# Trigonometrical ratios for various angles

The trigonometric ratios for common angles (0°, 30°, 45°, 60°, 90°) are often memorized and used frequently in calculations.

1. Sine ($\mathrm{sin}$):

• Definition:
• Common Values:
• $\mathrm{sin}\left({0}^{\circ }\right)=0$
• $\mathrm{sin}\left(3{0}^{\circ }\right)=\frac{1}{2}$
• $\mathrm{sin}\left(4{5}^{\circ }\right)=\frac{\sqrt{2}}{2}$
• $\mathrm{sin}\left(6{0}^{\circ }\right)=\frac{\sqrt{3}}{2}$
•

2. Cosine ($\mathrm{cos}$):

• Definition:
• Common Values:
• $\mathrm{cos}\left({0}^{\circ }\right)=1$
• $\mathrm{cos}\left(3{0}^{\circ }\right)=\frac{\sqrt{3}}{2}$
• $\mathrm{cos}\left(4{5}^{\circ }\right)=\frac{\sqrt{2}}{2}$
• $\mathrm{cos}\left(6{0}^{\circ }\right)=\frac{1}{2}$
• $\mathrm{cos}\left(9{0}^{\circ }\right)=0$

3. Tangent ($\mathrm{tan}$):

• Definition:
• Common Values:
• $\mathrm{tan}\left({0}^{\circ }\right)=0$
• $\mathrm{tan}\left(3{0}^{\circ }\right)=\frac{1}{\sqrt{3}}$
• $\mathrm{tan}\left(4{5}^{\circ }\right)=1$
• $\mathrm{tan}\left(6{0}^{\circ }\right)=\sqrt{3}$
• $\mathrm{tan}\left(9{0}^{\circ }\right)=\text{Undefined}$
•

4. Cosecant ($\mathrm{csc}$), Secant ($\mathrm{sec}$), Cotangent ($\mathrm{cot}$):

• Cosecant: $\mathrm{csc}\left(\theta \right)=\frac{1}{\mathrm{sin}\left(\theta \right)}$
• Secant: $\mathrm{sec}\left(\theta \right)=\frac{1}{\mathrm{cos}\left(\theta \right)}$
• Cotangent: $\mathrm{cot}\left(\theta \right)=\frac{1}{\mathrm{tan}\left(\theta \right)}$

Important Points:

• Trigonometric ratios are defined for all acute angles in a right-angled triangle.
• In the unit circle, trigonometric ratios can be extended to any angle.
• For complementary angles (angles that add up to 90 degrees), the sine and cosine ratios are complementary (i.e., $\mathrm{sin}\left(\theta \right)=\mathrm{cos}\left(9{0}^{\circ }-\theta \right)$ and vice versa).