Find the general solution for the equation 4tanx+cotx=5?????? and the steps please? THANYOO!
To find the general solution for the equation 4tan(x) + cot(x) = 5, you can follow these steps:
Step 1: Rewrite the equation using trigonometric identities. Since cot(x) is the reciprocal of tan(x), we can rewrite it as 1/tan(x):
4tan(x) + 1/tan(x) = 5
Step 2: Multiply through by tan(x) to remove the denominators:
4tan^2(x) + 1 = 5tan(x)
Step 3: Rearrange the equation to form a quadratic equation:
4tan^2(x) - 5tan(x) + 1 = 0
Step 4: Solve the quadratic equation. Let's denote tan(x) as a variable, say t. We can rewrite the quadratic equation as:
4t^2 - 5t + 1 = 0
Step 5: Factor the quadratic equation. Since the equation does not easily factor, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac))/(2a)
For our equation, a = 4, b = -5, and c = 1. Plugging in these values:
t = (5 ± √((-5)^2 - 4 * 4 * 1))/(2 * 4)
= (5 ± √(25 - 16))/(8)
= (5 ± √(9))/(8)
= (5 ± 3)/(8)
Step 6: Solve for t to find the possible values of tan(x):
t = (5 + 3)/(8) = 8/8 = 1
t = (5 - 3)/(8) = 2/8 = 1/4
Step 7: Find the corresponding values of x using the inverse tangent function (tan^(-1)):
Case 1: tan(x) = 1
x = tan^(-1)(1) ≈ 45° + nπ, where n is an integer.
Case 2: tan(x) = 1/4
x = tan^(-1)(1/4) ≈ 14.04° + nπ, where n is an integer.
Therefore, the general solution for the equation 4tan(x) + cot(x) = 5 is:
x ≈ 45° + nπ, where n is an integer,
or
x ≈ 14.04° + nπ, where n is an integer.
To find the general solution for the equation 4tanx + cotx = 5, you need to simplify the equation and solve for x. Here are the steps:
Step 1: Simplify the equation
Recall that cotx is the reciprocal of tanx. So, rewrite the equation 4tanx + cotx = 5 as 4tanx + 1/tanx = 5.
Step 2: Transform the equation
To eliminate the fractions, multiply the entire equation by tanx. You should obtain: 4tan^2(x) + 1 = 5tanx.
Step 3: Rearrange the equation
Move all terms to one side of the equation: 4tan^2(x) - 5tanx + 1 = 0.
Step 4: Factor the equation
We have a quadratic equation. To solve it, we need to factorize. The factors should be of the form: (tanx - a)(tanx - b) = 0. Find two numbers, "a" and "b," which multiply to give "c" (coefficient of tan^2(x)) and add up to give "b" (coefficient of tanx). In our equation, a = 1 and b = 1/4.
Therefore, the factored form is: (tanx - 1)(tanx - 1/4) = 0.
Step 5: Solve for x
Set each factor equal to zero and solve for x:
tanx - 1 = 0
tanx = 1
tanx - 1/4 = 0
tanx = 1/4
Step 6: Find the angles
To find the angles, take the inverse tangent (tan^-1) of both sides of each equation:
tan^-1(tanx) = tan^-1(1)
x = π/4 + nπ, where n is an integer
tan^-1(tanx) = tan^-1(1/4)
x = tan^-1(1/4) + nπ or x = -tan^-1(1/4) + nπ, where n is an integer
Therefore, the general solution for the equation 4tanx + cotx = 5 is:
x = π/4 + nπ or x = tan^-1(1/4) + nπ or x = -tan^-1(1/4) + nπ, where n is an integer.
Note: tan^-1 is also referred to as the inverse tangent function or arctan.
For these I generally like to change everything to sines and cosines
4tanx + cotx = 5
4sinx/cosx + cosx/sinx = 5
(4sin^2x + cos^2x)/sinxcosx = 5
4sin^2x + cos^2x = 5sinxcosx
4sin^2x - 5sinxcosx + cos^2x = 0
(4sinx - cosx)(sinx - cosx) = 0
4sinx = cosx
sinx/cosx = 1/4
tanx = 1/4
x = 14.0° or 194.0°
or
sinx = cosx
sinx/cosx = 1
tanx=1
x = 45° or 225°
the period of tanØ is 180°
general solutions:
45° + 180k°
14° + 180k°
notice that 45+180 = 225 , the other answer
and 14+180 = 194 , also the other answer, so we don't have to generalize those, they get hit anyway.