Properties of Triangles

1. Trigonometric Ratios:

• In a right-angled triangle, three fundamental trigonometric ratios relate the angles to the sides:
• Sine ($sin$):
• Cosine ($cos$):
• Tangent ($tan$):

2. Pythagorean Identity:

• In a right-angled triangle, the Pythagorean identity expresses the relationship between the trigonometric functions:
• ${\mathrm{sin}}^{2}\left(\theta \right)+{\mathrm{cos}}^{2}\left(\theta \right)=1$

3. Sine Law (Law of Sines):

• For any triangle, not necessarily right-angled, the Law of Sines relates the lengths of the sides to the sines of their opposite angles. For any triangle with sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$ respectively, the Law of Sines is given by:
• $\frac{a}{\mathrm{sin}\left(A\right)}=\frac{b}{\mathrm{sin}\left(B\right)}=\frac{c}{\mathrm{sin}\left(C\right)}$

4. Cosine Law (Law of Cosines):

• The Law of Cosines relates the lengths of the sides of any triangle to the cosine of one of its angles. For any triangle with sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$ respectively, the Law of Cosines is given by:
• ${c}^{2}={a}^{2}+{b}^{2}-2ab\mathrm{cos}\left(C\right)$

When the triangle is a right-angled triangle, the Law of Cosines simplifies to the Pythagorean Theorem.

• ${c}^{2}={a}^{2}+{b}^{2}$ (for $\mathrm{\angle }C=9{0}^{\circ }$)

5. Tangent of Half-Angle Formula:

• The tangent of half an angle can be expressed in terms of the sides of the triangle:

6. Extended Law of Sines:

• For a triangle, the Extended Law of Sines relates the circumradius ($R$) to the sides:
• $a=2R\mathrm{sin}\left(A\right)$
• $b=2R\mathrm{sin}\left(B\right)$
• $c=2R\mathrm{sin}\left(C\right)$

7. Half-Angle Formulas:

• The half-angle formulas express the trigonometric functions of half an angle in terms of the original angle:
• $\mathrm{sin}\left(\frac{\theta }{2}\right)=±\sqrt{\frac{1-\mathrm{cos}\left(\theta \right)}{2}}$
• $\mathrm{cos}\left(\frac{\theta }{2}\right)=±\sqrt{\frac{1+\mathrm{cos}\left(\theta \right)}{2}}$
• The tangent half-angle formula relates the tangent of an angle $A$ to the tangent of $A\mathrm{/}2$:
• $\mathrm{tan}\left(\frac{A}{2}\right)=±\sqrt{\frac{1-\mathrm{cos}\left(A\right)}{1+\mathrm{cos}\left(A\right)}}$

8. Double-Angle Formulas:

• The double-angle formulas express the trigonometric functions of twice an angle in terms of the original angle:
• $\mathrm{sin}\left(2\theta \right)=2\mathrm{sin}\left(\theta \right)\mathrm{cos}\left(\theta \right)$
• $\mathrm{cos}\left(2\theta \right)={\mathrm{cos}}^{2}\left(\theta \right)-{\mathrm{sin}}^{2}\left(\theta \right)$
• $\mathrm{tan}\left(2\theta \right)=\frac{2\mathrm{tan}\left(\theta \right)}{1-{\mathrm{tan}}^{2}\left(\theta \right)}$

9. Relationship Between Altitude and Sides:

• The altitude of a triangle from a vertex to the opposite side divides the triangle into two right-angled triangles. This relationship can be utilized in trigonometric calculations.

10. Projection Formulas:

• Horizontal Projection (onto the adjacent side):

• Vertical Projection (onto the opposite side):

• Projection onto Hypotenuse:

• ${\text{Projection}}_{\text{hypotenuse}}=\text{Hypotenuse}×\mathrm{cos}\left(\theta \right)$ or $\text{Hypotenuse}×\mathrm{sin}\left(9{0}^{\circ }-\theta \right)$

11. Napier's Analogy (Law of Tangents) in Trigonometry

Napier's Analogy, also known as the Law of Tangents, is a trigonometric formula developed by John Napier in the early 17th century. It provides a relationship between the sides and angles of a triangle, specifically when dealing with spherical triangles.

In a spherical triangle with sides $a$, $b$, and $c$, opposite angles $A$, $B$, and $C$, the Law of Tangents is given by:

• $\frac{\mathrm{tan}\left(\frac{A}{2}\right)}{\mathrm{tan}\left(\frac{B}{2}\right)}=\frac{\mathrm{tan}\left(\frac{a-b}{2}\right)}{\mathrm{tan}\left(\frac{a+b}{2}\right)}$

• $\frac{\mathrm{tan}\left(\frac{A}{2}\right)}{\mathrm{tan}\left(\frac{B}{2}\right)}=\frac{\mathrm{tan}\left(\frac{a}{2}\right)}{\mathrm{tan}\left(\frac{b}{2}\right)}$

• $A=\frac{1}{2}×a×b×\mathrm{sin}\left(\theta \right)$