How do I find all solutions to:
tanx+3cotx=4?
The answer choices are:
a.) x=arctan3+nPi
b.) x=arccot3+nPi
actually there are other answers ...
tanx + 3cotx = 4
tanx + 3/tanx = 4
tan^2x + 3 = 4tanx
tan^2x - 4tanx + 3 = 0
(tanx - 1)(tanx - 3) = 0
x = arctan 1 + npi or x = arctan3 + npi
Thanks! :D
To find all the solutions to the equation tan(x) + 3cot(x) = 4, we can use the following steps:
Step 1: Rewrite the equation in terms of sine and cosine.
Using the definitions of tangent and cotangent:
tan(x) = sin(x) / cos(x)
cot(x) = cos(x) / sin(x)
We can rewrite the equation as:
sin(x)/cos(x) + 3cos(x)/sin(x) = 4
Step 2: Multiply through by sin(x) * cos(x) to eliminate the denominators.
sin(x)*sin(x) + 3cos(x)*cos(x) = 4sin(x)*cos(x)
Simplifying further, we get:
sin^2(x) + 3cos^2(x) = 4sin(x)cos(x)
Step 3: Use the Pythagorean Identity for sine and cosine.
sin^2(x) + cos^2(x) = 1
Substitute this identity into the equation:
1 + 2cos^2(x) = 4sin(x)cos(x)
Step 4: Rearrange the equation.
2cos^2(x) - 4sin(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
cos(x) - 1 = 0
Solving these equations separately, we get:
cos(x) = 1/2
cos(x) = 1
Step 7: Take the inverse cosine (arccos) of both sides to solve for x.
For cos(x) = 1/2, the solutions are:
x = arccos(1/2) + 2nπ
x = π/3 + 2nπ
or
x = 5π/3 + 2nπ
For cos(x) = 1, the solutions are:
x = arccos(1) + 2nπ
x = 2nπ
Step 8: Finally, combine all possible values of x.
Combining the solutions, we have:
x = π/3 + 2nπ, 5π/3 + 2nπ, and 2nπ
Therefore, the correct answer choice is:
b.) x = arccot(3) + nπ
To find all solutions to the equation tan(x) + 3cot(x) = 4, we need to use trigonometric identities and algebraic manipulation. Let's break it down step by step:
1. Rewrite the equation using Pythagorean identities:
Recall that tan(x) = sin(x) / cos(x) and cot(x) = cos(x) / sin(x). Using these identities, we can rewrite the equation as:
sin(x)/cos(x) + 3cos(x)/sin(x) = 4
2. To simplify this expression, let's find the common denominator, which is cos(x) * sin(x). Multiply the first term (sin(x)/cos(x)) by sin(x)/sin(x) and the second term (3cos(x)/sin(x)) by cos(x)/cos(x):
sin^2(x) / (cos(x) * sin(x)) + 3cos^2(x) / (cos(x) * sin(x)) = 4
3. Combine the fractions by adding the numerators:
(sin^2(x) + 3cos^2(x)) / (cos(x) * sin(x)) = 4
4. Simplify the numerator of the fraction:
Using the identity sin^2(x) + cos^2(x) = 1, the above equation becomes:
(1 + 2cos^2(x)) / (cos(x) * sin(x)) = 4
5. Multiply both sides of the equation by (cos(x) * sin(x)) to clear the denominator:
1 + 2cos^2(x) = 4cos(x) * sin(x)
6. Rearrange the equation to form a quadratic equation:
Move all terms to one side of the equation:
2cos^2(x) - 4cos(x) * sin(x) + 1 = 0
7. Let's simplify this equation further:
Since 2cos^2(x) is common to both terms, we can factor it out:
2cos^2(x) - 4cos(x) * sin(x) + 1 = 2(cos^2(x) - 2cos(x) * sin(x) + 1/2)
8. We can recognize the expression in the parentheses as the square of a binomial:
2(cos(x) - sin(x)/√2)^2 = 0
9. Solve for the expression in parentheses:
Now we set the expression inside the square bracket equal to zero:
cos(x) - sin(x)/√2 = 0
10. Rearrange this equation to isolate cos(x):
cos(x) = sin(x)/√2
11. Divide both sides of the equation by sin(x):
cot(x) = 1/√2
12. Use the definition of the cotangent function:
Recall that cot(x) = 1/tan(x). So, we have:
1/tan(x) = 1/√2
13. Take the reciprocal of both sides:
tan(x) = √2
14. To find the solutions, we can take the inverse tangent of both sides:
x = arctan(√2)
15. Since tangent has a periodicity of π (180 degrees), we can add nπ to the solution to find all possible solutions:
x = arctan(√2) + nπ
So, the solution to the equation tan(x) + 3cot(x) = 4 is x = arctan(√2) + nπ.
Comparing this solution with the answer choices provided:
a.) x = arctan(3) + nπ is not the correct answer.
b.) x = arccot(3) + nπ is not the correct answer either.
Therefore, none of the answer choices match the correct solution.