fidn all values of tehta in [0, 360) that satisfy the equation.
√(3)tan(θ/2)=-1
I have four choices 300, 330, 150 , 120
and i got 120
√3 tan(θ/2) = -1
tan θ/2 = -1/√3
θ/2 = 150 or 330
θ = 300 or 660
θ=660 is equivalent to θ=300, since 660=300+360
tan 120 = -√3, not -1/√3
so the answer is 330
How am I going to help you if you don't read what I wrote?
i got it now. thanks
To find all values of theta in the interval [0, 360) that satisfy the equation √(3)tan(θ/2) = -1, you can follow these steps:
Step 1: Isolate the term containing the trigonometric function.
Start by squaring both sides of the equation to eliminate the square root:
(√(3)tan(θ/2))^2 = (-1)^2
This simplifies to:
3tan^2(θ/2) = 1
Step 2: Solve the equation for theta/2.
Divide both sides by 3:
tan^2(θ/2) = 1/3
Now, take the square root of both sides:
tan(θ/2) = ± √(1/3)
Step 3: Find the angle(s) where the tangent function matches the value of √(1/3).
Since we are looking for solutions in the interval [0, 360), we need to consider positive and negative values of √(1/3):
± √(1/3) ≈ ± 0.577
To find the angles, we can use the inverse tangent function (tan^(-1) or arctan). However, be cautious when using the inverse tangent function, as it returns values only in the range [-90, 90) degrees or (-π/2, π/2) radians. We'll need to take multiple steps to find all solutions.
Start by finding an angle (θ/2) whose tangent is approximately 0.577 using the inverse tangent function:
θ/2 ≈ tan^(-1)(0.577)
θ/2 ≈ 30.96° or 0.538 rad
To find other angles, we'll add or subtract multiples of 180° (π radians) to this result. This is because tangent has a period of 180°:
θ/2 = 30.96° + 180°k or 0.538 rad + πk, where k is an integer.
Finally, multiply theta/2 by 2 to solve for theta:
θ = 2(30.96° + 180°k) or 2(0.538 rad + πk)
Now, substitute different values of k to find all the solutions within the given interval [0, 360).
For k = 0:
θ = 2(30.96°) = 61.92°
For k = 1:
θ = 2(30.96° + 180°) = 391.92°
(Note: This value is outside the given interval [0, 360), so it is excluded.)
Therefore, the only value of theta that satisfies the equation in the given interval is 61.92° (or approximately 62°).
Note: None of the options you provided (300, 330, 150, 120) are correct. The correct answer is 61.92° (or approximately 62°).