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Two points on the same side of a tree are $19.8 \textrm m$ apart. The angles of elevation of the top of the tree are $21°$ from one point and $16°$ from the other point. Find the height of a tree in $16°$ and $21°$, and the distance of $21°$ to the tree.

Please somebody help me. I could not get even to the first step.

2 Answers 2

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Let AB be a tree. C and D are two points on one side of tree.

Point C is x m apart from tree so BC = x m. And D is (19.8 + x) m apart from tree so BD = (19.8 + x) m.

Angle made by point C to top of the tree is 21° and angle made by point D is 16°.

Then in triangle ABC,

$\tan 21° = \frac{AB}{BC}$

$AB = \tan 21° × BC$

In triangle ABD,

$\tan 16° = \frac{AB}{BD}$

$AB = \tan 16° × BD$

From above both equations,

$\tan 21° × BC = \tan 16° × BD$

$\tan 21° × (x) = \tan 16° × (19.8 + x)$

Find value of x. Then find value of AB.

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Let the tree's height be h.

Construct a triangle, with the tree being the leg, and the points lying along the horizontal leg, the angles of elevation being in different triangles.

Using an elementary definition (SOH-CAH-TOA) + reciprocals, you should see that $h\cot(16º)-h\cot(21º)=19.8$.

The tree should be approximately 22.44 meters tall.

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    @ Saketh Malyala, I could not understand this: (SOH- CAH-TOA) + reciprocals – 2017-02-08
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    @PierredeFermat Try drawing a diagram, everything will be clear. – 2017-02-08
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    @PierredeFermat It means "SineOppositeHypotenuse-CosineAdjacentHypotenuse-TangentOppositeAdjacent" and is a mnemonic device to help remember the definition of the elementary trigonometric functions. That being said, it is not universally taught (I haven't seen this specific one before), and extremely cryptic to put in an answer without explanation. And then to refuse to explain it when asked specifically is just not nice. – 2017-02-08
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    @Arthur Apologies. I wasn't on the site. – 2017-02-09