Solve for x in the equation:tan x=2sin x for -180<=x<=180
To solve the equation tan(x) = 2sin(x), we can rewrite it in terms of sine and cosine using the identity: tan(x) = sin(x)/cos(x).
Therefore, we have: sin(x)/cos(x) = 2sin(x).
Multiplying both sides by cos(x), we get: sin(x) = 2sin(x)cos(x).
Dividing both sides by sin(x), we have: 1 = 2cos(x).
Dividing both sides by 2, we get: 1/2 = cos(x).
So, x = arccos(1/2).
Using a calculator to find the inverse cosine of 1/2, we get two possible solutions:
x = 60 degrees or x = 300 degrees.
However, the range given is -180 degrees to 180 degrees. Therefore, the only solution in that range is x = 60 degrees.
To solve the equation tan x = 2sin x, we will use the properties of trigonometric functions. First, we'll rewrite the equation in terms of sine and cosine functions.
Recall that tan x = sin x / cos x.
Therefore, we have:
sin x / cos x = 2sin x
To simplify this equation, we'll multiply both sides by cos x:
sin x = 2sin x * cos x
Next, we can apply the trigonometric identity sin 2x = 2sin x * cos x:
sin x = sin 2x
Now, we have two cases to consider:
Case 1: sin x = sin 2x
In this case, x = 2x + 2kπ or x = π - 2x + 2kπ.
Simplifying each equation separately:
x - 2x - 2kπ = 0 or π - 3x - 2kπ = 0
Combining like terms:
- x - 2kπ = 0 or x - (π / 3) - 2kπ = 0
Rearranging terms:
x = - 2kπ or x = π / 3 - 2kπ
Case 2: sin x = -sin 2x
In this case, x = π - 2x + 2kπ or x = -π - 2x + 2kπ.
Simplifying each equation separately:
x + 2x - π = 0 or x + 2x + π = 0
Combining like terms:
3x - π = 0 or 3x + π = 0
Rearranging terms:
x = π / 3 or x = - π / 3
Therefore, the solution for x in the given equation tan x = 2sin x for -180° ≤ x ≤ 180° is:
x = -2kπ, π / 3 - 2kπ, π / 3, or -π / 3, where k is an integer.