Solve the following equation for values of theta between 0 and 360 degrees
2sec²theta-3tan theta - 1=0
Solve the following equation for values of theta between 0 and 360 degrees
8sin²theta + 6cos theta - 9= 0
2sec^2θ-3tanθ - 1 = 0
2(1+tan^2θ)-3tanθ-1 = 0
2tan^2θ - 3tanθ + 1 = 0
(2tanθ-1)(tanθ-1) = 0
tanθ = -1/2
tanθ = 1
now finish it off
same idea.
8sin^2θ + 6cosθ - 9= 0
8cos^2θ - 6cosθ + 1 = 0
(4cosθ - 1)(2cosθ - 1) = 0
cosθ = 1/4
cosθ = 1/2
and so on
To solve the equation 2sec²θ - 3tanθ - 1 = 0 for values of θ between 0 and 360 degrees, we can follow these step-by-step instructions:
Step 1: Simplify the equation
Replace sec²θ with 1 + tan²θ using the identity sec²θ = 1 + tan²θ:
2(1 + tan²θ) - 3tanθ - 1 = 0
Step 2: Distribute the 2
2 + 2tan²θ - 3tanθ - 1 = 0
Step 3: Combine like terms
Rearrange the terms in descending order of powers of tanθ:
2tan²θ - 3tanθ + 1 = 0
Step 4: Solve the quadratic equation
Since this equation is a quadratic equation with respect to tanθ, we can use the quadratic formula to solve for tanθ:
tanθ = (-b ± √(b² - 4ac)) / 2a
By comparing the equation 2tan²θ - 3tanθ + 1 = 0 with ax² + bx + c = 0, we can determine that:
a = 2, b = -3, c = 1
Plugging these values into the quadratic formula, we have:
tanθ = (-(-3) ± √((-3)² - 4(2)(1))) / (2(2))
Simplifying further:
tanθ = (3 ± √(9 - 8)) / 4
tanθ = (3 ± √1) / 4
tanθ = (3 ± 1) / 4
This gives us two possible solutions:
tanθ = (3 + 1) / 4 = 4 / 4 = 1
tanθ = (3 - 1) / 4 = 2 / 4 = 1/2
Step 5: Find θ values within the given range
Now we need to find the corresponding θ values for each of the two solutions. To do this, we apply the inverse tangent function (tan⁻¹).
For tanθ = 1:
θ = tan⁻¹(1)
θ ≈ 45 degrees
For tanθ = 1/2:
θ = tan⁻¹(1/2)
θ ≈ 26.565 degrees
Therefore, the solutions for θ within the given range of 0 to 360 degrees are approximately:
θ ≈ 45 degrees and θ ≈ 26.565 degrees.
for the second one, Step 1. Find the value of :
Given,
Divide the above equation by 5:
Step 2. Let
and
.(Consider)
are the solutions of the given equation.
or
Step 3. Take “sin” on both sides, we get
θ
=
2.678
or
2.3562
radians
Explanation:
As
sec
2
θ
=
1
+
tan
2
θ
2
sec
2
θ
+
3
tan
θ
=
1
can be written as
2
+
2
tan
2
θ
+
3
tan
θ
=
1
or
2
tan
θ
+
3
tan
θ
+
1
=
0
i.e.
(
2
tan
θ
+
1
)
(
tan
θ
+
1
)
=
0
Hence either
tan
θ
=
−
1
2
=
tan
(
π
−
0.4636
)
=
tan
2.678
(in radians)
or
tan
θ
=
−
1
=
tan
(
3
π
4
)
=
tan
2.3562
(in radians)
These are results in
[
0
,
π
]