\( \frac{\text { Example }}{\text { Prove that } \sqrt{2}} \sin \left(A+45^{\circ}\right)=\sin A+\cos A \)

To prove the equation \( \sqrt{2} \sin \left(A + 45^{\circ}\right) = \sin A + \cos A \), we can start by using the sine angle addition formula. Recall that \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). Substituting \( a = A \) and \( b = 45^{\circ} \): \[ \sin(A + 45^{\circ}) = \sin A \cos 45^{\circ} + \cos A \sin 45^{\circ} \] We know that \( \cos 45^{\circ} = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \). Therefore, we get: \[ \sin(A + 45^{\circ}) = \sin A \cdot \frac{\sqrt{2}}{2} + \cos A \cdot \frac{\sqrt{2}}{2} \] Multiplying both sides by \( \sqrt{2} \) gives: \[ \sqrt{2} \sin(A + 45^{\circ}) = \sqrt{2} \left( \sin A \cdot \frac{\sqrt{2}}{2} + \cos A \cdot \frac{\sqrt{2}}{2} \right) \] This simplifies to: \[ \sqrt{2} \sin(A + 45^{\circ}) = \sin A + \cos A \] Thus proving the given statement \( \sqrt{2} \sin \left(A+45^{\circ}\right)=\sin A+\cos A \). Now, if you find yourself confused about angle addition, remember this: visual aids can really help! Sketching the angles can make the relationships clearer. Also, don't forget that some folks mix up sine and cosine values at specific angles (hello, \( 30^{\circ} \), \( 45^{\circ} \), and \( 60^{\circ} \)), but practicing with triangles will help reinforce the correct values over time! For those wishing to dive deeper, consider exploring the connections between trigonometric identities and complex numbers. Euler’s formula, \( e^{ix} = \cos x + i \sin x \), perfectly intertwines these concepts, opening doors to a whole new world of mathematical beauty! Grab a good book or online resource about complex numbers and trigonometric functions to see the synergies!