Trigonometric Ratios in Quadrants by a Unit Circle

Trigonometric Ratios

Circles with Centre $(0,0)$

Consider a circle with centre $(0,0)$ and radius $r$ units. Suppose $(x,y)$ is any point on this circle.
Using this distance formula;
\( \begin{align} \displaystyle
\sqrt{(x-0)^2+(y-0)^2} &= r \\
\therefore x^2+y^2 &= r^2
\end{align} \)
$x^2+y^2 = r^2$ is the equation of a circle with centre $(0,0)$ and radius $r$.
The equation of the unit circle is $x^2+y^2=1$.
\( \begin{align}
\dfrac{x}{r} &= \cos \theta &\therefore x &= r \cos \theta \\
\dfrac{y}{r} &= \sin \theta &\therefore y &= r \sin \theta
\end{align} \)


In the first quadrant, the coordinates of $A$ are $(\cos \theta,\sin \theta)$.
$\dfrac{\sin \theta}{\cos \theta} = \tan \theta$

In the second quadrant, the coordinates of $B$ are $(-\cos \theta,\sin \theta)$.
$\dfrac{\sin \theta}{-\cos \theta} = -\tan \theta$

In the third quadrant, the coordinates of $C$ is $(-\cos \theta,-\sin \theta)$.
$\dfrac{-\sin \theta}{-\cos \theta} = \tan \theta$

In the fourth quadrant, the coordinates of $D$ are $(\cos \theta,-\sin \theta)$.
$\dfrac{\sin \theta}{-\cos \theta} = -\tan \theta$

In summary, the signs of $\sin \theta$, $\cos \theta$ and $\tan \theta$ in each quadrant are;

From the investigation above, you should notice that:

  • $\sin \theta$, $\cos \theta$ and $\tan \theta$ are positive in the first quadrant
  • Only $\sin \theta$ is positive in the second quadrant
  • Only $\tan \theta$ is positive in the third quadrant
  • Only $\cos \theta$ is positive in the fourth quadrant

We can use a letter to show which trigonometric ratios are positive in each quadrant. The A stands for $\text{all}$ of the ratios. You might like to remember them using;
“All Silly Turtles Crawl” or “All Stations To Central”.

Periodicity of Trigonometric Ratios

Since there are $2\pi$ radians or $360^{\circ}$ in a full revolution, if we add any integer multiple of $2 \pi$ to $\theta$, in radians, then the position of the point on the unit circle is unchanged.
\( \begin{aligned}
\cos \theta &= \cos (\theta + 2 \pi) \\
&= \cos (\theta + 4 \pi) \\
&= \cos (\theta + 6 \pi) \\
&= \cos (\theta + 8 \pi)
&\cdots
\end{aligned} \)
For example, $\cos 5\pi = \cos 3\pi = \cos \pi $.
\( \begin{aligned}
\sin \theta &= \sin (\theta + 2 \pi) \\
&= \sin (\theta + 4 \pi) \\
&= \sin (\theta + 6 \pi) \\
&= \sin (\theta + 8 \pi) \\
&\cdots
\end{aligned} \)
For example, $\sin \dfrac{9 \pi}{2} = \sin \dfrac{5 \pi}{2} = \sin \dfrac{\pi}{2}$.

Particularly, if we add any integers of $\pi$ to $\theta$, in radians, then the tangent value remains unchanged.
\( \begin{aligned}
\tan \theta &= \tan (\theta + \pi) \\
&= \tan (\theta + 2 \pi) \\
&= \tan (\theta + 3 \pi) \\
&= \tan (\theta + 4 \pi) \\
&\cdots
\end{aligned} \)
For example, $\tan \dfrac{7 \pi}{3} = \tan \dfrac{4 \pi}{3} = \tan \dfrac{\pi}{3}$.

Example 1

State the coordinates of $A$, correct to three significant figures.

\( \begin{align} \displaystyle
\cos 56^{\circ} &= 0.5591 \cdots = 0.559 \\
\sin 56^{\circ} &= 0.8290 \cdots = 0.829 \\
\therefore (0.599,0.829)
\end{align} \)

Example 2

State the coordinates of $B$, correct to three significant figures.

\( \begin{align} \displaystyle
\cos 145^{\circ} &= -0.8191 \cdots = -0.819 \\
\sin 145^{\circ} &= 0.5735 \cdots = 0.574 \\
\therefore (-0.819,0.574)
\end{align} \)

Example 3

State the coordinates of $C$, correct to three significant figures.

\( \begin{align} \displaystyle
\cos 235^{\circ} &= -0.5735 \cdots = -0.574 \\
\sin 235^{\circ} &= -0.8191 \cdots = -0.819 \\
\therefore (-0.574,-0.819)
\end{align} \)

Example 4

Use $\sin {(180^{\circ}-\theta)} = \sin \theta$, find the obtuse angle with the same sine as $45^{\circ}$.

\( \begin{align} \displaystyle
\sin 45^{\circ} &= \sin {(180^{\circ}-45^{\circ})} \\
&= \sin{135^{\circ}}
\end{align} \)

Example 5

Use $\cos {(\pi-\theta)} = \cos \theta$, find the obtuse angle with the same cosine as $\dfrac{\pi}{3}$.

\( \begin{align} \displaystyle
\cos \dfrac{\pi}{3} &= \cos {\Big({\pi}-\dfrac{\pi}{3}\Big)} \\
&= \cos {\dfrac{2\pi}{3}}
\end{align} \)

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