find the values of thiter between 0 degrees and 180 degrees such that 2cos thiter= 3sin thiter
To find the values of θ between 0 degrees and 180 degrees such that 2cosθ = 3sinθ, we can rearrange the equation:
2cosθ = 3sinθ
cosθ = (3/2)sinθ
We know that cosθ/sinθ is equal to cotθ, so we can rewrite the equation as:
cotθ = (3/2)
Now, we can determine the values of θ by taking the inverse cotangent of (3/2):
θ = arccot(3/2)
Using a calculator, we find that arccot(3/2) is approximately 33.69 degrees.
Therefore, the values of θ between 0 degrees and 180 degrees such that 2cosθ = 3sinθ are approximately 33.69 degrees.
To solve the equation 2cosθ = 3sinθ, we can use the trigonometric identity: cosθ = sin(90° - θ).
Substituting this into the equation, we have:
2sin(90° - θ) = 3sinθ
Expanding the equation, we get:
2sin90°cosθ - 2sinθcos90° = 3sinθ
2cosθ - 2 = 3sinθ
2cosθ - 3sinθ = 2
To solve this equation, we'll need to use trigonometric identities. Let's divide every term by 2:
(cosθ)/2 - (3sinθ)/2 = 1
Now, let's square both sides of the equation to eliminate the trigonometric functions:
((cosθ)/2)^2 - 2((cosθ)/2)((3sinθ)/2) + ((3sinθ)/2)^2 = 1^2
(cosθ)^2/4 - cosθsinθ + 9(sinθ)^2/4 = 1
4(cosθ)^2 - 8cosθsinθ + 9(sinθ)^2 = 4
Rearranging the equation, we get:
4(cosθ)^2 - 8cosθsinθ + 9(sinθ)^2 - 4 = 0
Using a trigonometric identity, we can simplify the equation further:
4(cosθ - 3sinθ/2)^2 = 4
(cosθ - 3sinθ/2)^2 = 1
Taking the square root of both sides, we have:
cosθ - 3sinθ/2 = ±1
Now, let's consider each case:
Case 1: cosθ - 3sinθ/2 = 1
cosθ - 3sinθ/2 = 1
Multiply every term by 2:
2cosθ - 3sinθ = 2
Rearranging the equation, we get:
2cosθ = 2 + 3sinθ
Dividing every term by 2:
cosθ = 1 + (3/2)sinθ
Using the identity cosθ = sin(90° - θ), we rewrite the equation as:
sin(90° - θ) = 1 + (3/2)sinθ
sin(θ - 90°) = 1 + (3/2)sinθ
By simplifying the equation, we have:
sin(θ - 90°) = 1 + (3/2)sinθ
sin(-θ) = 1 + (3/2)sinθ
By multiplying every term by -1, we get:
-sinθ = -1 - (3/2)sinθ
Rearranging the equation, we have:
(1/2)sinθ = -1
sinθ = -2
However, the range of sine function is [-1, 1], and since -2 is not in that range, there are no solutions for this case.
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Case 2: cosθ - 3sinθ/2 = -1
cosθ - 3sinθ/2 = -1
Multiply every term by 2:
2cosθ - 3sinθ = -2
Rearranging the equation, we get:
2cosθ = -2 + 3sinθ
Dividing every term by 2:
cosθ = -1 + (3/2)sinθ
Using the identity cosθ = sin(90° - θ), we rewrite the equation as:
sin(90° - θ) = -1 + (3/2)sinθ
sin(θ - 90°) = -1 + (3/2)sinθ
By simplifying the equation, we have:
sin(θ - 90°) = -1 + (3/2)sinθ
sin(-θ) = -1 + (3/2)sinθ
By multiplying every term by -1, we get:
-sinθ = 1 - (3/2)sinθ
Adding (3/2)sinθ to both sides, we have:
(1/2)sinθ = 1
sinθ = 2
Again, the range of sine function is [-1, 1], and since 2 is not in that range, there are no solutions for this case.
Therefore, the equation 2cosθ = 3sinθ has no solutions between 0 degrees and 180 degrees.