2 General definitions of trigonometric functions

We now define the trigonometric functions in a more general way than in terms of ratios of sides of a right-angled triangle. To do this we consider a circle of unit radius whose centre is at the origin of a Cartesian coordinate system and an arrow (or radius vector ) $OP$ from the centre to a point $P$ on the circumference of this circle. We are interested in the angle $\theta$ that the arrow makes with the positive $x$ -axis. See Figure 20.

Figure 20

Imagine that the vector $OP$ rotates in anti-clockwise direction . With this sense of rotation the angle $\theta$ is taken as positive whereas a clockwise rotation is taken as negative. See examples in Figure 21.

Figure 21

2.1 The sine and cosine of an angle

For $0\le \theta \le \frac{\pi }{2}$ (called the first quadrant) we have the following situation with our unit radius circle. See Figure 22.

Figure 22

The projection of $OP$ along the positive $x-$ axis is $OQ$ . But, in the right-angled triangle $OPQ$

$cos\theta =\frac{OQ}{OP}\text{or}OQ=OPcos\theta$

and since $OP$ has unit length $cos\theta =OQ\left(3\right)$

Similarly in this right-angled triangle

$sin\theta =\frac{PQ}{OP}\text{or}PQ=OPsin\theta$

but $PQ=OR$ and $OP$ has unit length

so $sin\theta =OR\left(4\right)$

Equation (3) tells us that we can interpret $$ cos θ as the projection of $OP$ along the positive $x$ -axis and $$ sin θ as the projection of $OP$ along the positive $y$ -axis .

We shall use these interpretations as the definitions of $sin\theta$ and $cos\theta$ for any values of $\theta$ .

2.2 Sine and cosine in the four quadrants

First quadrant $\left(0\le \theta \le 90^{\circ }\right)$

Figure 23

It follows from Figure 23 that $cos\theta$ decreases from 1 to 0 as $OP$ rotates from the horizontal position to the vertical, i.e. as $\theta$ increases from $0^{\circ }$ to $90^{\circ }$ .

$sin\theta =OR$ increases from 0 (when $\theta =0$ ) to 1 (when $\theta =90^{\circ }$ ).

Second quadrant $\left(90^{\circ }\le \theta \le 180^{\circ }\right)$

Referring to Figure 24, remember that it is the projections along the positive $x$ and $y$ axes that are used to define $cos\theta$ and $sin\theta$ respectively. It follows that as $\theta$ increases from $90^{\circ }$ to $180^{\circ },cos\theta$ decreases from 0 to $-1$ and $sin\theta$ decreases from 1 to 0.

Figure 24

Considering for example an angle of $135^{\circ }$ , referring to Figure 25, by symmetry we have:

$sin135^{\circ }=OR=sin45^{\circ }=\frac{1}{\sqrt{2}}cos135^{\circ }=O{Q}_2=-O{Q}_1=-cos45^{\circ }=-\frac{1}{\sqrt{2}}$

Figure 25

Task!

Without using a calculator write down the values of

$sin120^{\circ },sin150^{\circ },cos120^{\circ },cos150^{\circ }$ , $tan120^{\circ },tan150^{\circ }$ .

(Note that $tan\theta \equiv \frac{sin\theta }{cos\theta }$ for any value of $\theta$ .)

Answer

Third quadrant $\left(180^{\circ }\le \theta \le 270^{\circ }\right)$ .

Figure 26

Task!

Using the projection definition write down the values of $cos270^{\circ }$ and $sin270^{\circ }$ .

Answer

From the results of the last Task, with $\theta =180^{\circ }+x$ (see Figure 27) we obtain for all $x$ the relations:

$sin\theta =sin\left(180+x\right)=OR=-O{R}^{\prime }=-sinxcos\theta =cos\left(180+x\right)=OQ=-O{Q}^{\prime }=-cosx$

$\text{Hence}tan\left(180+x\right)=\frac{sin\left(180^{\circ }+x\right)}{cos\left(180^{\circ }+x\right)}=\frac{sinx}{cosx}=+tanx$ for all $x$ .

Figure 27 :

Fourth quadrant $\left(270^{\circ }\le \theta \le 360^{\circ }\right)$

Figure 28

From Figure 28 the results in Key Point 10 should be clear.

Task!

Write down (without using a calculator) the values of

$sin300^{\circ },sin\left(-60^{\circ }\right),cos330^{\circ },cos\left(-30^{\circ }\right).$

Describe the behaviour of $cos\theta$ and $sin\theta$ as $\theta$ increases from $270^{\circ }$ to $360^{\circ }$ .

Answer

Rotation beyond the fourth quadrant $\left(360^{\circ }<\theta \right)$

If the vector $OP$ continues to rotate around the circle of unit radius then in the next complete rotation $\theta$ increases from $360^{\circ }$ to $720^{\circ }$ . However, a $\theta$ value of, say, $405^{\circ }$ is indistinguishable from one of $45^{\circ }$ (just one extra complete revolution is involved).

So $sin\left(405^{\circ }\right)=sin45^{\circ }=\frac{1}{\sqrt{2}}\text{and}cos\left(405^{\circ }\right)=cos45^{\circ }=\frac{1}{\sqrt{2}}$

In general $sin\left(360^{\circ }+x^{\circ }\right)=sin{x}^{\circ },cos\left(360^{\circ }+x^{\circ }\right)=cos{x}^{\circ }$