Solve this trigonometric equation.give all positive values of the angle between 0deg and 360deg that will satisfy.
Give any approximate value to the nearest minute only.
3 sin theta - 4 cos theta = 2
theta = 209,55192°
theta = 76,70829°
Sum or difference of sin(θ) and cos(θ) can be solved in the following way, if you are familiar with the sum and difference formulae for sin(a±b).
Given
3sin(θ)-4cos(θ)=2
for a particular φ and amplitude A, we rewrite the above equation as:
Asin(θ-φ)=2
where A=sqrt(3²+4²)=5
or
sin(θ-&phi)=2/5....(1)
=> θ-&phi=asin(2/5)=23.5782°
or 180-23.5782=156.4218°. ...(2)
Expand (1) to get:
sin(θ)cos(&phi)-cos(θ)sin(&phi)=2/5
=> cos(φ)=3/5, sin(φ)=4/5
=> φ=53.1301° or 306.8699°...(3)
So solve for θ from (2) and (3) gives
θ=23.5782+53.1301=76.7083° or θ=156.4218+53.1301=209.5519°
To solve the trigonometric equation 3sin(theta) - 4cos(theta) = 2, we can use the identities for sine and cosine functions and simplify the equation.
First, let's express the equation in terms of either sine or cosine function. We can choose to express it in terms of sine by substituting the identity cos(theta) = sqrt(1 - sin^2(theta)).
3sin(theta) - 4sqrt(1 - sin^2(theta)) = 2
Now, let's square both sides of the equation to eliminate the square roots:
(3sin(theta))^2 - 2(3sin(theta))(4sqrt(1 - sin^2(theta))) + (4sqrt(1 - sin^2(theta))))^2 = 2^2
9sin^2(theta) - 24sin(theta)sqrt(1 - sin^2(theta)) + 16(1 - sin^2(theta)) = 4
Expanding the equation:
9sin^2(theta) - 24sin(theta)sqrt(1 - sin^2(theta)) + 16 - 16sin^2(theta) = 4
Combine like terms:
-7sin^2(theta) - 24sin(theta)sqrt(1 - sin^2(theta)) + 12 = 0
Now, we have a quadratic equation in terms of sin(theta). Let's substitute u = sin(theta) to make it easier to solve:
-7u^2 - 24u√(1 - u^2) + 12 = 0
To solve this quadratic equation, we can use the quadratic formula:
u = [-(-24) ± √((-24)^2 - 4(-7)(12))] / (2(-7))
Simplifying the quadratic formula:
u = [24 ± √(576 + 336)] / (-14)
u = [24 ± √912] / (-14)
Now, we have two possible values for u. To find the values of theta, we need to find the arcsin of these u values. Keep in mind that the range for arcsin is between -90 degrees and 90 degrees. We can use a calculator to find the approximate values to the nearest minute.
For the first value of u:
u = [24 + √912] / (-14) ≈ -0.933
Using the arcsin function on a calculator, arcsin(-0.933) ≈ -68.68 degrees ≈ 291.32 degrees
(Note: If you get a value greater than 90 degrees, subtract it from 180 degrees to get the corresponding angle in the range 0 to 90 degrees.)
For the second value of u:
u = [24 - √912] / (-14) ≈ 0.337
Using the arcsin function on a calculator, arcsin(0.337) ≈ 19.39 degrees
Therefore, the positive values of theta between 0 degrees and 360 degrees that satisfy the equation are approximately:
θ ≈ 19.39 degrees, 291.32 degrees (to the nearest minute)