Question 696991
NOTE: In math, when we see {{{cos(x)}}}, we assume {{{x}}} is measured in radians, not degrees.
 
{{{cos x - 3x + 1 = 0}}} <--> {{{cos x + 1 = 3x}}}
We know that the graphs of {{{y=cos(x)+1}}} and {{{y=3x}}} look like this:
{{{graph(300,150,-2,7,-2,2.5,cos(x)+1,3x)}}}
We need to look for the point where they intersect.
 
At {{{x=0.5}}}, {{{y=cos(x)+1}}} is greater than {{{y=3x}}} ,
{{{cos(0.5)+1=about}}}{{{1.8776>1.5=3*0.5}}} ,
but at {{{x=0.7}}}, {{{y=cos(x)+1}}} is lesser than {{{y=3x}}} ,
{{{cos(0.7)+1=about}}}{{{1.7648<2.1=3*0.7}}} ,
so the answer is between {{{x=0.5}}} and {{{x=0.7}}} .
For {{{x=0.6}}}, {{{y=cos(x)+1}}} is greater than {{{y=3x}}} , with
{{{cos(0.6)+1=about}}}{{{1.8253>1.8=3*0.6}}} ,
but they are very close, much closer that at {{{x=0.7}}} ,
so the answer is between {{{x=0.6}}} and {{{x=0.7}}} ,
but we expect it to be close to {{{x=0.6}}} .
For {{{x=0.61}}}, {{{y=cos(x)+1}}} is lesser than {{{y=3x}}} , with
{{{cos(0.61)+1=about}}}{{{1.8196<1.83=3*0.61}}} ,
so the answer is between {{{x=0.6}}} and {{{x=0.61}}} .
We may try {{{x=0.605}}, getting
{{{cos(0.605)+1=about}}}{{{1.8225>1.815=3*0.605}}} ,
where {{{y=cos(x)+1}}} is still a bit too high,
and then try {{{x=0.607}}}, getting
{{{cos(0.607)+1=about}}}{{{1.8214>1.8210=3*0.607}}} ,
where {{{y=cos(x)+1}}} is still a bit too high,
but very, very close.
We could next try {{{x=0.6071}}}, getting
{{{cos(0.6071)+1=about}}}{{{1.82131>1.8213=3*0.6071}}} ,
where {{{y=cos(x)+1}}} is still a hair too high.
We can then try {{{x=0.6072}}}, getting
{{{cos(0.6072)+1=about}}}{{{1.8212>1.8216=3*0.6072}}} ,
where {{{y=cos(x)+1}}} is a bit too low.
That tells us that the answer is between {{{x=0.6071}}} and {{{x=0.6072}}} .
We seemed to be closer at {{{x=0.6071}}}, so maybe we should try {{{x=0.60714}}} .
For {{{x=0.60714}}}, {{{y=cos(x)+1}}} is also a bit too low, with
{{{cos(0.60714)+1=about}}}{{{1.8213<1.82142=3*0.60714}} .
That tells us tha answer is between {{{x=0.60710}}} and {{{x=0.60714}}} ,
both of which round to {{{highlight(0.6071)}}} .