Trigonometry

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Trigonometry

What is trigonometry?

Trigonometry as the world itself from the Greek, meaning to measure a triangle. This branch of mathematics studies the relationship between side lengths and angles of triangles.

As we can see, Trigonometry is all about solving a triangle, which means finding all the missing parts of a triangle, basically all the side lengths and the angles.

 How do we solve triangles?

We know how to solve a triangle when is given the length of two sides in a right angle triangle by using the famous Pythagoras theorem”.

Also we know how to find the missing angle when we know an acute angle on the right triangle, simply by applying the rule that the sum of the angles on a triangle is 180°.

But, how do we solve a right triangle when we know only one length side and one angle?

First let’s recall what is a right angle triangle. A right angle triangle is a type of triangle that has a right angle that measures 90°.

Based on the acute angle we are going to name the sides on the right triangle.

 We know that the longest side on a right angle triangle is called the hypotenuse.

The side in which lies the angle that we know will always be called the adjacent.

The side in front of the angle will be called the opposite.

Depending on the angle we are given this two sides change the labeling.

Let’s suppose that our angle is labeled ϴ.

The sides:

trigonometry

Hypotenuse – the longest side.

Adjacent – the side in which lies the angle ϴ.

Opposite – in front of the angle ϴ.

How does trigonometry helps us solve a right angle triangle as in the case above?

Let’s study all the possible ratios we can have between sides length on our triangle above.

All ratios are: For our angle Ѳ:

1. $\displaystyle \frac{opposite}{hypotenuse}$

2. $ \displaystyle \frac{adjacent}{hypotenuse}$

3. $ \displaystyle \frac{opossite}{adjacent}$

4. $ \displaystyle \frac{hypotenuse}{opposite}$

5. $ \displaystyle \frac{hypotenuse}{adjacent}$

6. $ \displaystyle \frac{adjacent}{opposite}$

This 6 ratios we found represent six trigonometric functions of any angle in a right angle triangle. In our case the trigonometric functions of our angle Ѳ.

The first three ratios are the most basic and used in trigonometry and we call them Sine, Cosine, Tangent and label them as sin, cos and tan of the given angle.

trigonometry

The three others ratios are called Cosecant, Secant, Cotangent and we label them as csc, sec, cot of the given angle.

cscΘ = $ \displaystyle \frac{hypotenuse}{opposit}$

cscΘ = $ \displaystyle \frac{hypotenuse}{adjacent}$

cotΘ = $ \displaystyle \frac{adjacent}{opposit}$

A simply way how to remind the three basic trigonometric functions is by reminding the famous word “SOHCAHTOA”

We divide the word in three words with 3 letters.

SOH, CAH and TOA

– The first letter of each word represents the name of the function.

– The second letter represents what’s up the ratios sign.

– The third one represents what’s at the bottom of the ratio sign.

We write:

SOH 

sinΘ = $ \displaystyle \frac{\text{Opposite}}{\text{Hypotenuse}}$

CAH 

cosΘ = $ \displaystyle \frac{Adjacent}{\text{Hypotenuse}}$

TOA 

tanΘ = $ \displaystyle \frac{Opposite}{Adjacent}$

Example 1: Calculate the value of sine, cosine and tangent of the angle ϴ  on the triangle below.

trigonometry example 1

Solution: Firstly label the sides based on the angle Ѳ.

trigonometry 2

6 – The opposite

8 – The adjacent

10 – The hypotenuse

Secondly, write the sine, cosine and tangent ratios of our angle.

sinΘ = $ \displaystyle \frac{\text{Opposite}}{\text{Hypotenuse}}$

cosΘ = $ \displaystyle \frac{Adjacent}{\text{Hypotenuse}}$

tanΘ = $ \displaystyle \frac{Opposit}{Adjacent}$

Thirdly, apply the functions based on the side lengths that are given.

sinΘ = $ \displaystyle \frac{6}{10}=\frac{3}{5}$

cosΘ = $ \displaystyle \frac{8}{10}=\frac{4}{5}$

tanΘ = $ \displaystyle \frac{6}{8}=\frac{3}{4}$

Example 2: Find the x by using the right trigonometric function.

Solution: Based on the fig we know the hypotenuse and the angle 30° and we have to find the opposite(30°).

We see that the right function to use its sin30°

$ \displaystyle \sin {{30}^{\circ }}=\frac{x}{14}$

Using basic algebra we find the x.

$ \displaystyle x=\sin {{30}^{\circ }}\cdot 14$

$ \displaystyle x=\frac{1}{2}\cdot 14$
x = 7

Therefore, value of is x=7.

Example 3: The angle of approach of an airliner is 3°. If a plane is 305 meters above ground, how far should it be from the airfield?

Solution: We see that the right function to use is the function tangent.

$ \displaystyle \tan \theta =\frac{Opposite}{Adjacent}$

tan3° = $ \displaystyle =\frac{305}{x}$

x ⋅ tan3° = 305

x = $\displaystyle \frac{{305}}{{\tan {{3}^{{}^\circ }}}}$

$\displaystyle tg{{3}^{\circ }}=0.052$

x = $\displaystyle \frac{{305}}{{0.052}}=5865.38$

$\displaystyle \approx $ 5865

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