The Cosine Ratio in Trigonometry Mathematics

1. Image Source: http://rekkerd.org

2. Image Source: www.rkm.com.au - used with permission.
The graphs of the values of Sine and Cosine from
Right Triangles create shapes that are very much
the same as sound waves.
We can electronically make these sound wave shapes
and then alter their shape and sound with guitar pedals.

3. Image Source: Google Images.
Distorted Heavy Metal guitar sound occurs when
smooth Sine and Cosine Waves are mathematically
transformed into Square Shaped, Sawtooth,
or Triangle shaped waves.

4. Adjacent
Hypotenuse
ɵo
Adjacent - ADJ - A
ɵCosine =
ADJ
HYP
ɵCosine =
A / HɵCos =
The “Cosine” Ratio for a Right Triangle is defined as the
Adjacent Side Length divided by the Hypotenuse Length.
If we have several right triangles with the same Reference
Angle, the ratio of their Adjacent divided by the Hypotenuse
will be the same value for all of these Triangles.

5. Eg. If we look at the Adjacent / Hypotenuse
for each of the above three Triangles we get
Cos37o
= 4 / 5 = 8 / 10 = 12 / 15 = 0.8
37
o
37
o
37
o
5
4
10
8
15
12
The Cosine Ratio of Adjacent / Hypotenuse is the same value.

6. ɵo
ADJ
HYP
ɵCos =
These are the four formulas for
working with Cosine Triangles.
ADJ = HYP x Cosɵ
ɵ = Cos-1
ADJ
Cosɵ
ADJ
HYP
HYP =
We also use the special “Cos”
and “Cos-1
” calculator buttons
when solving Cosine Triangles.
Adjacent - ADJ - A

7. We use the special “Cos” and “Cos-1
” calculator
buttons when solving Cosine Triangle Questions.
Warning: Your calculator must be in “Degrees” DEG Mode.
Cos 60o
cos 60 enter 0.5
Cos 45o
cos 45 enter 0.7071
Cos 30o
cos 30 enter 0.8660
Note that we round off long decimal trig values
from the calculator to four decimal places.

8. To get “Cos-1
” on the calculator we use “2nd” or
“Fn” followed by the “Cos” calculator button.
Warning: Your calculator must be in “Degrees” DEG Mode.
30o
cos 0.8660 enterCos-1
(0.8660)
Note that we usually round off angle values
from the calculator to the nearest whole number
2nd
45o
cos 0.7071 enterCos-1
(0.7071) 2nd
60o
cos 1 enterCos-1
(1/2) 2nd n/d 2

9. 1. Label the Sides of the Triangle
2. Work out if unknown is ADJ, HYP, or the Angle.
3a. To find Unknown ADJ, use A = H x Cosɵ
3b. To find an Unknown HYP, use H = A / Cosɵ
3c. To find an Unknown Angle, use ɵ = Cos-1
(A / H )
4. Substitute values into the formula being used
5. Put values into a Calculator and Round Off Answer

10. To find Unknown ADJ, use A = H x Cosɵ
A = 10 x Cos35
A = 8.19520
A = 8.20
k = 8.20
o
k
35 o
ADJ k
35

11. To find Unknown HYP, use H = A / Cosɵ
H = 12 / Cos40
H = 15.66488
H = 15.66
m = 15.66
o
12
40
o
12
40
ADJ

12. To find Unknown Angle, use
β = Cos-1
(12 / 13)
β = 22.61986
β = 23o
β
ɵ = Cos-1
(ADJ / HYP)
β

13. ββ
To find Cosɵ, use
Cosβ = 12 / 13
Cosβ = 0.9230769
Cosβ = 0.9230
Cosɵ = ADJ / HYP
Find Cosβ

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