# Compound Angles

In trigonometry, compound angles are formed by combining two or more angles to create a new angle. Understanding the trigonometric relationships involving compound angles is essential for simplifying expressions and solving problems. Here are the key concepts and formulas related to compound angles:

1. Sum and Difference Formulas:

• Sine Sum and Difference:
• $\mathrm{sin}\left(\alpha ±\beta \right)=\mathrm{sin}\left(\alpha \right)\mathrm{cos}\left(\beta \right)±\mathrm{cos}\left(\alpha \right)\mathrm{sin}\left(\beta \right)$
• Cosine Sum and Difference:
• $\mathrm{cos}\left(\alpha ±\beta \right)=\mathrm{cos}\left(\alpha \right)\mathrm{cos}\left(\beta \right)\mp \mathrm{sin}\left(\alpha \right)\mathrm{sin}\left(\beta \right)$
• These formulas express the sine and cosine of the sum or difference of two angles.

2. Double-Angle Formulas:

• Sine Double-Angle:
• $\mathrm{sin}\left(2\theta \right)=2\mathrm{sin}\left(\theta \right)\mathrm{cos}\left(\theta \right)$
• Cosine Double-Angle:
• $\mathrm{cos}\left(2\theta \right)={\mathrm{cos}}^{2}\left(\theta \right)-{\mathrm{sin}}^{2}\left(\theta \right)$
• $\mathrm{cos}\left(2\theta \right)=2{\mathrm{cos}}^{2}\left(\theta \right)-1$
• $\mathrm{cos}\left(2\theta \right)=1-2{\mathrm{sin}}^{2}\left(\theta \right)$
• These formulas express the sine and cosine of double angles in terms of the original angle.

3.  Triple-Angle Formulas:

• The triple-angle formulas are extensions of the double-angle formulas.
• Sine Triple-Angle:
• $\mathrm{sin}\left(3\theta \right)=3\mathrm{sin}\left(\theta \right)-4{\mathrm{sin}}^{3}\left(\theta \right)$
• Cosine Triple-Angle:
• $\mathrm{cos}\left(3\theta \right)=4{\mathrm{cos}}^{3}\left(\theta \right)-3\mathrm{cos}\left(\theta \right)$
• These formulas provide expressions for the sine and cosine of three times an angle.

4. Tangent Sum and Difference:

• $\mathrm{tan}\left(\alpha ±\beta \right)=\frac{\mathrm{tan}\left(\alpha \right)±\mathrm{tan}\left(\beta \right)}{1\mp \mathrm{tan}\left(\alpha \right)\mathrm{tan}\left(\beta \right)}$
• This formula expresses the tangent of the sum or difference of two angles in terms of the tangents of the individual angles.

5. Product-to-Sum Formulas:

• Sine Product-to-Sum:
• $\mathrm{sin}\left(\alpha \right)\mathrm{sin}\left(\beta \right)=\frac{1}{2}\left[\mathrm{cos}\left(\alpha -\beta \right)-\mathrm{cos}\left(\alpha +\beta \right)\right]$
• Cosine Product-to-Sum:
• $\mathrm{cos}\left(\alpha \right)\mathrm{cos}\left(\beta \right)=\frac{1}{2}\left[\mathrm{cos}\left(\alpha -\beta \right)+\mathrm{cos}\left(\alpha +\beta \right)\right]$
• Sine-Cosine Product-to-Sum:
• $\mathrm{sin}\left(\alpha \right)\mathrm{cos}\left(\beta \right)=\frac{1}{2}\left[\mathrm{sin}\left(\alpha +\beta \right)+\mathrm{sin}\left(\alpha -\beta \right)\right]$
• These formulas allow you to express the product of trigonometric functions as a sum or difference.

6. Half-Angle Formulas:

• Sine Half-Angle:
• $\mathrm{sin}\left(\frac{\theta }{2}\right)=±\sqrt{\frac{1-\mathrm{cos}\left(\theta \right)}{2}}$
• Cosine Half-Angle:
• $\mathrm{cos}\left(\frac{\theta }{2}\right)=±\sqrt{\frac{1+\mathrm{cos}\left(\theta \right)}{2}}$
• These formulas express the sine and cosine of half-angles in terms of the original angle.

Example:

• Simplify $\mathrm{sin}\left(7{5}^{\circ }\right)$ using compound angle formulas:
• $\mathrm{sin}\left(7{5}^{\circ }\right)=\mathrm{sin}\left(4{5}^{\circ }+3{0}^{\circ }\right)$
• Applying the sum formula: $\mathrm{sin}\left(7{5}^{\circ }\right)=\mathrm{sin}\left(4{5}^{\circ }\right)\mathrm{cos}\left(3{0}^{\circ }\right)+\mathrm{cos}\left(4{5}^{\circ }\right)\mathrm{sin}\left(3{0}^{\circ }\right)$