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Angles, Arc Lengths and Trig Functions

In geometry the orientation of a point with respect to a reference point is very important. We have identified a fixed point as reference, called as origin. Even if the distances of two points from origin are same, it is not necessary that those are at the same place. The given points may be positioned different directions. The quantum of that difference in direction is measured with the concept of angles. The same situation happens even if a single point moves from one position to other maintaining a constant distance from a fixed point. Then its path is along an arc. The geometry of these two concepts is fused to create a new topic called as trigonometry.

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The change in direction of two points with respect to a fixed point is measured in terms of rotational displacement of one with respect to another. This is what is called as the angle between the two points. The preliminary unit of angle is degrees but later, in advanced studies, the same is changed into radians. 

When a point moves on a circle, the point comes back to the same position after a completing a full circle. We fix a constant measure of angle as 360 degrees, usually denoted as $360^{\circ}$. The length covered by the point from the initial position along the curve is the arc length of its path. 

Radius & Arc Length

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When a complete rotation is made, the total arc length is nothing but the circumference of circular path. With this approach we are going to establish a relation between the arc length, the radius and the angle made by the arc length. The arc length and the angle made by the same at the center (called as central angle of the arc) are in direct proportion. That is, if $l$ and $l'$ are two arc lengths in the same circle and if $\alpha$ and $\alpha'$ are their respective central angles in degrees, then,

$l : l'$ = $\alpha : \alpha'$ and therefore $(l)(\alpha')$ = $(\alpha)(l')$

Now, suppose we consider $l'$ as the entire circumference then $l'$ = $2 \pi r$ and $\alpha'$ is $360^{\circ}$, where '$r$' is the radius of the circle. 

In such a case, $(l)(360^{\circ})$ = $(\alpha)(2 \pi r)$ or $l$ = $\frac{\alpha}{180^{\circ}}$ $\pi\ r$.

Next let us bring in the concept of the unit of radian. If an arc of length equal to the measure of the radius, then the central angle of that arc is defined as $1$ radian. So, if an arc of curved length '$l$' makes a central angle of $\theta$ radians, then its length would be $\theta\ \times\ r$ units, Thus, the arc length formula in terms of radians is $l$ = $\theta\ r$.

The complete circumference of a circle is $2\ \pi\ r$. Applying the above formula, $2\ \pi\ r$ = $\theta\ r$ or $\theta$ = $2 \pi$. Hence the central angle for a complete circle is $2 \pi$  radians. That is also $360^{\circ}$ in the unit of degrees. Therefore, $2 \pi$  radians = $360^{\circ}$ or $\pi$ radians = $180^{\circ}$, which turns out to be an important formula for converting radians to degrees or vice-versa. Since $\pi$ is an irrational number, it is customary to express angles in radian measures in terms of $\pi$.

Trigonometry of Circles

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Let us first study the concept of two similar right triangles sharing one vertex as common.
Trigonometry of Circles
In the above diagram, the triangles $ABC$ and $ADE$ are similar both having vertex at $A$. Let the angle at $A$ be $\theta$.  As per the concept of similar triangles, the ratios of the sides $\frac{BC}{AB}$, $\frac{AC}{AB}$ and  $\frac{BC}{AC}$ in triangle $ABC$ are respectively equal to the ratios of the sides   $\frac{DE}{AD}$, $\frac{AE}{AD}$ and $\frac{DE}{AE}$ in triangle $ADE$ are equal, irrespective of what the size of triangle $ADE$ is, as long as the angle $\theta$ remains the same. Thus, in other words, the ratios of the sides in a right triangle are same for a particular base acute angle. In trigonometry these ratios are called as Sine (abbreviated as 'sin'), Cosine (abbreviated as 'cos') and Tangent (abbreviated as 'tan') ratios for a given angle. They are defined as follows.

