1)sin(x)/cos(x)+1=sqrt(3)(1+cos(x)/sin(x))
(sin(x)+cos(x))/cos(x)=
=sqrt(3)(sin(x)+cos(x))/sin(x)
1/cos(x)=sqrt(3)/sin(x)
tan(x)=sqrt(3)
x=Pi/3+Pi*n
2)cos^(x)---???
3)cos2(x)=cos(2x) or cos^2(x)?
tan(x)+1=sqrt(3)+sqrt(3)cot(x)as well as
sin(2x)= 2 cos^(x)
4cos2(x) = 8sin(x)cos(X)
(sin(x)+cos(x))/cos(x)=
=sqrt(3)(sin(x)+cos(x))/sin(x)
1/cos(x)=sqrt(3)/sin(x)
tan(x)=sqrt(3)
x=Pi/3+Pi*n
2)cos^(x)---???
3)cos2(x)=cos(2x) or cos^2(x)?
1) tan(x) + 1 = sqrt(3) + sqrt(3)cot(x)
Let's simplify this equation step by step:
Step 1: Rewrite cot(x) as 1/tan(x) using the reciprocal property.
tan(x) + 1 = sqrt(3) + sqrt(3)(1/tan(x))
Step 2: Multiply both sides of the equation by tan(x) to eliminate the denominator:
tan(x)(tan(x) + 1) = tan(x)(sqrt(3) + sqrt(3)(1/tan(x)))
tan^2(x) + tan(x) = sqrt(3)tan(x) + sqrt(3)
Step 3: Rearrange the terms:
tan^2(x) - sqrt(3)tan(x) + tan(x) - sqrt(3) = 0
Step 4: Combine like terms:
tan^2(x) - sqrt(3)tan(x) + (tan(x) - sqrt(3)) = 0
Step 5: Group the terms:
(tan^2(x) + tan(x)) - sqrt(3)(tan(x) - sqrt(3)) = 0
Step 6: Factor the trinomial:
(tan(x) + 1)(tan(x) - sqrt(3)) - sqrt(3)(tan(x) - sqrt(3)) = 0
Step 7: Simplify further:
(tan(x) + 1 - sqrt(3))(tan(x) - sqrt(3)) = 0
Step 8: Set each factor equal to zero and solve for x:
tan(x) + 1 - sqrt(3) = 0 OR tan(x) - sqrt(3) = 0
For the first equation, subtract 1 and sqrt(3) from both sides:
tan(x) = -1 + sqrt(3)
To find x, take the inverse tangent (arctan) of both sides using a calculator:
x = arctan(-1 + sqrt(3))
For the second equation, add sqrt(3) to both sides:
tan(x) = sqrt(3)
Again, take the inverse tangent of both sides using a calculator:
x = arctan(sqrt(3))
Therefore, the solutions are:
x = arctan(-1 + sqrt(3))
x = arctan(sqrt(3))
2) sin(2x) = 2cos^2(x)
Step 1: Recall the double-angle identity for sine:
sin(2x) = 2sin(x)cos(x)
Step 2: Substitute 2sin(x)cos(x) for sin(2x):
2sin(x)cos(x) = 2cos^2(x)
Step 3: Cancel out the common factor of 2:
sin(x)cos(x) = cos^2(x)
Step 4: Divide both sides by cos(x) to eliminate it:
sin(x) = cos(x)
Step 5: To solve this equation, either we can use the identities or examine the unit circle values.
Considering the unit circle, when sin(x) = cos(x), we have two possible solutions:
x = 45 degrees + n * 90 degrees or x = 225 degrees + n * 90 degrees
where n is an integer.
3) 4cos(2x) = 8sin(x)cos(x)
Step 1: Use the double-angle identity for cosine:
4(2cos^2(x) - 1) = 8sin(x)cos(x)
Step 2: Distribute the 4:
8cos^2(x) - 4 = 8sin(x)cos(x)
Step 3: Rearrange the terms:
8cos^2(x) - 8sin(x)cos(x) - 4 = 0
Step 4: Divide the whole equation by 4 to simplify it:
2cos^2(x) - 2sin(x)cos(x) - 1 = 0
Step 5: Factor the quadratic equation:
(2cos(x) - 1)(cos(x) - 1) = 0
Step 6: Set each factor equal to zero and solve for x:
2cos(x) - 1 = 0 or cos(x) - 1 = 0
For the first equation, add 1 and divide by 2:
cos(x) = 1/2
To find x, take the inverse cosine (arccos) of both sides using a calculator:
x = arccos(1/2)
For the second equation:
cos(x) = 1
Again, take the inverse cosine of both sides using a calculator:
x = arccos(1)
Therefore, the solutions are:
x = arccos(1/2)
x = arccos(1)