Find all solutions in the interval [0, pi], 2sec^2x+tan^2x-3=0
2sec^2x+tan^2x-3=0
2/cos^2x + sin^2x/cos^2x - 3 = 0
multiply by cos^2x
2 + sin^2x - 3cos^2x = 0
2 + (1 - cos^2x) - 3cos^2x = 0
3 = 4 cos^2x
cos^2 = 3/4
cosx = ± √3/2
x = 30° , 150° , 210° , or 330°
x = π/6, 5π/6 , 7π/6, or 11π/6
Thank u, helped a lot!☺
To solve the equation 2sec^2(x) + tan^2(x) - 3 = 0 in the interval [0, π], follow these steps:
Step 1: Rewrite the equation using trigonometric identities.
Recall that sec^2(x) = 1 + tan^2(x), which can be substituted into the equation:
2(1 + tan^2(x)) + tan^2(x) - 3 = 0
Simplify further:
2 + 2tan^2(x) + tan^2(x) - 3 = 0
3tan^2(x) - 1 = 0
Step 2: Factor the quadratic equation.
Since this is a quadratic equation in terms of tan^2(x), let's substitute tan^2(x) with a variable, t.
3t - 1 = 0
Step 3: Solve for t.
Solving the equation 3t - 1 = 0 for t gives:
t = 1/3
Step 4: Substitute back to find the values of tan(x).
Since tan^2(x) = t, substitute t with 1/3:
tan^2(x) = 1/3
Taking the square root of both sides:
tan(x) = ±√(1/3)
Step 5: Find the values of x in the interval [0, π].
Since the tan function is positive in both the first and third quadrants, you need to find the angles where tan(x) = √(1/3) in the interval [0, π].
To find the angles, use the inverse tangent function:
x = atan(√(1/3))
Calculating the value gives:
x ≈ 0.6155 radians
Since tan(x) is positive in both the first and third quadrants, the other possible angle is (π - x):
π - x ≈ 2.5265 radians
Therefore, the solutions in the interval [0, π] are:
x ≈ 0.6155 radians and x ≈ 2.5265 radians
To find all solutions in the interval [0, pi] to the equation 2sec^2(x) + tan^2(x) - 3 = 0, we can follow these steps:
Step 1: Start with the given equation: 2sec^2(x) + tan^2(x) - 3 = 0.
Step 2: Use the identity sec^2(x) = 1 + tan^2(x) to rewrite the equation: 2(1 + tan^2(x)) + tan^2(x) - 3 = 0.
Step 3: Simplify the equation: 2 + 2tan^2(x) + tan^2(x) - 3 = 0.
Step 4: Combine like terms: 3tan^2(x) - 1 = 0.
Step 5: Add 1 to both sides of the equation: 3tan^2(x) = 1.
Step 6: Divide both sides of the equation by 3: tan^2(x) = 1/3.
Step 7: Take the square root of both sides of the equation: tan(x) = ±√(1/3).
Step 8: Solve for x by taking the inverse tangent (or arctan) of both sides of the equation: x = arctan(±√(1/3)).
Step 9: The inverse tangent function has multiple solutions, so we need to consider both the positive and negative square root values.
Step 10: Use a calculator to find the approximate values for x.
For the positive square root: x ≈ 0.615, 1.527.
For the negative square root: x ≈ 2.618, 3.530.
Step 11: Check if each solution falls within the given interval [0, pi].
From the positive square root solutions, 0.615 and 1.527 are within the interval.
From the negative square root solutions, 2.618 and 3.530 are outside the interval.
So, the solutions in the interval [0, pi] are x ≈ 0.615 and x ≈ 1.527.