# Multiple and Submultiple Angles

Multiple Angles:

In trigonometry, multiple angles are angles that result from the repetition or combination of a given angle. Understanding the properties and relationships associated with multiple angles is essential in simplifying expressions, solving equations, and analyzing periodic functions. Here are key concepts related to multiple angles:

1. Definition:

• A multiple angle is formed by multiplying a given angle by an integer $n$.
• Examples: $2\theta$, $3\theta$, $4\theta$, where $\theta$ is the original angle.

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 trigonometric functions of double angles.

3. Triple-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:
• $\mathrm{cos}\left(3\theta \right)=4{\mathrm{cos}}^{3}\left(\theta \right)-3\mathrm{cos}\left(\theta \right)$
• Triple-angle formulas extend the concept of double angles to triple angles.

4. Quadruple-Angle Formula:

• Cosine Quadruple-Angle:
• $\mathrm{cos}\left(4\theta \right)=8{\mathrm{cos}}^{4}\left(\theta \right)-8{\mathrm{cos}}^{2}\left(\theta \right)+1$
• The quadruple-angle formula expresses the cosine of a quadruple angle in terms of the original angle.

5. General Formulas for Multiple Angles:

• Sine Multiple Angle:
• $\mathrm{sin}\left(n\theta \right)={\mathrm{sin}}^{n}\left(\theta \right)\mathrm{cos}\left(\frac{\left(n-1\right)\pi }{2}\right)-\left(\genfrac{}{}{0px}{}{n}{2}\right){\mathrm{sin}}^{n-2}\left(\theta \right){\mathrm{cos}}^{2}\left(\theta \right)+\left(\genfrac{}{}{0px}{}{n}{3}\right){\mathrm{sin}}^{n-3}\left(\theta \right){\mathrm{cos}}^{3}\left(\theta \right)-\dots$
• Cosine Multiple Angle:
• $\mathrm{cos}\left(n\theta \right)={\mathrm{cos}}^{n}\left(\theta \right)-\left(\genfrac{}{}{0px}{}{n}{2}\right){\mathrm{cos}}^{n-2}\left(\theta \right){\mathrm{sin}}^{2}\left(\theta \right)+\left(\genfrac{}{}{0px}{}{n}{3}\right){\mathrm{cos}}^{n-3}\left(\theta \right){\mathrm{sin}}^{3}\left(\theta \right)-\dots$
• These formulas provide a general expression for the trigonometric functions of multiple angles.

Submultiple Angles:

In trigonometry, submultiple angles refer to angles that result from dividing a given angle by an integer. Understanding the properties and relationships associated with submultiple angles is crucial for simplifying expressions and solving various mathematical problems. Here are key concepts related to submultiple angles:

1. Definition:

• A submultiple angle is formed by dividing a given angle by an integer $n$.
• Examples: $\frac{\theta }{2}$, $\frac{\theta }{3}$, $\frac{\theta }{4}$, where $\theta$ is the original angle.

2. 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 trigonometric functions of half angles.

3. Third-Angle Formula:

• Sine Third-Angle:
• $\mathrm{sin}\left(\frac{\theta }{3}\right)=\sqrt[3]{\frac{\mathrm{sin}\left(\theta \right)}{4-4{\mathrm{sin}}^{2}\left(\frac{\theta }{3}\right)}}$
•
• The third-angle formula expresses the sine of a third angle in terms of the original angle.

4. Fourth-Angle Formula:

• Cosine Fourth-Angle:
• $\mathrm{cos}\left(\frac{\theta }{4}\right)=\sqrt{\frac{1+\sqrt{2}\mathrm{cos}\left(\theta \right)+{\mathrm{cos}}^{2}\left(\theta \right)}{2\left(1+\sqrt{2}\mathrm{cos}\left(\theta \right)+{\mathrm{cos}}^{2}\left(\theta \right)\right)}}$
•
• The fourth-angle formula expresses the cosine of a fourth angle in terms of the original angle.

5. General Formulas for Submultiple Angles:

• Sine Submultiple Angle:
• $\mathrm{sin}\left(\frac{n\theta }{m}\right)={\mathrm{sin}}^{n}\left(\frac{\theta }{m}\right){\mathrm{cos}}^{n-1}\left(\frac{\theta }{m}\right)\mathrm{sin}\left(\frac{\left(m-1\right)\theta }{m}\right)$
• Cosine Submultiple Angle:
• $\mathrm{cos}\left(\frac{n\theta }{m}\right)={\mathrm{cos}}^{n}\left(\frac{\theta }{m}\right){\mathrm{sin}}^{n-1}\left(\frac{\theta }{m}\right)\mathrm{cos}\left(\frac{\left(m-1\right)\theta }{m}\right)$
• These formulas provide a general expression for the trigonometric functions of submultiple angles.