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Zorluk: KolayRight Triangle Trigonometry (SOHCAHTOA)

A 1010-foot ladder leans against a vertical wall. The base of the ladder is 66 feet away from the bottom of the wall, and the top of the ladder reaches a height of 88 feet up the wall. If θ\theta is the angle formed between the ladder and the ground, what is the value of cos⁡(θ)\cos(\theta)?

  1. A
    34\frac{3}{4}
  2. 35\frac{3}{5}Cevap
  3. C
    45\frac{4}{5}
  4. D
    43\frac{4}{3}
  5. E
    53\frac{5}{3}

Cevap

The correct answer is 35\frac{3}{5}, which represents the ratio of the adjacent side (66 feet) to the hypotenuse (1010 feet) for the angle θ\theta formed between the ladder and the ground.
The correct answer is the value 35\frac{3}{5}. The angle θ\theta is formed between the ladder (hypotenuse) and the ground (adjacent side). The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. Here, the adjacent side is 66 feet and the hypotenuse is 1010 feet. Therefore, cos⁡(θ)=610\cos(\theta) = \frac{6}{10}, which simplifies to 35\frac{3}{5}.

Adım Adım Çözüm

1
Identify the parts of the right triangle relative to the angle θ\theta formed between the ladder and the ground.
The hypotenuse is the length of the ladder (1010 feet). The side adjacent to θ\theta is the distance along the ground from the wall to the base of the ladder (66 feet). The side opposite to θ\theta is the height up the wall (88 feet).
To apply trigonometric ratios, we must first map the given dimensions of the scenario to the sides of a right triangle relative to the reference angle.
2
Recall the definition of the cosine ratio in a right triangle.
cos⁡(θ)=AdjacentHypotenuse\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}
We need to identify the mathematical definition of cosine to set up the calculation.
3
Substitute the identified side lengths into the cosine ratio and simplify the fraction.
cos⁡(θ)=610=35\cos(\theta) = \frac{6}{10} = \frac{3}{5}
Substituting the values of 66 feet for the adjacent side and 1010 feet for the hypotenuse gives 610\frac{6}{10}, which simplifies to 35\frac{3}{5} when both the numerator and denominator are divided by their greatest common divisor, 22.

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Right Triangle Trigonometry (SOHCAHTOA)
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