Find "x" with equation: cos3x + cos2x

Who are the experts?Our certified Educators are real professors, teachers, và scholars who use their academic expertise to lớn tackle your toughest questions. Educators go through a rigorous application process, and every answer they submit is reviewed by our in-house editorial team.

You watching: Find "x" with equation: cos3x + cos2x


*

To find the solutions here, you need lớn use double- & triple-angle formulae for cosine so that you can find what this equation looks lượt thích in terms of only `cosx`.

Let"s see what those formulae are:

` cos2theta = 2cos^2theta - 1 `

`cos3theta = 4cos^3theta - 3costheta`

Now,...


Start your 48-hour free trial lớn unlock this answer & thousands more. Enjoy 12guns.vn ad-không tính tiền và cancel anytime.

See more: Tăng Ram Cho Android Chưa Root Smartphone Chạy Nhanh Mượt Hơn


To find the solutions here, you need lớn use double- & triple-angle formulae for cosine so that you can find what this equation looks like in terms of only `cosx`.

Let"s see what those formulae are:

` cos2theta = 2cos^2theta - 1 `

`cos3theta = 4cos^3theta - 3costheta`

Now, let"s substitute `x` for `theta` & substitute these expressions inlớn the above sầu equation seen here:

`cosx*(2cos^2x- 1)*(4cos^3x-3cosx) = 1`

Let"s multiply all of this out using FOIL first, then by distributing `cosx`:

`cosx(8cos^5x - 6cos^3x - 4cos^3x + 3cosx) = 1`

`8cos^6x - 10cos^4x + 3 cos^2x = 1`

Subtract 1 from both sides:

`8cos^6x-10cos^4x +3cos^2x - 1 = 0`

We need lớn find some way to factor this so we can isolate `cosx`. This expression can be treated as the following polynomial:

`8a^3-10a^2+3a-1 = 0`

Using the fact that rational roots of polynomials take the size `+-p/q` where `p` is an integer factor of the constant term & `q` is a factor of the term of highest degree, we see the possible roots of the above sầu polynomial are `+-1, +-một nửa, +-1/4, and +-1/8`.

See more: Điện Thoại Không Cảm Ứng Được Lúc Không, Cách Khắc Phục Điện Thoại Mất Cảm Ứng

We can start with the factor `(a-1)`. Doing some polynomial long division, we see that it factors inlớn the following equation:

`(a-1)(8x^2-2x+1) = 0`

We no longer have sầu to guess & kiểm tra. We can use the quadratic formula lớn find the other roots:

`a = (-2+-sqrt(4-4*1*8))/16 = (-2+-sqrt(-28))/16=-1/8 +-i sqrt7/8`

Apparently the other two roots are complex. The cấp độ of math required lớn solve sầu for cosines resulting in complex numbers is likely outside the scope of your course, so, I will not solve this equation for the complex roots.

Therefore, we will say we can divide by the expression `8a^2-2a+1` in our original polynomial to lớn yield the following result:

`a-1 = 0`

Therefore, we find the following:

`cos^2x - 1 = 0`

Incidentally, this is equivalent khổng lồ saying the following based on trigonometric identities:

`-sin^2x = 0`

Implying:

`sin^2x = 0`

Considering the only way for `sin^2x` lớn be zero is that `sinx` is zero, as well, we can reduce this to the following equation:

`sinx = 0`

This equation is easy khổng lồ solve sầu, and the solution is well-known! Our result is the following: