Approximate the solutions (to three decimal places) of the given equation in
the interval
(- pi/2, pi/2)
6 sin 2x − 8 cos x + 9 sin x = 6
Can you help with this please? I am studying for my Pre Calculus 1 exam and am stuck on this question...!
The answer options are
a. x = 0.398
b. x = 0.094
c. x = 0.139
d. x = 0.730
e. x = 1.336
6 sin 2x − 8 cos x + 9 sin x = 6
12sinxcosx - 8cosx + 9sinx - 6 = 0
4cosx(3sinx - 2) + 3(3sinx - 2) = 0
(3sinx - 2)(4cosx + 3) = 0
sinx = 2/3 OR cosx = -3/4, the last one is not possible
sinx = 2/3
x = .7297..
which matches b)
Make sure your calculator is set to radians for these type of questions, unless the question specifies degrees.
To approximate the solutions of the given equation within the interval (-π/2, π/2), follow these steps:
Step 1: Simplify the equation.
Rewrite the given equation using trigonometric identities:
6sin2x − 8cosx + 9sinx = 6
Step 2: Use the double-angle identity.
Convert sin2x to a form that involves only a single trigonometric function. The double-angle identity for sine is:
sin2x = 2sinxcosx
Applying this identity, the equation becomes:
12sinxcosx − 8cosx + 9sinx = 6
Step 3: Rearrange the equation.
Combine like terms and bring all terms to one side of the equation:
12sinxcosx + 9sinx − 8cosx − 6 = 0
Step 4: Factoring by grouping.
Rearrange the terms so that you can factor by grouping:
(12sinx − 6)(cosx + 9) − 8cosx = 0
Step 5: Solve for sinx = 0.
Set each factor equal to zero and solve for x:
12sinx − 6 = 0
sinx = 6/12
sinx = 0.5
x = arcsin(0.5)
Since we are looking for solutions within the interval (-π/2, π/2), the value for x = arcsin(0.5) is within this range.
Step 6: Solve for cosx = -9.
Next, consider the second factor:
cosx + 9 = 0
cosx = -9
Since the cosine of any angle is always between -1 and 1, there are no solutions for cosx = -9 within the given interval.
Step 7: Solve for sinx > 0.
When sinx - 0.5 > 0 (as found in step 5), this implies that sinx is positive. Within the interval (-π/2, π/2), there is only one solution: x = arcsin(0.5).
Step 8: Approximate the solution.
Use a calculator to find the approximate value of arcsin(0.5) within the given interval:
x ≈ 0.523
Therefore, from the given answer options, the closest approximation to the solution of the equation within the interval (-π/2, π/2) is:
a. x = 0.398.