Solve the equation of the interval (0, 2pi)
cosx=sinx
I squared both sides to get :cos²x=sin²x
Then using tri indentites I came up with
cos²x=1-cos²x
Ended up with 2cos²x=1
Would the answer be cos²x=1/2???
so far so good, divide both sides by 2,
now take √ of both sides,
cos x = ± 1/√2
so x = 45, 135, 225 and 315º
the problem is that you squared both sides, so each answer must be verified.
only 60º and 300º work
so x = 45 or x= 225, or in radians
x = pi/4 or x = 5pi/4
another ways that would have avoided the above problems, and the way I would have done it, is to divide both sides by cosx
sinx/cosx = 1
tanx = 1
now x = 45º or x = 180+45 = 225º
or x = pi/4 or x = 5pi/4
the 6th line down which was
<< only 60º and 300º work >>
should have said :
only 45º and 225º work
So my best way to solve
(Tanx +1)(cosx +1)=0
Would be to divide by either cos or tan..instead of squaring and using an identity?
You don't have to do any more before proceeding.
since you have a product equal to zero, simply set each of its factors to zero, so ...
tanx + 1 = 0 or cosx + 1 = 0
1. tanx = -1 , so x must be in quadrants II or IV and
x = 180-45 or x = 360-45
x = 135 or 315
2. cosx = -1
x = 180
so x = 135 , 180 or 315 which in radians is
x = 3pi/4 , pi , 7pi/4
To solve the equation cos(x) = sin(x) on the interval (0, 2π), you followed the correct steps until you arrived at the equation 2cos²x = 1. However, the next step will be slightly different.
To isolate cos²x, divide both sides of the equation by 2:
(2cos²x)/2 = 1/2
Simplifying, you get:
cos²x = 1/2
Now, to find the value(s) of x that satisfy this equation on the given interval, take the square root of both sides:
√(cos²x) = √(1/2)
Since cos²x is positive on (0, 2π), you can drop the negative square root:
cosx = √(1/2)
To find the values of x that satisfy this equation, use the unit circle or trigonometric identities.
On the unit circle, the cosine function is positive in the first (0 to π/2) and fourth (3π/2 to 2π) quadrants. So, you need to find the angles in these quadrants where cos(x) = √(1/2).
In the first quadrant, cos(x) = √(1/2) is true for x = π/4.
In the fourth quadrant, cos(x) = √(1/2) is true for x = 7π/4.
Therefore, on the interval (0, 2π), the solution to the equation cos(x) = sin(x) is x = π/4 and x = 7π/4.