6. (i) Without using a calculator, find the exact value of (sin 22.5° + cos 22.5°)² You must show each stage of your working - Edexcel - A-Level Maths Pure - Question 27 - 2013 - Paper 1

We start with the equation:

$ext{cos } 2θ + ext{sin } θ = 1$

Using the double angle formula for cosine:

$ext{cos } 2θ = 1 - 2 ext{sin}² θ$

Substituting this into the equation gives:

$1 - 2 ext{sin}² θ + ext{sin} θ = 1$

Simplifying, we have:

$-2 ext{sin}² θ + ext{sin} θ = 0$

Rearranging it, we find:

$2 ext{sin}² θ - ext{sin} θ = 0$

Thus, it can be written in the form:

$k ext{sin}² θ - ext{sin} θ = 0$

where the value of k is 2.

From the previous step, we derived:

$2 ext{sin}² θ - ext{sin} θ = 0$

Factoring gives:

$ext{sin} θ (2 ext{sin} θ - 1) = 0$

This implies two cases:

1. $ext{sin} θ = 0$
2. $$$$

ightarrow ext{sin} θ = rac{1}{2}$$

For the first case, ext{sin} θ = 0 occurs at: $θ = 0°, 180°$
Since we require 0 < θ < 360°: $θ = 180°$

For the second case, ext{sin} θ = rac{1}{2} occurs at: $θ = 30°, 150°$

Thus, the complete solution set for 0 < θ < 360° is: $θ = 30°, 150°, 180°$.