The trigonometric expression tan (x) has a value of three-fourths when csc (x) = five-thirds. Hence, 4th option is the right choice.
The values of all trigonometric functions dependent on the value of the ratio of sides of a right-angled triangle are known as trigonometric ratios. The trigonometric ratios of a right-angled triangle's sides about any of its acute angles are known as that angle's trigonometric ratios.
The three sides of the right-angled triangle are:
Hypotenuse (the longest side)
Perpendicular (opposite side to the angle)
Base (Adjacent side to the angle)
The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec or csc), and secant (sec).
The trigonometry ratios for a specific angle ‘θ’ is given below:
Trigonometric Ratios:
Sin θ = Perpendicular/Hypotenuse
Cos θ = Base/Hypotenuse
Tan θ = Perpendicular/Base or Sin θ/Cos θ
Cot θ = Base/Perpendicular or 1/tan θ
Sec θ = Hypotenuse/Base or 1/cos θ
Cosec θ or csc θ = Hypotenuse/Perpendicular or 1/sin θ
According to the Pythagoras theorem, we can say that in a right-angled triangle:
Hypotenuese² = Base² + Perpendicular²
We are given csc (x) = 5/3, x being an acute angle (0° < x < 90°).
We are asked to find the trigonometric ratio in x, whose value = 3/4.
We know csc (x) = hypotenuse/perpendicular.
So we let the hypotenuse be 5k and the perpendicular be 3k. Let the base be b.
By Pythagoras' Theorem,
Hypotenuese² = Base² + Perpendicular²
or, (5k)² = b² + (3k)²
or, 25k² = b² + 9k²
or, b² = 25k² - 9k² = 16k² = (4k)²
or, b = 4k.
Now, we find each trigonometric expression with respect to x:
Sin x = Perpendicular/Hypotenuse = 3k/5k = 3/5
Cos x = Base/Hypotenuse = 4k/5k = 4/5
Tan x = Perpendicular/Base = 3k/4k = 3/4
Cot x = Base/Perpendicular = 4k/3k = 4/3
Sec x = Hypotenuse/Base = 5k/4k = 5/4
csc x is already given as 5/3.
We can see that tan (x) = 3/4, the required value.
∴ The trigonometric expression tan (x) has a value of three-fourths when csc (x) = five-thirds. Hence, 4th option is the right choice.
Learn more about trigonometric expressions at
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