Answer :
The amount of fence that Priya would need (in meters) is 19 meters approximately.
What are the six trigonometric ratios?
Trigonometric ratios for a right angled triangle are from the perspective of a particular non-right angle.
In a right angled triangle, two such angles are there which are not right angled(not of 90 degrees).
The slant side is called hypotenuse.
From the considered angle, the side opposite to it is called perpendicular, and the remaining side will be called base.
From that angle (suppose its measure is θ),
[tex]\sin(\theta) = \dfrac{\text{Length of perpendicular}}{\text{Length of Hypotenuse}}\\\cos(\theta) = \dfrac{\text{Length of Base }}{\text{Length of Hypotenuse}}\\\\\tan(\theta) = \dfrac{\text{Length of perpendicular}}{\text{Length of base}}\\\\\cot(\theta) = \dfrac{\text{Length of base}}{\text{Length of perpendicular}}\\\\\sec(\theta) = \dfrac{\text{Length of Hypotenuse}}{\text{Length of base}}\\\\\csc(\theta) = \dfrac{\text{Length of Hypotenuse}}{\text{Length of perpendicular}}\\[/tex]
For this case, we've to find the length of the fencing she needs to cover her garden.
Length of fencing needed = Perimeter of the considered right triangle = sum of its sides' lengths
Using the image attached below, from the perspective of angle A which is of 50 degrees from inside, we get:
- |AB| = base length = 5 meters (given)
- |BC| = perpendicular length = p meters (say)
- |AC| = hypotenuse length = h meters (say)
- Using tangent ratio,
[tex]\tan(m\angle A) = \tan(50^\circ) = \dfrac{p}{5}\\\\p = 5 \tan(50^\circ) \approx 5.96 \: \rm m[/tex]
- Using sin ratio,
[tex]\sin(m\angle A) = \sin(50^\circ) = \dfrac{p}{h}\\\\h = \dfrac{p}{\sin(50^\circ)} \approx \dfrac{5.96}{\sin(50^\circ)} \approx 7.78 \: \rm m[/tex]
Thus, perimeter of the triangle = 5 + p + h ≈ 18.74 m ≈ 19 m
Thus, the amount of fence that Priya would need (in meters) is 19 meters approximately.
Learn more about trigonometric ratios here:
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