Trigonometry: further identities, equations, solutions

1️⃣ Additional Trigonometric Identities

Beyond the basic sine, cosine and tangent, there are many useful identities that simplify equations and help in proofs.

• Double‑Angle Identities – Think of a spinning arrow that doubles its angle:
  • $\sin 2\theta = 2\sin\theta\cos\theta$
  • $\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$
• Half‑Angle Identities – Splitting the angle in half:
  • $\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{2}}$
  • $\cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}}$
• Product‑to‑Sum Identities – Turning products into sums:
  • $\sin A \sin B = \frac{1}{2}\bigl[\cos(A-B) - \cos(A+B)\bigr]$
  • $\cos A \cos B = \frac{1}{2}\bigl[\cos(A-B) + \cos(A+B)\bigr]$
  • $\sin A \cos B = \frac{1}{2}\bigl[\sin(A+B) + \sin(A-B)\bigr]$
• Sum‑to‑Product Identities – Turning sums into products:
  • $\sin A + \sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}$
  • $\cos A + \cos B = 2\cos\frac{A+B}{2}\cos\frac{A-B}{2}$

2️⃣ Solving Trigonometric Equations

To solve equations, isolate the trigonometric function, use identities, and remember the general solution.

1. Rewrite the equation using a single trigonometric function if possible.
2. Apply the relevant identity to simplify.
3. Set the simplified expression equal to zero or solve for the angle.
4. Use the general solution:
   • $\sin x = k \;\Rightarrow\; x = (-1)^n\arcsin k + n\pi$
   • $\cos x = k \;\Rightarrow\; x = \pm\arccos k + 2n\pi$
   • $\tan x = k \;\Rightarrow\; x = \arctan k + n\pi$

Example:

$$\sin 2x = \sqrt{3}\cos x$$

Use $\sin 2x = 2\sin x\cos x$:

$$2\sin x\cos x = \sqrt{3}\cos x \;\Rightarrow\; \cos x(2\sin x - \sqrt{3}) = 0$$

So either $\cos x = 0$ or $2\sin x = \sqrt{3}$.

Solutions:

• $\cos x = 0 \;\Rightarrow\; x = \frac{\pi}{2} + n\pi$
• $\sin x = \frac{\sqrt{3}}{2} \;\Rightarrow\; x = \frac{\pi}{3} + 2n\pi$ or $x = \frac{2\pi}{3} + 2n\pi$

3️⃣ Applications & Examples 🚀

Trigonometric identities are handy in geometry, physics and real‑world problems.

4️⃣ Examination Tips 📚