Solve the trigonometric equation 3sinθ=1.2 for 0≤θ≤4π
3sin = 1.2
sinØ = .4
(set your calculator to RAD)
Ø = .41152 or π-.41152 which is 2.73008
but you wanted from 0 to 4π
so add a period , or 2π, to each of the above answers
Ø = .41152 , 2.73008 , .6947, and 9.01326
To solve the trigonometric equation 3sinθ = 1.2, we need to find the values of θ that satisfy this equation within the given range of 0≤θ≤4π.
We know that sinθ = opposite/hypotenuse in a right triangle. In this case, we are given that 3sinθ = 1.2.
To isolate sinθ, we divide both sides of the equation by 3:
sinθ = 1.2/3
sinθ = 0.4
Now, we need to find the values of θ that satisfy this equation. We can do this by taking the inverse sine (or arcsin) of both sides of the equation:
θ = arcsin(0.4)
Using a calculator or a trigonometric table, we find that arcsin(0.4) is approximately 0.4115 radians (or about 23.6 degrees).
However, we need to find all the possible values of θ within the given range of 0≤θ≤4π. To do this, we can add integer multiples of 2π to the initial solution:
θ = 0.4115 + 2πn where n is an integer.
Now, we need to find the values of θ within the given range. Since 0≤θ≤4π, we can substitute different integer values for n and check if the resulting values of θ fall within the specified range. We continue this process until we find all possible solutions.
For n = 0, we have:
θ = 0.4115 + 2π(0) = 0.4115
For n = 1, we have:
θ = 0.4115 + 2π(1) = 6.1706
For n = 2, we have:
θ = 0.4115 + 2π(2) = 12.9297
For n = 3, we have:
θ = 0.4115 + 2π(3) = 19.6889
Since 19.6889 is greater than 4π, we stop here.
So, the solutions to the trigonometric equation 3sinθ = 1.2 within the given range are:
θ = 0.4115, 6.1706, 12.9297
Note: The values of θ are in radians.