Angles and sides in trigonometry: Tangent
Last but not the least is the trigonometric function called tangent.
Look at the triangle above. Do you notice the following?
* The unknown angle, α
* The side adjacent to the unknown angle (15.3 in)
* The side opposite to the unknown angle (6.37 in)
We need the value of angle, α. Notice that the sides of the triangle are labeled appropriately 'adjacent
side' and opposite' relative to the unknown angle α.
To simplify our discussion, we will simply call the 'length of the adjacent side'simply the 'adjacent'. The other
side will simply be referred to as 'opposite'.
The value for the tan of angle α is defined as the value that results when you divide the opposite side by
the adjacent. The formula is written below:
tan(α) = opposite / adjacent
Or simply:
tan(α) = opp / adj
From the diagram above, we can easily solve the unknown angle with the tangent formula:
tan(α) = 6.37 inches / 15.3 in
tan(α) = 0.42
α = tan1 0.42
α = 23°
Note that the inverse sign is used above. This value can be found using your calculator.
Remember:
1) If any two values i.e. the 2 sides or, an angle and a side, are given, the missing angle or side can always
be found by simply substituting the correct values in the formula.
2) It always helps to draw the diagram to get an accurate picture of what's being asked.
3) Use the calculator to enter the values of cosine and inverse cosine.
Try the following questions

Felix was asked by his dad to measure the height of their orange tree in the backyard. He walks exactly 121
feet from the base of the tree and looks up. The angle from the ground to the top of the tree is 33°. How tall is
the orange tree?
Answer: 78.65 ft
Explanation:
tan 33° = x / 121
0.65 x 121 = x
x = 78.65 ft

Jerry is walking along a straight road when he notices the top of a glass building subtending at an angle
73o with the ground at the point where he is standing. If the height of the tower is 23 m, then how far is he
from the base of the building?
Explanation:
tan 73o = 23 / x
23 / 3.27 = x
X = 7 m

Felix was asked by his dad to measure the height of their orange tree in the backyard. He walks exactly 121
feet from the base of the tree and looks up. The angle from the ground to the top of the tree is 33°. How tall is
the orange tree?
Answer: 60°
Explanation:
tan x = 60 / 34
tan x = 1.76
x = tan 1.76
x = 60°

Paul and Matt are avid UFO fanatics. They observe a flying object not too far which is approximately 58° north
of where they are standing. Judging from its close proximity to the distant trees, they estimate it to be 860
m from where they are. How far up the sky is the object?
Answer: 1376 m
Explanation:
Tan 58° = x / 860
1.6 x 860 = x
x = 1376 m