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Difficulty: MediumRight Triangle Trigonometry (SOHCAHTOA)

A surveyor standing at point AA on level ground measures the angle of elevation to the top, point CC, of a vertical observation tower. The base of the tower is at point BB, forming right triangle ABCABC with the right angle at vertex BB. The horizontal distance along the ground from point AA to base BB is 4040 meters. If sin⁡(∠A)=941\sin(\angle A) = \frac{9}{41}, what is the height, in meters, of the tower?

  1. 99Answer
  2. B
    36041\frac{360}{41}
  3. C
    4040
  4. D
    4141
  5. E
    1609\frac{160}{9}

Answer

The height of the tower is 99 meters.
By definition of SOHCAHTOA, sin⁡(∠A)=oppositehypotenuse=BCAC=941\sin(\angle A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{9}{41}. Since cos⁡(∠A)=adjacenthypotenuse=ABAC\cos(\angle A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AB}{AC}, and using the Pythagorean triple 99-4040-4141, we find cos⁡(∠A)=4041\cos(\angle A) = \frac{40}{41}. Given that the adjacent side AB=40AB = 40 meters, the hypotenuse ACAC must be 4141 meters, which means the opposite side (tower height BCBC) is 99 meters.

Step-by-Step Solution

1
Identify the given trigonometric ratio and sides of the right triangle.
In right triangle ABCABC with right angle at BB, the adjacent side to ∠A\angle A is AB=40AB = 40, the opposite side is BCBC (height of the tower), and the hypotenuse is ACAC. We are given sin⁡(∠A)=BCAC=941\sin(\angle A) = \frac{BC}{AC} = \frac{9}{41}.
SOHCAHTOA defines sine as the ratio of the opposite side to the hypotenuse.
2
Determine the relationship between the sides using the Pythagorean theorem or cosine ratio.
Since sin⁡(∠A)=941\sin(\angle A) = \frac{9}{41}, we know cos⁡(∠A)=ABAC=1−(941)2=1681−811681=16001681=4041\cos(\angle A) = \frac{AB}{AC} = \sqrt{1 - \left(\frac{9}{41}\right)^2} = \sqrt{\frac{1681 - 81}{1681}} = \sqrt{\frac{1600}{1681}} = \frac{40}{41}.
The Pythagorean identity sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1 relates sine and cosine for any acute angle in a right triangle.
3
Calculate the length of the hypotenuse and the height of the tower.
Setting cos⁡(∠A)=40AC=4041\cos(\angle A) = \frac{40}{AC} = \frac{40}{41} gives AC=41AC = 41 meters. Then, BC=41×sin⁡(∠A)=41×941=9BC = 41 \times \sin(\angle A) = 41 \times \frac{9}{41} = 9 meters.
Substituting the known adjacent length of 4040 meters into the ratio yields the exact vertical height.

Key Concept

Right Triangle Trigonometry (SOHCAHTOA) and Pythagorean Triples
Estimated Time:1m 0s
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Topics

Right Triangle Trigonometry (SOHCAHTOA)

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