If cos^2 theta -6 sin 2 theta =0, one value of theta is a)0 degrees b) 90 degrees c)180 degrees or D)255 degrees.
Please show me how to do it and not just give answer cause I have two more questions that are similar. Thanks so much.
cos^2 Ø - 6sin 2Ø = 0
cos^2Ø - 12sinØcosØ = 0
cosØ(cosØ - 12sinØ) = 0
cos Ø = 0 or 12sinØ = cosØ
Ø = 90 degrees or ......
To solve the equation cos^2θ - 6sin2θ = 0, we can use the trigonometric identity sin2θ = 2sinθcosθ.
Substituting this into the equation, we get:
cos^2θ - 6(2sinθcosθ) = 0
Simplifying further:
cos^2θ - 12sinθcosθ = 0
Using another trigonometric identity cos^2θ = 1 - sin^2θ:
(1 - sin^2θ) - 12sinθcosθ = 0
Rearranging the terms:
1 - sin^2θ - 12sinθcosθ = 0
Now, we can factor out sinθ from the last two terms:
1 - sin^2θ - 12sinθ(1 - sin^2θ) = 0
Expanding and simplifying:
1 - sin^2θ - 12sinθ + 12sin^3θ = 0
Rearranging the terms:
12sin^3θ - sin^2θ - 12sinθ + 1 = 0
We now have a cubic equation in terms of sinθ. Unfortunately, solving cubic equations generally involves complex methods and can be quite complex. However, we can attempt to find the value of θ using a calculator or software, or via estimation.
For the given answer choices, let's substitute each value of θ and check which one satisfies the equation:
a) θ = 0 degrees:
cos^2(0) - 6sin(2(0)) = 0 - 0 = 0 (satisfied)
b) θ = 90 degrees:
cos^2(90) - 6sin(2(90)) = 0 - 6sin(180) = 0 - 0 = 0 (satisfied)
c) θ = 180 degrees:
cos^2(180) - 6sin(2(180)) = 1 - 6sin(360) = 1 - 0 = 1 (not satisfied)
d) θ = 255 degrees:
cos^2(255) - 6sin(2(255)) = 0 - 0 = 0 (satisfied)
From the given answer choices, we can see that options a) and b) satisfy the equation. Therefore, the values of θ that satisfy cos^2θ - 6sin2θ = 0 are θ = 0 degrees and θ = 90 degrees.
To find the value of theta, we can start by simplifying the given equation.
We're given:
cos^2(theta) - 6sin(2theta) = 0
Let's start by replacing sin(2theta) with its equivalent expression using the double angle formula for sin:
cos^2(theta) - 6 * 2 * sin(theta)cos(theta) = 0
Simplifying further:
cos^2(theta) - 12sin(theta)cos(theta) = 0
Now, let's try to factor out common terms. Notice that there is a common factor of cos(theta) in both terms on the left side. Factorizing it gives us:
cos(theta) * (cos(theta) - 12sin(theta)) = 0
Now, we have two possibilities for the equation to be true:
1. cos(theta) = 0
2. cos(theta) - 12sin(theta) = 0
For the first possibility, cos(theta) = 0, we can easily determine the possible values of theta. In this case, theta can be 90 degrees or 270 degrees, as cos(90 degrees) = 0 and cos(270 degrees) = 0.
For the second possibility, cos(theta) - 12sin(theta) = 0, we need to solve the equation. By rearranging the terms, we get:
cos(theta) = 12sin(theta)
Dividing both sides by cos(theta):
1 = 12tan(theta)
Now, we can find the value of theta using the inverse tangent function (tan^(-1)). Taking the inverse tangent of both sides, we get:
tan^(-1)(1) = tan^(-1)(12tan(theta))
Since tangent is a periodic function, we add or subtract multiples of 180 degrees to find the general solutions. In this case, tan^(-1)(1) is equal to 45 degrees.
Therefore, we have the additional possible values for theta:
theta = 45 degrees + k * 180 degrees, where k is an integer.
To summarize:
- The values of theta for cos(theta) = 0 are 90 degrees and 270 degrees.
- The values of theta for cos(theta) - 12sin(theta) = 0 are 45 degrees + k * 180 degrees, where k is an integer.
In this specific case, the answer options are a) 0 degrees, b) 90 degrees, c) 180 degrees, and d) 255 degrees. Based on the solutions we derived, the correct answer is b) 90 degrees.