APPLICATION OF DOImLE. ANGLE IDENIITIES 4.1 Without the use of tables of a calculator prove that \( \frac{1-\cos 2 A}{\sin 2 A}=\tan A \) 42

To prove \( \frac{1-\cos 2A}{\sin 2A}=\tan A \), we can start by using angle identities. Recall that \( \cos 2A = 1 - 2\sin^2 A \) and \( \sin 2A = 2 \sin A \cos A \). This gives us: \[ 1 - \cos 2A = 1 - (1 - 2\sin^2 A) = 2\sin^2 A \] Now, substituting these into the left-hand side: \[ \frac{1 - \cos 2A}{\sin 2A} = \frac{2\sin^2 A}{2\sin A \cos A} = \frac{\sin A}{\cos A} = \tan A \] Thus, we have shown that \( \frac{1 - \cos 2A}{\sin 2A} = \tan A \). --- Understanding trigonometric identities can be incredibly useful in simplifying problems in physics and engineering. For example, in wave motion, angles and their ratios often define the behavior of waves in different mediums. Mastering these identities gives you powerful tools to not just solve equations but also better understand periodic behavior in real-world systems from acoustics to optics! Additionally, a common mistake when working with trigonometric identities is neglecting to simplify fully or to misinterpret the angle transformations. Always double-check each step and remember to apply the correct identities. Visual aids and diagrams can often help in affirming that each transformation leads logically to the end goal.