Question

express sqrt 3 cosx-sinx in the form Rsin (x-a)

Answer 1

Answer:

$-\sqrt{10} \times sin(x - \theta )$

Here, R = -√10 and a = θ

Step-by-step explanation:

Question :

Express 3cosx - sinx in the form of Rsin(x-a)

To Find :

The value of R and the form

Solution :

Here, in the question, we have 3 and -1 as the coefficients of cosx and sinx respectively.

So, to have a format of triangle with sides 3 and 1, hypotenuse will be of √10. As Hypotenuse² = 3² + 1² = 9 + 1 = 10

So, Multiplying and dividing the question with √10, we get

$: \implies \sqrt{10} \times \dfrac{(3 \cos x - \sin x)}{\sqrt{10} }$

$: \implies \sqrt{10} \times \bigg( \dfrac{3\cos x}{\sqrt{10}} - \dfrac{sinx}{\sqrt{10}} \bigg)$

Let us assume,

$\sin \theta = \dfrac{3}{\sqrt{10} } \ ; \ so, \ \cos \theta = \dfrac{1}{\sqrt{10} }$

So, The equation changes of the form

$: \implies \sqrt{10} \times \bigg( \sin \theta \cos x - \cos \theta \sin x\bigg)$

We know that,

$\boxed{sin(A-B) = sinAcosB-cosAsinB}$

So, the form changes to

$: \implies \sqrt{10} \times \bigg( \sin \theta \cos x - \cos \theta \sin x\bigg)$

$: \implies \sqrt{10} \times sin(\theta - x)$

$: \implies -\sqrt{10} \times sin(x - \theta )$

So, this is of the form  Rsin(x-a)

where, R = -√10 and a = θ

Hope it helps!!

Answer 2

Step-by-step explanation:

express sqrt 3 cosx-sinx in the form Rsin (x-a)