Trigonometric Ratios

Circles with Cnetre $(0,0)$

Consider a circle with centre $(0,0)$ and radius $r$ units. Suppose $(x,y)$ is any point on this circle.
Using ths 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 $A$ is $(\cos \theta,\sin \theta)$.
$\dfrac{\sin \theta}{\cos \theta} = \tan \theta$


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


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 $D$ is $(\cos \theta,-\sin \theta)$.
$\dfrac{\sin \theta}{-\cos \theta} = -\tan \theta$


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 trigonometryc 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 3 significant figures.

Example 2

State the coordinates of $B$, correct to 3 significant figures.


Example 3

State the coordinates of $C$, correct to 3 significant figures.

Example 4

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

Example 5

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






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