If tan = 4/3, find the value of sin θ + cos θ.

If tan = 4/3, find the value of sin θ + cos θ (both sin θ and cos θ are positive).

Answer :

Let ∆ABC be a right angled

∠ACB = θ

Given that, tan θ = 4/3

(AB/BC = 4/3)

tan θ = 4/3

(AB/BC = 4/3)

Let AB = 4x,

then BC = 3x

In right angled ∆ABC

AC² = AB² + BC²

⇒ AC² = AB² + BC²

⇒ AC² = AB² + BC²

⇒ AC² = AB² + BC²

⇒ (AC² = (4x)² + (3x)²

⇒ AC² = 16x² + 9x²

⇒ AC² = 25x²

⇒ AC² = (5x)²

⇒ AC = 5x

Sin θ = perpendicular/Hypotenuse

= AB/AC

= 4x/5x

= 4/5

cos θ = Base/Hypotenuse

= BC/AC

= 3x/5x

= 3/5

sin θ + cos θ

= 4/5 + 3/5

= (4 + 3)/5

= 7/5

Hence,

Sin θ + cos θ = 7/5 = 1 2/5

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