Filter a family

To solve $\sin x=1/2$ to $[0,4\pi)$, first obtain the families $x=\pi/6+2k\pi$ and $x=5\pi/6+2k\pi$. Then impose the limits of the range. The solutions are:

$$\frac\pi6,\quad\frac{5\pi}6,\quad\frac{13\pi}6,\quad\frac{17\pi}6.$$

The inclusion sign of the ends matters. After squaring or multiplying by a variable expression, substitute the answers into the initial equation.

A combination of sine and cosine

To solve $a\sin x+b\cos x=c$, with $(a,b)\ne(0,0)$, set $R=\sqrt{a^2+b^2}$ and choose $\varphi$ with:

$$\cos\varphi=\frac aR,\qquad\sin\varphi=\frac bR.$$

This is possible because the squares of these two numbers add up to 1. Using the addition formula:

$$a\sin x+b\cos x=R\sin(x+\varphi).$$

The equation has a real solution exactly when $|c|\le R$. For example, $\sin x+\cos x=1$ is equivalent to $\sin(x+\pi/4)=\sqrt2/2$. Returns $x=2k\pi$ or $x=\pi/2+2k\pi$.

Sums that become products

A sum of sines or cosines can be transformed into a product. At $\sin3x+\sin x=0$, we get $2\sin2x\cos x=0$. Solve each zero factor and bring the families together, removing repetitions. Do not divide by a factor that can be zero.

Fourth powers

Squaring $\sin^2x+\cos^2x=1$ gives:

$$\sin^4x+\cos^4x=1-2\sin^2x\cos^2x =1-\frac12\sin^22x.$$

Therefore, $\sin^4x+\cos^4x=A$ is equivalent to $\sin^22x=2(1-A)$. There is only a solution if $1/2\le A\le1$. Extract the two possible signs of the sine and solve.

Sixth powers

Use $u^3+v^3=(u+v)^3-3uv(u+v)$ with $u=\sin^2x$ and $v=\cos^2x$. Like $u+v=1$:

$$\sin^6x+\cos^6x=1-\frac34\sin^22x.$$

The equation with right-hand side $A$ has a solution if, and only if, $1/4\le A\le1$.

In everyday life

  • Spring. The position of a mass on a spring, $a\operatorname{sen}\omega t + b\cos\omega t$, becomes a single sinusoid of amplitude $\sqrt{a^2 + b^2}$.
  • Waves that add up. Two sounds of the same frequency out of phase add up into a single wave; the amplitude depends on the lag.
  • Answers in the range. In physical problems, only solutions in a certain range are useful, such as launch angles between $0^\circ$ and $90^\circ$.
$a\operatorname{sen}t + b\cos t$ on a spring: a single oscillation of amplitude $\sqrt{a^2 + b^2}$.

Exercises with resolution

1. Solve exercise C.243 from reference: $\sin^4x+\cos^4x=3/4$.

Check resolution 1

Identity gives $\sin^22x=1/2$, then $2x=\pi/4+k\pi/2$. Logo $x=\pi/8+k\pi/4$, $k\in\mathbb Z$.

2. Solve exercise C.244 from reference: $\sin^6x+\cos^6x=7/16$.

Check resolution 2

$1-\frac34\sin^22x=7/16$ gives $\sin^22x=3/4$. So, $2x=\pm\pi/3+k\pi$, or $x=\pm\pi/6+k\pi/2$.

In the interactive book

The same subjects, with all the demonstrations and the complete list of exercises, are in the interactive Trigonometry book:

This class is a shorter introduction to the topic; In the book, the theory is accompanied by solved and proposed exercises from the chapter.

Stage index · ← Previous class · Next class →