Find the solutions to the following equation on the interval (-inf, +inf). Round your answer to the nearest ten thousandth. Enter answers in increasing order.
5) sqrt3 sec theta + sec theta cot theta = 0
sqrt3 sec theta + sec theta cot theta = 0
Do you mean
√(3secθ+secθcotθ)=0?
I read it as
√3 secθ+secθcotθ=0
then:
secθ(√3 + cotθ) = 0
secθ = 0 <------ no solution
or
cotθ = -1/√3
tanθ = -1/√3
θ = 150° or 330° OR θ = 5π/6 or 11π/6 radians
looks good at Wolfram:
https://www.wolframalpha.com/input/?i=graph+y+%3D+%E2%88%9A3+sec%CE%B8%2Bsec%CE%B8cot%CE%B8
I will let you state the general solution since the domain is the set of all real θ's
To solve the equation sqrt(3) sec(theta) + sec(theta) cot(theta) = 0, we can start by factoring out the term sec(theta).
Factor the equation:
sec(theta)(sqrt(3) + cot(theta)) = 0
To find the solutions, we need to set each factor equal to zero and solve for theta.
Factor 1: sec(theta) = 0
To find when sec(theta) = 0, we know that sec(theta) is the reciprocal of cos(theta). So, cos(theta) = 1 / sec(theta) = 1 / 0 which is undefined. Therefore, there are no solutions when sec(theta) = 0.
Factor 2: sqrt(3) + cot(theta) = 0
To simplify this term, we can express cot(theta) in terms of sin(theta) and cos(theta). Cotangent is the reciprocal of tangent, so cot(theta) = 1 / tan(theta) = cos(theta) / sin(theta).
sqrt(3) + cos(theta) / sin(theta) = 0
To eliminate the fraction, we can multiply both sides by sin(theta):
sqrt(3)sin(theta) + cos(theta) = 0
Now, let's square the equation to get rid of the square root:
(3sin^2(theta)) + 2sqrt(3)sin(theta)cos(theta) + cos^2(theta) = 0
Using the trigonometric identity sin^2(theta) + cos^2(theta) = 1, we can simplify further:
3 + 2sqrt(3)sin(theta)cos(theta) = 0
Rearranging the equation:
2sqrt(3)sin(theta)cos(theta) = -3
Divide both sides by 2sqrt(3):
sin(theta)cos(theta) = -3 / (2sqrt(3))
Using the double angle formula, we can rewrite sin(theta)cos(theta) as (1/2)sin(2theta):
(1/2)sin(2theta) = -3 / (2sqrt(3))
Now, solve for sin(2theta):
sin(2theta) = -3 / sqrt(3)
Multiply both sides by sqrt(3) and simplify:
sqrt(3)sin(2theta) = -3
Since sin(2theta) is a known value, we can find the two possible values of 2theta and then divide by 2 to find the values of theta.
Case 1: sin(2theta) = -3
This equation does not have any real solutions since the value of sin(theta) cannot exceed 1.
Therefore, there are no solutions to the equation sqrt(3) sec(theta) + sec(theta) cot(theta) = 0 on the interval (-∞, +∞).