Properties of Triangles

1. Trigonometric Ratios:

  • In a right-angled triangle, three fundamental trigonometric ratios relate the angles to the sides:
    • Sine (sin): sin(θ)=Opposite sideHypotenuse
    • Cosine (cos): cos(θ)=Adjacent sideHypotenuse
    • Tangent (tan): tan(θ)=Opposite sideAdjacent side

2. Pythagorean Identity:

  • In a right-angled triangle, the Pythagorean identity expresses the relationship between the trigonometric functions:
    • sin2(θ)+cos2(θ)=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:
    • asin(A)=bsin(B)=csin(C)

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:
    • c2=a2+b22abcos(C)

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

  • c2=a2+b2 (for C=90)

5. Tangent of Half-Angle Formula:

  • The tangent of half an angle can be expressed in terms of the sides of the triangle:
    • tan(θ2)=Opposite sideAdjacent side

6. Extended Law of Sines:

  • For a triangle, the Extended Law of Sines relates the circumradius (R) to the sides:
    • a=2Rsin(A)
    • b=2Rsin(B)
    • c=2Rsin(C)

7. Half-Angle Formulas:

  • The half-angle formulas express the trigonometric functions of half an angle in terms of the original angle:
    • sin(θ2)=±1cos(θ)2
    • cos(θ2)=±1+cos(θ)2
    • The tangent half-angle formula relates the tangent of an angle A to the tangent of A/2:
      • tan(A2)=±1cos(A)1+cos(A)

8. Double-Angle Formulas:

  • The double-angle formulas express the trigonometric functions of twice an angle in terms of the original angle:
    • sin(2θ)=2sin(θ)cos(θ)
    • cos(2θ)=cos2(θ)sin2(θ)
    • tan(2θ)=2tan(θ)1tan2(θ)

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):

    • Projectionhorizontal=Adjacent side×cos(θ)
  • Vertical Projection (onto the opposite side):

    • Projectionvertical=Adjacent side×sin(θ)
  • Projection onto Hypotenuse:

    • Projectionhypotenuse=Hypotenuse×cos(θ) or Hypotenuse×sin(90θ)

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:

  • tan(A2)tan(B2)=tan(ab2)tan(a+b2)

  • tan(A2)tan(B2)=tan(a2)tan(b2)

  • A=12×a×b×sin(θ)