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Trigonometry
In these notes, we will present the concepts of trigonometry from basics to
advance
...
Let us replace the 𝑠𝑖𝑛𝛼 and 𝑐𝑜𝑠𝛼 with the
length of the triangle
...
(∗)
𝑐2
2

2

𝐵𝑢𝑡 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑃𝑦𝑡ℎ𝑎𝑔𝑜𝑟𝑒𝑎𝑛 𝑇ℎ𝑒𝑜𝑟𝑒𝑚 𝑤𝑒 ℎ𝑎𝑣𝑒 𝑡ℎ𝑎𝑡: 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐
...


 Exercises:
1
...

((sin 𝛼)2 )3 + ((cos 𝛼)2 )3 = ((sin 𝛼)2 + (cos 𝛼)2 )(((sin 𝛼)2 )2 +
((cos 𝛼)2 )2 − (sin 𝛼)2 (cos 𝛼)2 ) = 1 ∗ (((sin 𝛼)2 + (cos 𝛼)2 )2 −
2(sin 𝛼)2 (cos 𝛼)2 − (sin 𝛼)2 (cos 𝛼)2 ) = 1 − 3(sin 𝛼)2 (cos 𝛼)2
...
Prove that: 13 − 12 cos 𝛼 − 4(sin 𝛼)2 = (2 cos 𝛼 − 3)2
4

3
...
𝐹𝑖𝑛𝑑 cos 𝛼 , tan 𝛼 𝑎𝑛𝑑 cot 𝛼
...

sin 𝜃
1

sin 𝛼
-1

cos 𝜃

𝛼
cos 𝛼

1

-1
So, as it can be easily seen, the vertical projection of the angle in the
vertical axes represents the sin 𝛼 and the one in horizontal axes the
cos 𝛼
...
The blue,
green and red line form a right triangle, where the green
represents cos 𝛼, the blue sin 𝛼 and the red, the radius of the circle,
is equal to 1 unit
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o Many other formulas are very easily understood just taking a brief
look at the circle
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I
...
cos(180° − 𝑎) = − cos 𝑎
III
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cos(90° − 𝑎) = sin 𝑎
V
...
cos(90° + 𝑎) = sin 𝑎
VII
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cos(360° + 𝑎) = cos 𝑎
Therefore, we can continue writing many other consequent formulas
...
Try to figure out the rule for tan(180° − 𝑎) , tan(90° − 𝑎) and all the
others
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Look at the circle and explain all the formulas written above
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o The formulas that we are going to introduce in the following pages
can be observed more than clearly in the circle
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Find the value of 𝐬𝐢𝐧 𝟏𝟗𝟓𝟎°
...
So the point is
to express 1950 as 360*x + c
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5

2
...

We shall just use the formulas and try to find a connection between
the terms of expression
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 Work by yourself:
5

3
...

4
...




Getting deeper into Trigonometric Circle

cotg 𝜃

cotg 𝛼

1
sin 𝜃

tan 𝛼

-1

𝛼

cos 𝜃
1

-1
tan 𝜃
6

 As introduced in the scheme, the trigonometric circle is not used
only for sine and cosine, but also for tangent and cotangent
...
On the other hand, the vertical one,
tangent in the right hand side, is the tangent axe
...

Moreover, it can clearly be observed that |tan 𝛼| ≥
|sin 𝛼| and |cot 𝛼| ≥ |cos 𝛼|
...
You can understand this,
as soon as you have solved a couple of exercises proving trigonometric
formulas
...
But between them there is a strong and wellknown relation, which is determined by the rule:

𝝅 = 𝟏𝟖𝟎°
Which means that to transform and angles’ mass from radian to
degree, we use the following formula:
𝛼° =

𝛼(𝑟𝑎𝑑𝑖𝑎𝑛)
𝜋

∙ 180° Or 𝛼(𝑟𝑎𝑑𝑖𝑎𝑛) =
7

𝛼°
180°

∙𝜋

So let us use it in an example:
Is given the mass of an angle (in radian) equal to 3∙π/4
...

Solution
From the conclusion above:
𝛼° =

𝛼(𝑟𝑎𝑑𝑖𝑎𝑛)
3∙𝜋
∙ 180° ↔ 𝛼° =
∙ 180° = 120°
...




Basic values of trigonometric functions

𝟎

𝝅
𝟔

𝝅
𝟒

𝝅
𝟑

𝝅
𝟐

𝐬𝐢𝐧 𝜶

0

1
2

√2
2

√3
2

1

√3
2

𝐜𝐨𝐬 𝜶

1

√3
2

√2
2

1
2

0

−

𝐭𝐚𝐧 𝜶

0

√3
3

1

√3

∞

−√3

-1

−

𝐜𝐨𝐭 𝜶

∞

√3

1

√3
3

0

−

√3
3

-1

8

𝟐∙

𝝅
𝟑

1
2

𝟑∙

𝝅
𝟒

𝟓∙
1
2

√2
2

−

𝝅
𝟔

𝝅

0

√2
√3
−
2
2

-1

√3
3

0

−√3

∞

 TASK: Find the value of these function for the rest of the angles
from 𝜋 𝑡𝑜 2𝜋
...

 Exercises and tasks for each topic

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