If $$ \cos x+\sin x=k $$, express the following in terms $$ k $$ : $$ \cos \left(x-45^{\circ}\right) $$

Question

UpStudy AI · Accepted Answer

To express $$ \cos \left(x - 45^{\circ}\right) $$ in terms of $$ k $$, we can use the cosine subtraction formula:

$$
\cos(a - b) = \cos a \cos b + \sin a \sin b
$$

In our case, let $$ a = x $$ and $$ b = 45^{\circ} $$. Thus, we have:

$$
\cos \left(x - 45^{\circ}\right) = \cos x \cos 45^{\circ} + \sin x \sin 45^{\circ}
$$

We know that:

$$
\cos 45^{\circ} = \sin 45^{\circ} = \frac{\sqrt{2}}{2}
$$

Substituting these values into the equation gives:

$$
\cos \left(x - 45^{\circ}\right) = \cos x \cdot \frac{\sqrt{2}}{2} + \sin x \cdot \frac{\sqrt{2}}{2}
$$

This can be factored as:

$$
\cos \left(x - 45^{\circ}\right) = \frac{\sqrt{2}}{2} (\cos x + \sin x)
$$

Given that $$ \cos x + \sin x = k $$, we can substitute $$ k $$ into the equation:

$$
\cos \left(x - 45^{\circ}\right) = \frac{\sqrt{2}}{2} k
$$

Thus, the expression for $$ \cos \left(x - 45^{\circ}\right) $$ in terms of $$ k $$ is:

$$
\cos \left(x - 45^{\circ}\right) = \frac{\sqrt{2}}{2} k
$$
$$ \cos \left(x - 45^{\circ}\right) = \frac{\sqrt{2}}{2} k $$

If \( \cos x+\sin x=k \), express the following in terms \( k \) : \( \cos \left(x-45^{\circ}\right) \)