Solve the equation
cos θ + 4 cos 2θ = 3,
giving all solutions in the interval 0◦ ≤ θ ≤ 180◦
how do we solve it ?
.

Question

Delilah Aguilar · Accepted Answer

Let #x=cos theta# and remember #cos 2 theta= 2 cos ^2 theta - 1 = 2 x^2-1# so our equation becomes

#x + 4 (2x^2-1)=3 # or #x+8x^2-4=3# or #0=8x^2+x-7=(x+1)(8x-7)# so #cos theta = -1 or cos theta=7/8 # so

#theta = 180^circ +360^circ k or theta=pm text{Arc}text{cos}(7/8) + 360^circ k quad # integer #k#

and you can work out the ones in the range.

Asher Friedman · Answer

undefined To solve the equation $ \cos \theta + 4 \cos 2\theta = 3 $ for $ 0^\circ \leq \theta \leq 180^\circ $:

1. Apply the double-angle identity: $ \cos 2\theta = 2\cos^2 \theta - 1 $.
2. Substitute $ \cos 2\theta $ with $ 2\cos^2 \theta - 1 $ in the equation.
3. Rearrange the equation and combine like terms to form a quadratic equation in terms of $ \cos \theta $.
4. Solve the quadratic equation for $ \cos \theta $ using standard methods like factoring, completing the square, or using the quadratic formula.
5. After finding the values of $ \cos \theta $, determine corresponding values of $ \theta $ within the given interval $ 0^\circ \leq \theta \leq 180^\circ $.

Following these steps will yield the solutions for the given equation within the specified interval.

Solve the equation cos θ + 4 cos 2θ = 3, giving all solutions in the interval 0◦ ≤ θ ≤ 180◦ how do we solve it ? .