After take-off, planes ascend at an angle to reach their cruising altitude. This angle of elevation is related to the opposite and adjacent sides of a right triangle through the tangent ratio, which is the opposite side divided by the adjacent side. For any right triangle with the same angle, the tangent ratio of opposite over adjacent will be the same value. Solving tangent triangle problems involves labeling sides, determining what is unknown (opposite, adjacent, or angle), and using the appropriate tangent formula along with a calculator set to degrees mode.

1. Image Source: Jeff Wilson Photography

2. After take-off, a plane ascends at a specific Angle of
Elevation so that it can fly a certain distance, and attain
a certain height, to thereby reach the cruising altitude
for its assigned flight path.
Opposite and Adjacent of a Right Triangle are involved
with this angle in a relationship called a “Tangent” Ratio.

3. Opposite
Adjacent
ɵo
Adjacent - ADJ - A
Opposite-OPP-O
ɵTangent =
OPP
ADJ
ɵTangent =
O / AɵTan =
The “Tangent” Ratio for a Right Triangle is defined as the
Opposite Side Length divided by the Adjacent Length.
If we have several right triangles with the same Reference
Angle, the ratio of their Opposite divided by the Adjacent
will be the same value for all of these Triangles.

4. Eg. If we look at the Opposite / Adjacent
for each of the above three Triangles we get
Tan37o
= 3 / 4 = 6 / 8 = 9 / 12 = 0.75
37
o
37
o
37
o
3
4
6
8
9
12
The Tangent Ratio of Opposite / Adjacent is the same value.

5. ɵo
OPP
ADJ
ɵTan =
These are the four formulas for
working with Tangent Triangles.
OPP = ADJ x Tanɵ
ɵ = Tan-1
OPP
Tanɵ
OPP
ADJ
ADJ =
We also use the special “Tan”
and “Tan-1
” calculator buttons
when solving Tangent Triangles.
Adjacent - ADJ - A
Opposite-OPP-O

6. We use the special “Tan” and “Tan-1
” calculator
buttons when solving Tangent Triangle Questions.
Warning: Your calculator must be in “Degrees” DEG Mode.
Tan 60o
tan 60 enter 1.7321
Tan 45o
tan 45 enter 1
Tan 30o
tan 30 enter 0.5774
Note that we round off long decimal trig values
from the calculator to four decimal places.

7. To get “Tan-1
” on the calculator we use “2nd” or
“Fn” followed by the “Tan” calculator button.
Warning: Your calculator must be in “Degrees” DEG Mode.
60o
tan 1.7321 enterTan-1
(1.7321)
Note that we usually round off angle values
from the calculator to the nearest whole number
2nd
35o
tan 0.7071 enterTan-1
(0.7071) 2nd
37o
tan 3 enterTan-1
(3/4) 2nd n/d 4

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

9. To find Unknown OPP, use O = A x Tanɵ
O = 10 x Tan35
O = 7.002
O = 7.00
m = 7
o
m
10
35 o
OPP
m
ADJ 10
35

10. To find Unknown Angle, use
β = Tan-1
(5 / 12)
β = 22.6198
β = 23o
5
β
ɵ = Tan-1
(OPP / ADJ)
β

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