Using the Sine Ratio in Trigonometry Mathematics

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When a plane descends it has a speed at an angle,
as well as decreasing height, and its flight path forms
the hypotenuse of a right triangle.
The pilot needs to use the correct angle and speed
to make it safely onto the runway at the right location.

3. Opposite
Hypotenuse
ɵo
Opposite-OPP-O
ɵSine =
OPP
HYP
ɵSine =
O / HɵSin =
The “Sine” Ratio for a Right Triangle is defined as the
Opposite Side Length divided by the Hypotenuse Length.
If we have several right triangles with the same reference
Angle, the ratio of their Opposite divided by the Hypotenuse
will be the same value for all of these Triangles.

4. Eg. If we look at the Opposite / Hypotenuse
for each of the above three Triangles we get
Sin37o
= 3 / 5 = 6 / 10 = 9 / 15 = 0.6
37
o
37
o
37
o
5
3
10
6
15
9
The Sine Ratio of Opposite / Hypotenuse is the same value.

5. Opposite
Hypotenuse
ɵo
Opposite-OPP-O
ɵSine =
OPP
HYP
ɵSin =
O / HɵSin =
The “Sine” formula can be re-arranged to make a formula
for the “Opposite” “O”, by multiplying both sides by “H”
OPP
HYP
ɵSin =HYP x x HYP
OPP = HYP x Sinɵ

6. ɵo
Opposite-OPP-O
OPP
HYP
ɵSin =
The “OPP” formula can be re-arranged to make a formula
for the “Hypotenuse” “H”, by dividing both sides by “Sinɵ”
Sinɵ
OPP = HYP x Sinɵ
Sinɵ
OPP = HYP x Sinɵ OPP
Sinɵ
HYP =

7. ɵo
Opposite-OPP-O
OPP
HYP
ɵSin =
There is a fourth and final
Formula for finding the Angle.
It is called the “Inverse
“Sine” formula.
OPP = HYP x Sinɵ
ɵ = Sin-1
OPP
Sinɵ
OPP
HYP
HYP =

8. ɵo
Opposite-OPP-O
OPP
HYP
ɵSin =
These are the four formulas for
working with Sine Triangles.
OPP = HYP x Sinɵ
ɵ = Sin-1
OPP
Sinɵ
OPP
HYP
HYP =
We also use the special “Sin”
and “Sin-1
” calculator buttons
when solving Sine Triangles.

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

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

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

12. To find Unknown OPP, use O = H x Sinɵ
O = 10 x Sin40
O = 6.42787
O = 6.43
d = 6.43
o
d
40 o
OPP
d
40

13. To find Unknown HYP, use H = O / Sinɵ
H = 12 / Sin50
H = 15.66488
H = 15.66
k = 15.66
o
12
50
o
12
50
OPP

14. To find Unknown Angle, use
ɵ = Sin-1
(12 / 13)
ɵ = 67.38013
ɵ = 67o
ɵ ɵ
ɵ = Sin-1
(OPP / HYP)

15. To find Sinɵ, use
Sinɵ = 12 / 13
Sinɵ = 0.9230769
Sinɵ = 0.9230
ɵ ɵ
Sinɵ = OPP / HYP
Find Sinɵ

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