I need help. I have tried to do this problem.

Question

I have tried to do this problem. I want to make sure I am doing the steps and processes the right way.

Solve the equation, giving the exact solutions which lie in [0, 2π).

tan(x)=cos(x)

Answer 1

Try graphing cosx and tanx from 0 to 2 pi or 0 to 360 degrees.

The only intersections are between 0 and pi/2 and between pi/2 and pi

Carefully plot them, and they intersect at close to 38.13 degrees and 141.87 degrees

or in 0.665 radians and 2.476 radians

You might visualize this better by drawing a unit circle and draw a line from the origin to the circle

until the opposite side over the adjacent side looks close to the adjacent side over the hypotenuse

of a right triangle, where the line from the origin to the circle is the hypotenuse of that right triangle.

a=adjacent o=opposite h=hypotenuse tanx = o/a cosx=a/h

They appear close for a little less than 45 degrees or a little more than 135 degrees

Or a little less than pi/4 radians or a little more than 3pi/4 radians

To get an exact value:

Try algebra with a unit circle and hypotenuse=1 then cosx=a/h=a/1=a and tanx=o/a

a=o/a a²=o also hypotenuse squared equals sum of the other two sides

squared, so h²=1²=a²+o² = o + o² = 1 rewrite as a standard quadratic

equation and use the quadratic formula on o² + o - 1 = 0 Solve for o

o=.62 a = the square root of .62 or .785 = cosx then cos^-1(.785)= 38.13 degrees

but also 180-38.13= 141.87 degrees

rounding errors may increase or decrease that by a hundredth or so

Answer 2

tan (x) = cos(x)

sin(x)/cos(x) = cos(x)

sin(x) = cos²(x)

sin(x) = 1-sin²(x) <-----here I used the Pythagorean identity: sin²x + cos²x = 1

sin²(x) + sin(x) -1 = 0

Recognize that this is a quadratic equation in standard form if you let sin(x) equal a random variable. So let sin(x) = u

u² + u -1 = 0

Use the quadratic formula and you should get:

u = (-1 (+ or -) √5)/2

This gives the results

u = 0.618034 or -1.61803

However, remember that u = sin(x). So

Sin(x) = 0.618034 or -1.61803

We know that the range of the Sine function is [-1,1] so our only viable answer choice is 0.618034

Sin (x) = 0.618034

x = arcsin(0.618034)

x = 0.666239

However we're not done. The sine function is positive in both quadrant 1 and quadrant 2. Our quadrant 1 answer is x = 0.666239, which is also our reference angle. Quadrant 2 answer is:

π - (reference angle) = answer

π - 0.666239 = 2.475353

So your two answers on the interval [0,2π] are x = 0.666239 or 2.475353