If someone could help me with these that would be great! :)
Directions: Solve the equations.
1.) tan4x=1
Location on Unit Circle:
Period:
General Solution:
2.) sec xcsc x=2 csc x
Location on Unit Circle:
Period:
General Solution:
tan π/4 = 1, so
4x = π/4
(1/√2,1/√2)
period = π/4
x = π/16 + n(π/4)
for the second, you just have
cscx =0
or secx = 2
and proceed as above
Sure! I'd be happy to help you with these equations.
1.) tan4x = 1
To solve this equation, we need to find the values of x that satisfy the equation. Let's start by finding the general solution.
The tangent function has a period of π, which means that it repeats every π radians. So, we can write the equation as:
tan(4x) = tan(1π)
To find the values of x, we need to take the inverse tangent (also known as arctan or tan^(-1)) of both sides.
arctan(tan(4x)) = arctan(tan(1π))
Since the inverse tangent function and tangent function cancel each other out, we are left with:
4x = 1π
Now, we can solve for x by dividing both sides by 4:
x = 1π/4
So, the general solution is x = π/4.
Note that we can add any integer multiple of the period (π) to the general solution to get more values that satisfy the equation.
Location on Unit Circle: The location on the unit circle for x = π/4 is (cos(π/4), sin(π/4)).
Period: The period of this equation is π.
General Solution: The general solution is x = π/4 + nπ, where n is an integer.
2.) sec(x)csc(x) = 2csc(x)
To solve this equation, we'll start by dividing both sides by csc(x):
sec(x) = 2
Since sec(x) is the reciprocal of cos(x), we can rewrite the equation as:
1/cos(x) = 2
To get rid of the fraction, we can multiply both sides of the equation by cos(x):
1 = 2cos(x)
Now, let's isolate cos(x) by dividing both sides by 2:
cos(x) = 1/2
To find the values of x, we need to take the inverse cosine (also known as arccos or cos^(-1)) of both sides.
arccos(cos(x)) = arccos(1/2)
Again, since the inverse cosine function and cosine function cancel each other out, we are left with:
x = arccos(1/2)
The value of arccos(1/2) is π/3.
Location on Unit Circle: The location on the unit circle for x = π/3 is (cos(π/3), sin(π/3)).
Period: The period of this equation is 2π, since cos(x) has a period of 2π.
General Solution: The general solution is x = π/3 + 2πn, where n is an integer.
I hope this helps! Let me know if you have any further questions.