Law of tangent formula

Trigonometry includes angles and triangles. Trigonometric functions are:

  • Sine,
  • Cosine,
  • Tangent,
  • Cosecant,
  • Secant,
  • Cotangent.

The tangent( in trigonometry) is defined as an angle in a right-angled triangle which has a ratio of perpendicular and base. The tangent of an angle x is written as tan x.

Formula of Law of Tangent

The formula of a tangent in a right triangle PQR, where side opposite angle P, Q , R are p, q , r respectively.

  • p-q/p+q ={ tan (P-Q)/2 }/{ tan (P+Q)/2}
  • q-r/q+r ={ tan (Q-R)/2 }/{ tan (Q+R)/2}
  • r-p/r+p ={ tan (R-P)/2 }/{ tan (R+P)/2}

Example of Tangent Formula

Problem : If in a triangle ABC, ∠B = 90∘, ∠C = 30∘. If the side opposite to ∠B is 4 cm. Find the value of the side opposite to ∠C.

Solution:

b-c/b+c ={ tan (B-C)/2 }/{ tan (B+C)/2}

4-c/4+c ={ tan (90 – 30)/2 }/{ tan (90 +30)/2}

4-c/4+c ={ tan 30}/{ tan (60)}

4-c/4+c ={ 1/√3}/{ √3}

4-c/4+c =⅓

3(4-c) = 4 + c

12 -3c = 4 + c

12 – 4 = c +3c

8 = 4c

8/4 = c

c =2

Example 2: If b = 3 cm, B = 30°, A = 60° in a right triangle ABC, right angled at C, then find the sides opposite to angle A, i.e. a.

Solution:
Given,
b = 3 cm, B = 30°, A = 60°
Using the law of tangent,

a – b/a + b = [tan (A-B)/2]/tan (A + B)/2]

a – 3/a + 3 = [tan (60°- 30°)/2]/tan (60° + 30°)/2]

a – 3/a + 3 = tan 15/tan 45

a – 3/a + 3 = (2 – √3)/1

a – 3 = (2 – √3)(a + 3)

2a + 6 – √3a -3√3 -a + 3 = 0

a(1-√3) + 9 -3√3 = 0

a = (3√3 – 9)/(1 – √3)

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