(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!!

Guest Apr 10, 2015

#1**+5 **

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.

geno3141
Apr 10, 2015

#3**+5 **

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.

geno3141
Apr 10, 2015

#5**+5 **

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:

.

Alan
Apr 10, 2015

#6**+5 **

√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

CPhill
Apr 10, 2015