# prove that this expression equals the area of a specific triangle [closed]

Prove that the area of a triangle with one of its angles $$\frac{\pi}4$$ rad, and the side opposite to this angle is $$2$$cm equals $$\sin(2\theta)-\cos(2\theta)+1$$, where $$\theta$$ is the angle adjacent to the $$2$$cm side.

and here is a picture to make things cleaner: • Use the sine rule to find another side. Aug 14, 2022 at 7:08

Let $$a = 2, A = \frac {\pi} 4, B = \theta$$

Then,

$$\frac{b}{\sin \theta} = \frac{2}{\frac{\sqrt{2}}2} \implies b = 2\sqrt 2 \sin \theta$$

Then,

$$\sin C = \sin(A+B) = \frac {\sqrt{2}}2 (\sin \theta + \cos \theta)$$

Then,

$$S = \frac 12 ab\sin C =\frac 12 \times 2\times 2\sqrt 2\sin \theta\times \frac {\sqrt{2}}2 (\sin \theta + \cos \theta)\\ = 2\sin \theta (\sin \theta + \cos \theta)=2\sin^2\theta+2\sin \theta \cos \theta = 1-\cos 2\theta + \sin 2\theta$$

• I really appreciate for your answer! Aug 14, 2022 at 9:28
• @AmrWaleed You didn't mark the points with letters so I marked them. A is the known angle, a is the known side and B is $\theta$, mark these on the picture you provided and you are set. Aug 14, 2022 at 9:29
• I asked about the step of law of sines, but i just recognized what you did there, thanks anyway Aug 14, 2022 at 9:32