got step 2 to equal 1.107 -- because it's cosine, you get two answers both + and - 1.107
So, after you get your first answer, use -1.107 and finish to get your second answer.
cosine is positive in the first and fourth quadrants, so the angle whose cosine is 1/√5 is not just 1.107, but also -1.107. However, this angle can also be expressed as 2pi - 1.107. Using 2pi - 1.107 should give you the other solution in the range 0 to pi. (Most calculators/software will just give one value when asked for cos-1 .)
See image:
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√5cos(2x + 0.464) = 1 let 2x + 0.464 = θ divide both sides by √5
cos θ = 1/√5 using the cosine inverse, we have
cos-1(1/√5) = 1.107 and 5.176 = θ
So
2x + 0.464 = 1.107 subtract 0.464 from both sides
2x = .643 divide both sides by 2
x = .3215 or about .322 rounded
And
2x + 0.464 = 5.176
2x = 4.712
x = 2.356 or about 2.36 rounded