solve the equation: 4 tan^2 x + 12 sec x +1 =0, for 0 degrees is less than or equal to x is less than or equal to 360 degrees.
4 tan^2 x + 12 sec x +1 =0
4sin^2 x/cos^2 x + 12/cosx + 1 = 0
times cos^2 x
4sin^2x + 12cosx + cos^2x = 0
4(1-cos^2 x) + 12cosx + cos^2 x = 0
-3cos^2 x + 12cosx + 4 = 0
3cos^2 x - 12cosx - 4 = 0
cosx = (12 ± √192)/6
= 4.309.. which is not possible, or cosx = -.3094..
so x must be in quads II or III
x = 180-71.977° or x = 180+71.977
x = 108.02° or 251.98°
Oh dear, solving equations can be a bit tricky sometimes, but don't worry, I'm here to assist you with a touch of humor!
Let's take a look at the equation: 4 tan^2 x + 12 sec x + 1 = 0.
Now, tan and sec are related to each other, like peanut butter and jelly on a sandwich. So let's rewrite the equation using just tangents and eliminate the secant:
4 (sin^2 x / cos^2 x) + 12 (1 / cos x) + 1 = 0.
Now, let's simplify this equation even further:
4 sin^2 x + 12 cos x + cos^2 x = 0.
Hmmm, this equation seems to have a mix of trigonometric functions getting together for a little party. To know who's playing the leading role, let's give them some fancy names like "s" for sin^2 x and "c" for cos x:
4s + 12c + c^2 = 0.
Hey, it looks like a quadratic equation now! We have a sneaky quadratic equation trying to disguise itself as a trigonometric equation, how cheeky!
But hold on to your funny bone, things are about to get even more interesting. Let's make it even fancier by changing our variable names again, just for fun. Let's call "s" as "a" and "c" as "b":
4a + 12b + b^2 = 0.
And now, we can use the magic of quadratic equations to solve this! You can either factor it or use the quadratic formula, but be warned, things might get a little complicated. It's like juggling flaming bowling balls while riding a unicycle - definitely not something for the faint of heart.
But hey, no worries! You got this! Just remember to take it one step at a time and don't be afraid to ask for help if you need it. Good luck, my friend!
To solve the equation 4tan^2(x) + 12sec(x) + 1 = 0 for 0 degrees ≤ x ≤ 360 degrees, we can start by making use of the trigonometric identity: sec^2(x) = 1 + tan^2(x).
1. Rewrite the equation using the trigonometric identity:
4tan^2(x) + 12sec(x) + 1 = 0
4tan^2(x) + 12(1 + tan^2(x)) + 1 = 0
2. Simplify the equation:
4tan^2(x) + 12 + 12tan^2(x) + 1 = 0
16tan^2(x) + 13 = 0
3. Rearrange the equation to isolate tan^2(x):
16tan^2(x) = -13
4. Divide both sides by 16:
tan^2(x) = -13/16
5. Take the square root of both sides (remembering to consider both positive and negative roots):
tan(x) = ±√(-13/16)
6. Calculate the square root of -13/16:
tan(x) = ±√(-13)/√(16)
tan(x) = ±√(-13)/4
Note: Since the square root of a negative number is not real, there are no exact solutions for this equation within the given range.
Thus, there are no solutions for the equation 4tan^2(x) + 12sec(x) + 1 = 0 for 0 degrees ≤ x ≤ 360 degrees.
To solve the equation 4 tan^2(x) + 12 sec(x) + 1 = 0 for 0 degrees ≤ x ≤ 360 degrees, we can use a combination of trigonometric identities and algebraic techniques. Here's the step-by-step solution:
Step 1: Rewrite the equation using trigonometric identities.
Since tan^2(x) = sec^2(x) - 1, we can substitute sec^2(x) - 1 for tan^2(x) in the equation:
4(sec^2(x) - 1) + 12 sec(x) + 1 = 0
Simplifying this equation, we get:
4 sec^2(x) + 12 sec(x) - 3 = 0
Step 2: Substitute sec(x) = 1/cos(x) in the equation.
Rewriting the equation in terms of cos(x) instead of sec(x), we have:
4 (1/cos^2(x)) + 12 (1/cos(x)) - 3 = 0
Multiply the entire equation by cos^2(x) to eliminate the denominators, giving:
4 + 12 cos(x) - 3 cos^2(x) = 0
Step 3: Rearrange the equation to the standard quadratic form.
Rearrange the terms to obtain a quadratic equation:
-3 cos^2(x) + 12 cos(x) + 4 = 0
Step 4: Simplify the quadratic equation by dividing the equation by -1.
Since multiplying the entire equation by -1 does not affect the solutions, we can divide the equation by -1 to simplify:
3 cos^2(x) - 12 cos(x) - 4 = 0
Step 5: Solve the quadratic equation.
To solve the quadratic equation, we can either factor it or use the quadratic formula.
By factoring:
(3 cos(x) + 2)(cos(x) - 2) = 0
Setting each factor equal to zero:
3 cos(x) + 2 = 0 or cos(x) - 2 = 0
For 3 cos(x) + 2 = 0:
Subtracting 2 from both sides:
3 cos(x) = -2
Dividing both sides by 3:
cos(x) = -2/3
Using the inverse cosine function:
x = arccos(-2/3)
For cos(x) - 2 = 0:
Adding 2 to both sides:
cos(x) = 2
Using the inverse cosine function:
x = arccos(2)
Step 6: Determine the values of x within the specified range.
To find the values of x within the given range (0 degrees ≤ x ≤ 360 degrees), we need to evaluate the inverse cosine functions.
For x = arccos(-2/3):
Using a scientific calculator, find the primary angle with the same cosine value:
x ≈ 131.8 degrees
Since cosine is an even function, there is a corresponding angle in the second quadrant:
x ≈ 360 - 131.8 ≈ 228.2 degrees
For x = arccos(2):
This equation has no real solutions, as the range of the cosine function is -1 to 1, and 2 is outside this range.
Therefore, the solutions to the equation 4 tan^2(x) + 12 sec(x) + 1 = 0 for 0 degrees ≤ x ≤ 360 degrees are approximately:
x ≈ 131.8 degrees and x ≈ 228.2 degrees.