If 7 sinθ + 24 cosθ = 25, then what is the value of (sin θ + cos θ)?

Concept:

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

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

sin2 θ + cos2 θ = 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 + cos² θ = 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) 

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

= \(\rm \frac{31}{25}\)