BodhiAI

Nov. 9, 2019
Angles of a Triangle and the Related Trigonometrical Identities

a) If A, B, C be the angles of a triangle, then A + B + C = 1800

A + B = 1800 - C, B + C = 1800 - A, C + A = 1800 - B

sin (A+B) = sin (1800-C) = sin C,

Similarly, sin

Important

The sine of the sum of two angles of a triangle is equal to the sine of the remaining angle.

Again,

Important

The cosine of the sum of two angles of a triangle is equal to (-1) times the cosine of the remainins angle

b) If A, B, C be the angles of a triangle, then

Similarly,

Important

The sine of half the sum of the two angles of a triangle is equal to cosine of half the remaining angle.

Again,

SImilarly,

Important

The cosine of half the sum of two angles of a triangle is equal to sine of half the remaining angle.

c) When the angles A, B, C satisfy some given relation, many identical relations hold between their trigonometrical ratios. The following important identities should be remembered.

(i) OR

(ii)

(iii) sin2A + sin2B + sin2C = 4 sinA sinB sinC

(iv)

(v) cos 2A + cos2B + cos2C = -1 - 4cos Acos B cosC

(vi) cosA + cosB + cosC = 1+ 4

(vii) sinB sinC cosA + sinC sinA cosB + sinA sinB cosC = 1+cosA cosB cosC

(viii) sinA cosB cosC + sinB cosC cosA+ sinC cosA cosB =sinA sinB sinC