Question

Rewrite in terms of sin(x) and cos(x).

sin(x+[7pi/4])

Answer 1

using the addition formulae for sin

• sin(x + y ) = sinxcosy + cosxsiny

sin (x + (7π/4))

= sinx cos (7π/4) + cosxsin (7π/4)

[ the related acute angle for 7π/4 is π/4 ]

= sinxcos (π/4) + cosx (- sin (π/4 )

= sinx × √2 / 2 - cosx × √2 /2

= √2 / 2(sinx - cosx )

Answer 2

$\mathbf{sin(x + \frac{7\pi}{4}})}$ in terms of sin(x) and cos(x) s $\mathbf{ \frac{\sqrt 2}{2} (sin(x) - cos(x))}$

The expression is given as:

$\mathbf{sin(x + \frac{7\pi}{4}})}$

Using sine rule, we have:

$\mathbf{sin(a + b ) = sin(a)cos(b) + cos(a)sin(b)}$

So, the expression becomes

$\mathbf{sin(x + \frac{7\pi}{4} ) = sin(x)cos(\frac{7\pi}{4}) + cos(x)sin(\frac{7\pi}{4})}$

7π/4 is corresponding to π/4.

So, we have:

$\mathbf{sin(x + \frac{7\pi}{4} ) = sin(x)cos(\frac{\pi}{4}) + cos(x)sin(-\frac{\pi}{4})}$

Evaluate the sine and the cosine of the angle

$\mathbf{sin(x + \frac{7\pi}{4} ) = sin(x) \times \frac{\sqrt 2}{2} - cos(x)\times \frac{\sqrt 2}{2} }$

Factor out √2 / 2

$\mathbf{sin(x + \frac{7\pi}{4} ) = \frac{\sqrt 2}{2} (sin(x) - cos(x))}$

Hence, the equivalent of $\mathbf{sin(x + \frac{7\pi}{4}})}$ is $\mathbf{ \frac{\sqrt 2}{2} (sin(x) - cos(x))}$

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