Increasing and decreasing ratios

In the last les­son we com­pared val­ues such as $\sin 1$ and $\sin 3$ by look­ing at the graphs. This les­son makes that rea­son­ing pre­cise. We define increas­ing and decreas­ing func­tions, find exactly where sine, cosine and tan­gent increase and decrease, and use those inter­vals to com­pare two val­ues of the same ratio, includ­ing the tricky case where the two angles lie on dif­fer­ent parts of the graph.

Increas­ing and decreas­ing func­tions#

A func­tion $f$ is increas­ing on an inter­val if, for any two num­bers $x_1$, $x_2$ in it,

$$x_1 < x_2 \implies f(x_1) < f(x_2)$$

Its graph climbs as you move to the right. A func­tion is decreas­ing on an inter­val if

$$x_1 < x_2 \implies f(x_1) > f(x_2)$$

and its graph falls as you move to the right.

So, on an inter­val where $f$ is increas­ing, the larger input gives the larger out­put; where it is decreas­ing, the larger input gives the smaller out­put.

Where sine, cosine and tan­gent increase#

Read­ing from the graphs over one full period:

The pat­tern repeats every period. Sine increases up to its peak at $\displaystyle \frac{\pi}{2}$, decreases to its low­est point at $\displaystyle \frac{3\pi}{2}$, then increases again. Cosine decreases from its peak at $0$ to its low­est point at $\pi$, then increases.

Use­ful dec­i­mal val­ues: $\displaystyle \frac{\pi}{2} \approx 1.57$, $\pi \approx 3.14$, $\displaystyle \frac{3\pi}{2} \approx 4.71$, $2\pi \approx 6.28$.

Com­par­ing two val­ues#

1. Locate both angles (in radi­ans) using the dec­i­mals above.
2. If both lie in the same inter­val of increase or decrease, com­pare directly.
3. If not, use signs, or an allied-angle result, to move one angle into the same inter­val as the other.

Exam­ple 1#

Which is greater, $x = \cos 1$ or $y = \cos 2.5$?

Both $1$ and $2.5$ lie in $(0, \pi)$, where cosine is decreas­ing. Since $1 < 2.5$, $\cos 1 > \cos 2.5$. So $x > y$.

Exam­ple 2#

Which is greater, $x = \cos 2.5$ or $y = \cos 3.2$?

Here $2.5$ is in $(0, \pi)$ but $3.2$ is just past $\pi$, in $(\pi, 2\pi)$, so the two angles are on dif­fer­ent sides of the low­est point. Move $3.2$ back into $(0, \pi)$ using $\cos(2\pi - \theta) = \cos\theta$:

$$\cos 3.2 = \cos(2\pi - 3.2) \approx \cos 3.08$$

Now $2.5$ and $3.08$ both lie in $(0, \pi)$, where cosine decreases, and $2.5 < 3.08$. So $\cos 2.5 > \cos 3.08 = \cos 3.2$, that is, $x > y$. (Check: $\cos 2.5 \approx -0.80$ and $\cos 3.2 \approx -1.00$.)

Exam­ple 3#

Which is greater, $x = \sin 2.5$ or $y = \sin 3.2$?

$2.5$ lies in $(0, \pi)$, so $\sin 2.5 > 0$. $3.2$ lies in $(\pi, 2\pi)$, so $\sin 3.2 < 0$. A pos­i­tive num­ber is greater than a neg­a­tive one: $x > y$.

Exam­ple 4#

Which is greater, $\sin 1$ or $\cos 1$?

Write $\displaystyle \cos 1 = \sin\left(\frac{\pi}{2} - 1\right) \approx \sin 0.57$. Both $0.57$ and $1$ lie in $\displaystyle \left(0, \frac{\pi}{2}\right)$, where sine increases, so $\sin 1 > \sin 0.57 = \cos 1$.

Prac­tice#

1. On which inter­vals within $[0, 2\pi]$ is $\sin x$ decreas­ing, and on which is $\cos x$ increas­ing?
2. Which is greater, $\sin 0.5$ or $\sin 1.2$?
3. Which is greater, $\cos 0.4$ or $\cos 1.4$?
4. Which is greater, $\sin 2$ or $\sin 3$?
5. Which is greater, $\tan 1$ or $\tan 1.5$?
6. Which is greater, $\cos 3.5$ or $\cos 4$?
7. Which is greater, $\sin 3$ or $\sin 4$?
8. Which is greater, $\cos 1.2$ or $\sin 1.2$?

Answers#