5) (a) Show that the equation cos2x=sinx can be written in the form 2sin^2x+sinx-1=0 (b) Hence,solve cos2x=sinx where -pi leqslant xleqslant pi

Question

Answer 1

(a) To show that the equation

can be written in the form

, we start by using the double-angle identity for cosine:

Given

, we substitute

with

:

Next, we use the Pythagorean identity

to express

in terms of

:

Substitute

into the equation:

Simplify the equation:

Rearrange the equation to the standard quadratic form:

Thus, the equation

can be written in the form

.(b) To solve

where

, we solve the quadratic equation

.Let

. The quadratic equation becomes:

We solve this quadratic equation using the quadratic formula

, where

,

, and

:

So, the solutions for

are:

Now, we find the corresponding values of

:1. For

:

2. For

:

Thus, the solutions to

in the interval

are:

Soru