Question

If the pair of equations x + y = √2 and x sin θ + y cos θ = 1 has infinitely many solutions, then θ = _____

Answer 1

Answer:

I HOPE IT WILL HELP YOU

Step-by-step explanation:

Q =45 is right answer

Answer 2

Given:

The pair of equations x + y = √2 and x sin θ + y cos θ = 1 has infinitely many solutions.

To find:

The value of θ.

Solution:

First of all, we need to understand that if a pair of linear equations ax + by = c and px + qy = r has infinitely many solutions then

$\frac{a}{p} = \frac{b}{q} = \frac{c}{r}$

So,

As given in the question, x + y = √2 and x sin θ + y cos θ = 1 has infinitely many solutions. Thus, using the above formula, we have,

$\frac{1}{sinθ} = \frac{1}{cosθ} = \frac{√2}{1}$

$\frac{1}{sinθ} = \frac{√2}{1} \: and \: \frac{1}{cosθ} = \frac{√2}{1}$

$sin θ = cos θ = \frac{1}{ \sqrt{2} } \: \: (i)$

From trigonometric ratios, we know that

$sin 45° = cos 45° = \frac{1}{ \sqrt{2} } \: \: (ii)$

On comparing both (i) and (ii), we have

θ = 45°

Hence, the value of θ = 45°.