$Sin\ \theta$ = $\frac{oppsoite\ side}{hypotenuse}$;  $Cos\ \theta$ = $\frac{adjacent\ side}{hypotenuse}$;  $Tan\ \theta$ = $\frac{oppsoite\ side}{adjacent\ side}$

The descriptions 'opposite side' and 'adjacent side' are with reference to the angle considered. It also implies that, 

$Tan\ \theta$ = $\frac{Sin\ \theta}{Cos\ \theta}$  

Now, to find these ratios for different angles, we need to draw different diagrams. But to condense into a simpler process, let us construct a circle and draw the triangles inside the same circle. Look at the following figure.
Ratios for Different Trig Angles
A circle with center $O$ is drawn with radius '$r$'. The center may also be considered the origin of the coordinate system with axes dividing the circle into four quadrants. Consider any point $P$ on the circle with radius $OP$ making an angle $\theta 1$ with the positive $x$-axis. Then, as we discussed earlier,

$Cos\ \theta_1$ = $\frac{OR}{OP}$ and $Sin\ \theta_1$ = $\frac{PR}{OP}$

Now, as per coordinate geometry $OR$ = $x_1$ and $PR$ = $y_1$. Therefore, in such a case,

$Cos\ \theta_1$ = $\frac{x_1}{r}$ and $Sin\ \theta_1$ = $\frac{y_1}{r}$

Suppose we consider a circle with radius as $1$ unit, then as a special case, 

$Cos\ \theta_1$ = $x_1$ and $Sin\ \theta_1$ = $y_1$.

Next we will consider any other point $Q$, where the radius is making an angle $\theta_2$. By similar argument we can say,

$Cos\ \theta_2$ = $x_2$ and $Sin\ \theta_2$ = $y_2$.

Thus, if a circle of unit radius is drawn with center as the origin of coordinate system, then the $x$-coordinate of any point gives the cosine ratio of the angle made the radius at that point and the $y$-coordinate of the same point gives the sine ratio of the angle made the radius at that point. This special type of circle is called as unit circle and it is a very important tool in trigonometry. A unit circle labeled for various standard angles is shown below.
Unit Circle Diagram
It is obvious that the ratio of $y$-coordinate and the $x$ coordinate of any point gives the value of the tangent ratio of the angle made by the radius at that point.


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Example 1: 

Convert $210^{\circ}$ to radians. 


The conversion formula is, 

$180^{\circ}$ = $\pi$ radians. Therefore, $210^{\circ}$ = $(\frac{210^{\circ}}{180^{\circ}})$ $\pi$ = $(\frac{7 \pi}{6})$ radians.
Example 2: 

A man walks along the circumference of a circular park with radius of $140$ ft.  When he moves from a point $A$ to another point $B$ on the circular path he has covered an angle of $108^{\circ}$ at the center of the park. Find the distance walked by the person between these two points. (Use the approximate value of $\pi$ as 22/7)


First convert the angle of $108^{\circ}$ into radians. 

$108^{\circ}$ = $(\frac{108^{\circ}}{180^{\circ}})$ $\pi$ = $(\frac{3 \pi}{5})$ radians.

The distance walked by the man between the two given point is the arc length of a circle with radius of $140$ ft. for a central angle of $(\frac{3 \pi}{5})$ radians. 

Therefore, the required distance = $(\frac{3 \pi}{5})$ $\times\ (140)$ = $(\frac{66}{35})$ $(140\ ft)$ = $264$ ft.
Example 3: 

A man of $6$ feet tall is walking towards a building. He sees the top of a building at an angle of $45^{\circ}$ and also sees the top of the flag post above the building at an angle of $60^{\circ}$ (these angles are known as angle of elevations). If the height of the flag post is $22$ feet, find the height of the building to the nearest foot.

Trig Function Example

The above picture describes the situation. The man is at a distance of $X$ feet from the building. We can see that, 

$\frac{h}{X}$ = $tan\ 45^{\circ}$ and $\frac{(h + 22)}{X}$ = $tan\ 60^{\circ}$.

Dividing both the second equation by the first, we get,

$\frac{(h + 22)}{(h)}$ = $\frac{tan\ 60^{\circ}}{tan\ 45^{\circ}}$

From the unit circle we can find tan $60^{\circ}$ = $(\frac{\frac{\sqrt3}{2}}{\frac{1}{2}})$ = $\sqrt{3}$ and tan $45^{\circ}$ = $\frac{(\frac{\sqrt2}{2})}{(\frac{\sqrt2}{2})}$ = $1$

Therefore, $\frac{(h + 22)}{(h)}$ = $\frac{\sqrt{3}}{1}$ = $\sqrt{3}$ or $(\frac{22}{h})$ = $(\sqrt{3}\ -\ 1)$

So, $h$ = $\frac{22}{(\sqrt{3} - 1)}$ $\approx\ 30$ feet.

The total height of the building $H$ = $h + 6$ = $30 + 6$ = $36$ feet.
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