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tan^2(u)= -3-3sec(u); [pi/2,pi]
To solve the equation tan^2(u) = -3 - 3sec(u) in the interval [π/2, π], follow these steps:
Step 1: Rewrite the equation
Start by substituting sec(u) with 1/cos(u) to rewrite the equation as follows:
tan^2(u) = -3 - 3/cos(u)
Step 2: Convert tan^2(u) to sin^2(u)/cos^2(u)
Use the identity tan^2(u) = sin^2(u) / cos^2(u) to rewrite the equation further:
sin^2(u) / cos^2(u) = -3 - 3/cos(u)
Step 3: Multiply through by cos^2(u)
Multiply both sides of the equation by cos^2(u) to eliminate the fraction:
sin^2(u) = -3cos^2(u) - 3
Step 4: Use the Pythagorean Identity sin^2(u) + cos^2(u) = 1
Apply the Pythagorean Identity sin^2(u) + cos^2(u) = 1 to rewrite the equation:
1 - cos^2(u) = -3cos^2(u) - 3
Step 5: Simplify the equation
Combine like terms and move all terms to one side of the equation:
4cos^2(u) = 4
Step 6: Solve for cos(u)
Divide both sides of the equation by 4 to solve for cos^2(u):
cos^2(u) = 1
Step 7: Take the square root
Take the square root of both sides of the equation to solve for cos(u):
cos(u) = ±1
Step 8: Determine the values of u in the given interval
Since the interval is [π/2, π], we need to find the values of u that satisfy cos(u) = ±1 within that range.
For the value cos(u) = 1, we have u = π (since π/2 is not in the given interval).
For the value cos(u) = -1, we have u = π/2.
Step 9: Finalize the solution
The solution to the equation tan^2(u) = -3 - 3sec(u) in the interval [π/2, π] is:
u = π and u = π/2.
To solve the equation tan^2(u) = -3 - 3sec(u) over the interval [π/2, π], we need to follow a few steps:
Step 1: Rewrite the equation using trigonometric identities if necessary.
Step 2: Solve the resulting equation algebraically.
Step 3: Check if the solutions are within the given interval.
Let's go through each step in detail:
Step 1: Rewrite the equation using trigonometric identities if necessary.
The equation we are given is:
tan^2(u) = -3 - 3sec(u)
To rewrite the equation, we can use the identity: sec^2(u) = 1 + tan^2(u). Rearranging this identity, we get: tan^2(u) = sec^2(u) - 1.
Substituting this identity into the equation, we have:
sec^2(u) - 1 = -3 - 3sec(u)
Step 2: Solve the resulting equation algebraically.
Let's simplify the equation by combining like terms:
sec^2(u) + 3sec(u) - 4 = 0
This is a quadratic equation in terms of sec(u). To solve it, we can set sec(u) = x, which gives us:
x^2 + 3x - 4 = 0
Factoring the quadratic equation, we have:
(x + 4)(x - 1) = 0
Setting each factor equal to zero, we get:
x + 4 = 0 --> x = -4
x - 1 = 0 --> x = 1
Remember that x represents sec(u), so we need to find the corresponding values for u.
To get u, we can take the inverse secant (arcsec) of each solution:
u = arcsec(-4) and u = arcsec(1)
Step 3: Check if the solutions are within the given interval.
Since the given interval is [π/2, π], we need to check if the solutions u = arcsec(-4) and u = arcsec(1) are within this interval.
To find the values of u within the interval, we can use the range of arcsec(x), which is [0, π] or [π, 2π].
For u = arcsec(-4):
The range of arcsec(x) is [0, π], so u = arcsec(-4) is not within the interval [π/2, π]. Therefore, it is not a valid solution.
For u = arcsec(1):
The range of arcsec(x) is [0, π], so u = arcsec(1) is within the interval [π/2, π]. Therefore, it is a valid solution.
Therefore, the only valid solution within the interval [π/2, π] is u = arcsec(1).
Please note that depending on the specific context or question given, there might be additional steps or considerations, but this is the general approach to solving the given equation over the given interval.
To solve the equation tan^2(u) = -3 - 3sec(u) over the interval [pi/2, pi], follow these steps:
1. Start by multiplying both sides of the equation by cos^2(u) to get rid of sec(u) and convert tan^2(u) and sec(u) to sin(u) and cos(u), respectively.
cos^2(u)*tan^2(u) = cos^2(u)*(-3 - 3sec(u))
cos^2(u)*tan^2(u) = -3cos^2(u) - 3cos^2(u)*sec(u)
2. Use the trigonometric identity tan^2(u) = sin^2(u)/cos^2(u) to simplify the equation further.
sin^2(u)/cos^2(u) = -3cos^2(u) - 3cos^2(u)*sec(u)
sin^2(u) = -3cos^4(u) - 3cos^2(u)*sin(u)
3. Rearrange the equation by moving all the terms to one side to form a quadratic equation.
sin^2(u) + 3cos^4(u) + 3cos^2(u)*sin(u) = 0
4. Factor out sin(u) from the equation.
sin(u)*(sin(u) + 3cos^2(u)) + 3cos^4(u) = 0
5. Divide both sides of the equation by cos^4(u) to isolate the trigonometric functions.
(sin(u)*(sin(u) + 3cos^2(u)))/cos^4(u) + 3 = 0
6. Use the trigonometric identity sin^2(u) + cos^2(u) = 1 to simplify the equation further.
(sin(u)*(sin(u) + 3(1 - sin^2(u))))/cos^4(u) + 3 = 0
(sin(u)*(sin(u) + 3 - 3sin^2(u)))/cos^4(u) + 3 = 0
(sin(u)*(3 - 2sin^2(u)))/cos^4(u) + 3 = 0
7. Multiply both sides of the equation by cos^4(u) to get rid of the denominators.
sin(u)*(3 - 2sin^2(u)) + 3cos^4(u) = 0
8. Expand the equation by distributing sin(u) to each term.
3sin(u) - 2sin^3(u) + 3cos^4(u) = 0
9. Use the trigonometric identity cos^2(u) = 1 - sin^2(u) to substitute for cos^4(u).
3sin(u) - 2sin^3(u) + 3(1 - sin^2(u))^2 = 0
10. Simplify the equation by squaring and distributing.
3sin(u) - 2sin^3(u) + 3(1 - 2sin^2(u) + sin^4(u)) = 0
3sin(u) - 2sin^3(u) + 3 - 6sin^2(u) + 3sin^4(u) = 0
11. Combine like terms and rearrange the equation.
3sin^4(u) - 2sin^3(u) - 6sin^2(u) + 3sin(u) + 3 = 0
12. Now you have a quartic equation in terms of sin(u). You can use numerical methods or advanced algebraic techniques to find the roots of the equation.
Note: The steps provided above may not lead to an explicit solution for u due to the presence of a quartic equation.