If r sinθ= $$\frac{7}{2 }$$​and r cos⁡θ=$$\frac{7√3}{2}$$​, then what will be the value of r?

Correct option is D

Given:

We are given:
​$$r \sin \theta = \frac{7}{2}\ \\r \cos \theta = \frac{7 \sqrt{3}}{2}$$​​
We need to find the value of r .

Concept Used:

To find r , we can use the identity:
​$$r^2 = (r \sin \theta)^2 + (r \cos \theta)^2$$​​
which allows us to express r in terms of the given values of $$r \sin \theta \ and \ r \cos \theta$$​.

Solution:

$$r \sin \theta = \frac{7}{2} \ and\ r \cos \theta = \frac{7 \sqrt{3}}{2}$$​into the identity:

​$$r^2 = \left( \frac{7}{2} \right)^2 + \left( \frac{7 \sqrt{3}}{2} \right)^2\\\ \\r^2 = \frac{49}{4} + \frac{49 \times 3}{4}\\\ \\r^2 = \frac{49}{4} + \frac{147}{4} = \frac{196}{4}\\\ \\r = \sqrt{\frac{196}{4}} = \sqrt{49} = 7$$​​

The value of r is 7.