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In the given figure, what is the value of $3 tan θ − 2 sin α + 4 cos α$ ?

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125

135

127

25

answer is A.

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Detailed Solution

Given a figure.
Question Image

In right triangle RSQ applying Pythagoras theorem to find RS, we get,

R S 2 = S Q 2 + R Q 2 ⇒ R S 2 = (6) 2 + (8) 2 ⇒ R S 2 = 36 + 64 ⇒ R S 2 = 100 ⇒ R S = ± 10 ⇒ R S = 10

As negative length can be discarded.
Applying Pythagoras theorem in right triangle QPR to find PQ, we get,

P R 2 = P Q 2 + R Q 2 ⇒ (17) 2 = P Q 2 + (8) 2 ⇒ P Q 2 = 289 − 64 ⇒ P Q 2 = 225 ⇒ P Q = ± 15 ⇒ P Q = 15

As negative length can be discarded.
Cosine is a ratio of the side adjacent to the angle

α

to the hypotenuse, thus, applying, we get,

cos α = S Q S R ⇒ cos α = 610 ⇒ cos α = 35 ...... (1)

Sine is a ratio of the side opposite to the angle

α

to the hypotenuse, thus, applying, we get,

sin α = R Q S R ⇒ sin α = 810 ⇒ sin α = 45 ...... (2)

Tangent is a ratio of the side opposite to the angle

θ

to the side adjacent to the angle

θ

Thus, applying, we get,

tan θ = R Q P Q ⇒ tan θ = 815 ...... (3)

Putting values from equations (1), (2), (3), we get,

3 tan θ − 2 sin α + 4 cos α = 3 × 815 − 2 × 45 + 4 × 35 ⇒ 3 tan θ − 2 sin α + 4 cos α = 85 − 85 + 125 ⇒ 3 tan θ − 2 sin α + 4 cos α = 125

Thus, the value of

3 tan θ − 2 sin α + 4 cos α

125

.
Hence, option 1 is correct.

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