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If 7 sinθ + 24 cosθ = 25, then what is the value of (sin θ + cos θ)?
1. 1
2. \(\dfrac{26}{25}\)
3. \(\dfrac{6}{5}\)
4. \(\dfrac{31}{25}\)

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Correct Answer - Option 4 : \(\dfrac{31}{25}\)

Concept:

\(\rm \sin \theta = \frac{Perpendicular}{Hypotenuse}\)

\(\rm \cos \theta = \frac{Base}{Hypotenuse} \)

sinθ + cosθ = 1 

Calculation:

7 sinθ + 24 cosθ = 25

Dividing by 25 on both the sides, we get

\(\rm \frac{7}{25}\)sinθ + \(\rm \frac{24}{25}\)cosθ = 1      ....(i)

We know that,

sin2 θ + cosθ = 1  

sin θ.sin θ + cos θ.cos θ = 1      ....(ii)

On comparing equ (i) and (ii)

sin θ = \(\rm \frac{7}{25}\) 

cos θ = \(\rm \frac{24}{25}\)

Now, (sinθ + cosθ) 

\(\rm \frac{7}{25}\)\(\rm \frac{24}{25}\)

\(\rm \frac{31}{25}\)

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