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August 12, 2023
Trigonometry was invented because its need arose in astronomy. Since then the astronomers have used it, for instance, to calculate distances from the Earth to the planets and stars. In this page we are going to learn about various Formulas for Height and Distances which will include different types of trigonometric ratios and figures. For your knowledge
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Then,
Harmonic progression is the series when the reciprocal of the terms are in AP.
For example, \frac{1}{a}, \frac{1}{ (a + d)}, \frac{1}{(a + 2d)}…… are termed as a harmonic progression as a, a + d, a + 2d are in Arithmetic progression.
Question 1: Varun Observed that from a point 355 meters away from the foot of a tower, the top of the tower is at an angle of elevation of 30^{o}, then the height (in meters) of the tower is?
Answer: Tan 30° = X/355
= \frac{1}{\sqrt{3}} = X/355
=X= 614.87
Question 2: A man is standing in his balcony, which is on the third floor of the building and is at the height of 10m. His angle of elevation at the top of the opposite building is 60^{o} ,and the angle of depression of the base is 30^{o} . Determine the height of the building opposite to him?
Answer: Let AB the building opposite to the man.Let us assume he is standing at point CTherefore, we can say that ∠BCD = 600∠ACD = 300Let BD = hIn the given ADC, tan 300 = 10/CD
Question 3: A man stands at the corner of a farm, which is square. The angle of elevation of a scarecrow which presents diagonally opposite corner is60^{o} . When he walks backward in the straight line fir 80ft. The angle of elevation of the scarecrow now becomes 30^{o}. Calculate the field’s area.
Answer: Tan 600 = PQ/QR = √3PQ= √3QRTan 300 = 1/√3 = PQ/SQ|= PQ/ 80+ QR80+ QR= √3PQ80+ QR = 3QRQR = 40ft
It can also be seen from the figure that the man at point R and the scarecrow at point Q are diagonally opposite to one another.Therefore, QR is the diagonal of the square farm.Diagonal =side x /√2Therefore, 40 = side x /√2Side = 40 √2Therefore, area = (side)2 = (40 √2)2= 800 sq feet
Question 4: Adam is standing in the Sun. He is 6ft tall and is casting a shadow of 4ft. A flag is also present near the man casting a shadow of 36ft. Calculate the height of the flag.Answer: The angle of elevation for the Sun will be the same for both the cases.Therefore, the ratio of object to shadow will be the same for both the cases as per the proportionality rule.Object height/ Shadow height = 6/ 4 = H /36Therefore, H = 54 ft = Height of the flag
Question 5: When a boy looks from the foot and the top of a tower at the roof of a building, the angles of elevation and depression are 27^{o}and 63^{o}, The height of this building is 40m, then calculate the height of the tower given that tan 630 =2.
Answer: Let the tower be ABLet the building of height 40m be CDIn the given triangle ACD, AC/DC = cot 270= cot (90-63)AC/40 = tan 630 = 2AC = 80mNow, DE = AC = 80mAlso, in triangle BED, tan 630 = BE/DE2= BE/80Therefore, BE = 160 mTherefore, the height of the tower can be calculated as AE + EB= 40 +160 = 200m
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