How do i solve this equation over the interval [0degrees,360)
cot(theta)+2csc(theta)=5
change it to
cosØ/sinØ + 2/sinØ = 5
multiply by sinØ
cosØ + 2 = 5sinØ
square both sides
cos^2 Ø + 4cosØ + 4 = 25sin^2 Ø = 25(1 - cos^2 Ø)
cos^2Ø + 4cosØ + 4 = 25 - 25cos^2Ø
26cos^2Ø + 4cosØ - 21 = 0
cosØ = (-4 ± √2200)/52
= .82508 or -.978926
if cosØ = .82508, then Ø = 34.4° or 325.6°
if cosØ =-.978926, Ø = 168.2° or 191.8°
BUT... since we squared our equation all answers have to be verified in the original equation
x = 34.4° , LS = 5.0005 , good
x = 325.6 , LS = -3.5 ≠ RS , no good
x = 168.2 , LS = 4.999 , good
x = 191.8 , LS = -5 , no good
so x = 34.4° or x = 168.2°
To solve the equation cot(theta) + 2csc(theta) = 5 over the interval [0 degrees, 360 degrees), you can follow these steps:
Step 1: Rewrite cot(theta) and csc(theta) in terms of sin(theta) and cos(theta).
Using the definitions of cot(theta) and csc(theta):
cot(theta) = cos(theta) / sin(theta)
csc(theta) = 1 / sin(theta)
Step 2: Rewrite the equation using sin(theta) and cos(theta).
cos(theta) / sin(theta) + 2 / sin(theta) = 5
Step 3: Find a common denominator for the fractions.
(cos(theta) + 2) / sin(theta) = 5
Step 4: Cross-multiply the equation.
cos(theta) + 2 = 5 * sin(theta)
Step 5: Rewrite sin(theta) in terms of cos(theta) using the Pythagorean identity.
1 - cos^2(theta) = sin^2(theta)
1 - cos^2(theta) = 1 - cos^2(theta) / cos^2(theta)
1 - cos^2(theta) = cos^2(theta) - cos^4(theta)
Step 6: Rearrange the equation to solve for cos(theta).
2cos^4(theta) - 6cos^2(theta) + 3 = 0
Step 7: Factor the equation.
(cos^2(theta) - 1)(2cos^2(theta) - 3) = 0
Step 8: Solve for cos(theta) using each factor.
cos^2(theta) - 1 = 0
cos^2(theta) = 1
cos(theta) = ±1
2cos^2(theta) - 3 = 0
cos^2(theta) = 3/2
cos(theta) = ±√(3/2)
Step 9: Determine the values of theta using the unit circle or a calculator.
For cos(theta) = 1:
theta = 0 degrees
For cos(theta) = -1:
theta = 180 degrees
For cos(theta) = √(3/2):
theta = 30 degrees, 330 degrees
For cos(theta) = -√(3/2):
theta = 150 degrees, 210 degrees
Therefore, the solutions to the equation cot(theta) + 2csc(theta) = 5 over the interval [0 degrees, 360 degrees) are:
theta = 0 degrees, 30 degrees, 150 degrees, 180 degrees, 210 degrees, 330 degrees.
To solve the given equation, cot(theta) + 2csc(theta) = 5, over the interval [0 degrees, 360 degrees), we'll follow these steps:
Step 1: Simplify the equation using trigonometric identities.
Step 2: Transform the equation into a single trigonometric function.
Step 3: Solve the equation using algebraic techniques.
Step 4: Check the solutions within the given interval.
Let's go through each step in detail:
Step 1: Simplify the equation using trigonometric identities.
Recall the following identities:
- cot(theta) = cos(theta) / sin(theta)
- csc(theta) = 1 / sin(theta)
Rewriting the given equation using these identities, we get:
cos(theta) / sin(theta) + 2 / sin(theta) = 5
Step 2: Transform the equation into a single trigonometric function.
To eliminate the fractions in the equation, we can multiply the entire equation by sin(theta). This gives us:
cos(theta) + 2 = 5sin(theta)
Step 3: Solve the equation using algebraic techniques.
Rearranging the equation, we get:
cos(theta) - 5sin(theta) = -2
Now, we'll square both sides of the equation to eliminate the trigonometric functions, yielding:
cos^2(theta) - 10sin(theta)cos(theta) + 25sin^2(theta) = 4
Recall the Pythagorean identity: sin^2(theta) + cos^2(theta) = 1. We can substitute this identity into the equation above:
1 - 10sin(theta)cos(theta) + 25sin^2(theta) = 4
Rearranging this equation, we have:
25sin^2(theta) - 10sin(theta)cos(theta) - 3 = 0
Step 4: Check the solutions within the given interval.
Now, we can solve this quadratic equation for sin(theta). Once we find the values of sin(theta), we can determine the corresponding values of theta.
To find the solutions, we can apply the quadratic formula:
sin(theta) = [-(-10) ± sqrt((-10)^2 - 4(25)(-3))] / 2(25)
Simplifying this equation, we have:
sin(theta) = [10 ± sqrt(100 + 300)] / 50
sin(theta) = (10 ± sqrt(400)) / 50
sin(theta) = (10 ± 20) / 50
There are two possibilities:
1. sin(theta) = (10 + 20) / 50 = 30 / 50 = 3 / 5
2. sin(theta) = (10 - 20) / 50 = -10 / 50 = -1 / 5
Now, to find the corresponding angles in the interval [0 degrees, 360 degrees), we can use the inverse sine function (sin^-1) to find theta.
1. sin^-1(3/5) ≈ 36.87 degrees (approximately 36.87 degrees)
2. sin^-1(-1/5) ≈ -11.54 degrees (approximately -11.54 degrees)
Therefore, the solutions within the interval [0 degrees, 360 degrees) for the equation cot(theta) + 2csc(theta) = 5 are approximately 36.87 degrees and -11.54 degrees.