Solve tan2x = 5sin2x
Giving all solutions in the interval 0 ≤ x ≤ 180°
posting for anyone else who comes across this thread:
tan2x=5sin2x
tan2x=5*tan2x*cos2x
divide both sides by tan(2x)
1=5cos2x
0.2=cos2x
cos^-1 both sides
78.46..=2x
39.2=x
draw cos2x graph. as symmetrical next value = 180-39.2.
and don't forget the values where both = 0
To solve the equation tan(2x) = 5sin(2x), we can use trigonometric identities to rewrite both sides of the equation and then solve for x.
Using the identity tan(2x) = 2tan(x) / (1 - tan^2(x)), we can rewrite the equation as:
2tan(x) / (1 - tan^2(x)) = 5sin(2x).
Next, we can rewrite sin(2x) using the identity sin(2x) = 2sin(x)cos(x):
2tan(x) / (1 - tan^2(x)) = 5(2sin(x)cos(x)).
Simplifying further:
2tan(x) / (1 - tan^2(x)) = 10sin(x)cos(x).
Multiplying both sides by (1 - tan^2(x)) to eliminate the denominator:
2tan(x) = 10sin(x)cos(x)(1 - tan^2(x)).
Using the identity tan^2(x) = sin^2(x) / cos^2(x), we can rewrite the equation as:
2tan(x) = 10sin(x)cos(x)(1 - sin^2(x) / cos^2(x)).
Simplifying further:
2tan(x) = 10sin(x)cos(x)(cos^2(x) - sin^2(x)) / cos^2(x).
Distributing:
2tan(x) = 10sin(x)cos^3(x) - 10sin^3(x)cos(x) / cos^2(x).
Dividing both sides by 2tan(x):
1 = 5sin(x)cos^2(x) - 5sin^3(x) / sin(x)cos(x).
Simplifying further:
1 = 5cos^2(x) - 5sin^2(x) / cos(x).
Using the identity cos^2(x) - sin^2(x) = cos(2x), we can rewrite the equation as:
1 = 5cos(2x) / cos(x).
Multiplying both sides by cos(x) to eliminate the denominator:
cos(x) = 5cos(2x).
Now, we have a simpler equation to solve. We can proceed as follows:
1. If cos(x) = 0, then x = 90°.
2. If cos(x) ≠ 0, divide both sides by cos(x):
1 / cos(x) = 5cos(2x) / cos(x).
Simplifying further:
sec(x) = 5cos(2x).
Using the identity cos(2x) = 2cos^2(x) - 1, we can rewrite the equation as:
sec(x) = 5(2cos^2(x) - 1).
Substituting sec(x) = 1 / cos(x):
1 / cos(x) = 10cos^2(x) - 5.
Rearranging the equation:
10cos^3(x) - 5cos(x) - 1 = 0.
To solve this cubic equation, we can either use numerical methods or graphical methods. Unfortunately, it cannot be solved algebraically in a straightforward manner.
Therefore, to find the solutions in the given interval of 0 ≤ x ≤ 180°, we will need to use numerical or graphical methods to approximate the values of x that satisfy the equation.
To solve the equation tan(2x) = 5sin(2x), we can use trigonometric identities to express everything in terms of a single trigonometric ratio.
First, we will use the double-angle identity for tangent: tan(2x) = 2tan(x) / (1 - tan^2(x)).
Now we can substitute this expression into the equation:
(2tan(x) / (1 - tan^2(x))) = 5sin(2x).
Next, we will use the double-angle identity for sine: sin(2x) = 2sin(x)cos(x).
Substituting this into the equation, we have:
(2tan(x) / (1 - tan^2(x))) = 5(2sin(x)cos(x)).
Next, we can simplify this equation by dividing both sides by 2 and multiplying both sides by (1 - tan^2(x)):
tan(x) / (1 - tan^2(x)) = 5sin(x)cos(x).
Now, let's express cosine in terms of sine using the Pythagorean identity: cos^2(x) = 1 - sin^2(x).
Substituting this into the equation, we have:
tan(x) / (1 - tan^2(x)) = 5sin(x)(√(1 - sin^2(x))).
Now let's express tan(x) in terms of sine and cosine: tan(x) = sin(x) / cos(x).
Substituting this into the equation, we get:
(sin(x) / cos(x)) / (1 - (sin(x) / cos(x))^2) = 5sin(x)(√(1 - sin^2(x))).
Simplifying this equation, we have:
(sin(x) / cos(x)) / (1 - sin^2(x) / cos^2(x)) = 5sin(x)(√(1 - sin^2(x))).
Multiplying both sides by cos^2(x), we get:
sin(x) / (1 - sin^2(x)) = 5sin(x)(√(1 - sin^2(x))).
Now we can cross-multiply and simplify:
sin(x) = 5 sin^2(x) √(1 - sin^2(x)).
Let's solve this equation:
1. First, we notice the condition 0 ≤ x ≤ 180°. Since sin(x) is always between -1 and 1, we can eliminate the negative solution range.
2. If sin(x) equals 0, the equation becomes 0 = 0, which is a true statement. Hence, sin(x) = 0 is a solution to the equation.
3. If sin(x) is not equal to 0, we can divide by sin(x) on both sides:
1 = 5 sin(x) √(1 - sin^2(x)).
4. Squaring both sides of the equation, we get:
1 = 25 sin^2(x) (1 - sin^2(x)).
5. Expanding and simplifying the equation, we have:
1 = 25 sin^2(x) - 25 sin^4(x).
6. Rearranging the terms, we get:
25 sin^4(x) - 25 sin^2(x) + 1 = 0.
This equation is a quadratic equation in terms of sin^2(x). Let y = sin^2(x), we have:
25y^2 - 25y + 1 = 0.
We can solve this quadratic equation for y using the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / 2a,
where a = 25, b = -25, and c = 1.
Using these values, we get:
y = (-(-25) ± √((-25)^2 - 4(25)(1))) / (2(25)).
Simplifying further:
y = (25 ± √(625 - 100)) / 50,
y = (25 ± √525) / 50.
This gives two possible values for y:
y = (25 + √525) / 50,
y = (25 - √525) / 50.
7. Now, let's solve for sin(x) using the values of y:
If y = (25 + √525) / 50, then sin(x) = √y.
If y = (25 - √525) / 50, then sin(x) = √y.
8. Finally, we can solve for x by taking the inverse sine of both sides:
If y = (25 + √525) / 50, then x = arcsin(√y) + kπ, where k is an integer.
If y = (25 - √525) / 50, then x = arcsin(√y) + kπ, where k is an integer.
These are the solutions for the equation tan(2x) = 5sin(2x) in the interval 0 ≤ x ≤ 180°.
tan2x = sin2x
sin2x/cos2x = sin2x
sin2x(cos2x-1) = 0
so,
sin2x=0
or
cos2x=1
That help?