# Trigonometrical Ratios of Allied Angles

In trigonometry, allied angles are angles that share the same terminal side. Knowing the trigonometric ratios (sine, cosine, tangent, cosecant, secant, and cotangent) of these angles can simplify problem-solving. Let's explore the trigonometrical ratios of allied angles.

1. Co-Function Identities:

• Sine and Cosine:
• $\mathrm{sin}\left(\theta \right)=\mathrm{cos}\left(90\mathrm{°}-\theta \right)$
• $\mathrm{cos}\left(\theta \right)=\mathrm{sin}\left(90\mathrm{°}-\theta \right)$
• These identities highlight the relationship between sine and cosine of complementary angles.

2. Double-Angle Identities:

• 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 identities express the sine and cosine of double angles in terms of the original angle.

3. Sum and Difference Formulas:

• Sine Sum and Difference:
• $\mathrm{sin}\left(\alpha +\beta \right)=\mathrm{sin}\left(\alpha \right)\mathrm{cos}\left(\beta \right)+\mathrm{cos}\left(\alpha \right)\mathrm{sin}\left(\beta \right)$
• $\mathrm{sin}\left(\alpha -\beta \right)=\mathrm{sin}\left(\alpha \right)\mathrm{cos}\left(\beta \right)-\mathrm{cos}\left(\alpha \right)\mathrm{sin}\left(\beta \right)$
• Cosine Sum and Difference:
• $\mathrm{cos}\left(\alpha +\beta \right)=\mathrm{cos}\left(\alpha \right)\mathrm{cos}\left(\beta \right)-\mathrm{sin}\left(\alpha \right)\mathrm{sin}\left(\beta \right)$
• $\mathrm{cos}\left(\alpha -\beta \right)=\mathrm{cos}\left(\alpha \right)\mathrm{cos}\left(\beta \right)+\mathrm{sin}\left(\alpha \right)\mathrm{sin}\left(\beta \right)$
• These formulas allow the calculation of trigonometric ratios for the sum or difference of two angles.

4. 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.

5. Coterminal Angle Relationships:

• Angles that differ by a multiple of 360 degrees or $2\pi$ radians are coterminal.
• Trigonometric ratios for coterminal angles are the same.