If $\sin(x) + \cos(x) = \frac14$, then find the value of $\sin^3(x) + \cos^3(x)$

Question

If $$\sin(x) + \cos(x) = \frac14$$ then find the value of $$\sin^3(x) + \cos^3(x)$$

I tried it as follows $\sin^3(x) = \sin(x)(1 -\cos^2(x))$, and then taking out the common, but it became very messy and hard after some time.

Is there a simple solution for this?

Answer 1

You don’t need a lot of Trigonometry for this question. One basic identity and a bit of algebra will do it for you.

Suppose $a +b = 1/4$

Also we know $(a+b)^3 = a^3 + b^3 + 3ab(a+b)$ So, $a^3 + b^3 = (a+b)^3 - 3ab(a+b)$

We know the value of $a+b$ in this question. All we now have to do is find value of $ab$.

As you know,

$(a+b)^2 = a^2 + b^2 + 2ab$

We also know that $sin^2 x + cos^2 x = 1$

So, $1/16 = 1 + 2ab $

We get $ab = -15/32$

If we now calculate the value of $a^3 + b^3$ using all the values we have, we get $47/128$ as final answer.

Answer 2

Write $c=\cos x$ and $s=\sin x$.

Then $$c^3+s^3=(c+s)(c^2-cs+s^2)=(c+s)(1-cs)$$ so if you know $c+s$ and $cs$, you know $c^3+s^3$. But $$(c+s)^2=c^2+2cs+s^2=1+2cs$$ so if you know $c+s$, you know $cs$.

Answer 3

We know $p=s+c$ and $q=s^2+c^2$. Let $r=sc$. Here is another way of using standard relations between symmetric expressions in two variables to reach the answer.

We have that the polynomial $f(x)=(x-s)(x-c)=x^2-px+r=0$ has the roots $s$ and $c$.

Now this means that $0=f(s)+f(c)=q-p^2+2r=0$ so that $2r=p^2-q$ (as you can check directly, so we have evaluated $r$)

Also $0=sf(s)+cf(c)=s^3+c^3-pq+pr$ which gives $s^3+c^3=p(q-r)$, again this is checkable directly against what others have suggested.

This method can be used to give a recurrence relation for sums of powers of $s$ and $c$ using $s^nf(s)+c^nf(c)=0$ with $(s^{n+2}+c^{n+2})-p(s^{n+1}+c^{n+1})+r(s^n+c^n)=0$. It would therefore be particularly helpful if you were asked for $s^4+c^4$ and $s^5+c^5$ too.

If you work it through you will find that this is not algebraically demanding, and doesn't require remembering identities or manipulating substantial expressions in $s$ and $c$.