Advertisements
Advertisements
Question
If A + B + C = 180◦, prove that sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
Advertisements
Solution
LHS = (sin 2A + sin 2B) + sin 2C
= 2 sin(A + B) cos(A – B) + 2 sin C cos C
[sin(A + B) = sin(180° – C) = sin C]
= 2 sin C cos(A – B) + 2 sin C cos C
= 2 sin C [cos(A – B) + cos C]
{cos C = cos[180° – (A + B)] = – cos (A + B)}
= 2 sin C [cos(A – B) – cos(A + B)]
= `2sin"C"{2sin (2"A")/2 sin (2"B")/2}`
= 4 sin A sin B sin C
= R.H.S
APPEARS IN
RELATED QUESTIONS
Find the values of tan(1050°)
`(5/7, (2sqrt(6))/7)` is a point on the terminal side of an angle θ in standard position. Determine the six trigonometric function values of angle θ
If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2` find the value of sin(x + y)
Find cos(x − y), given that cos x = `- 4/5` with `pi < x < (3pi)/2` and sin y = `- 24/25` with `pi < y < (3pi)/2`
Prove that sin(45° + θ) – sin(45° – θ) = `sqrt(2) sin θ`
Prove that cos(A + B) cos C – cos(B + C) cos A = sin B sin(C – A)
Prove that cos(A + B) cos(A – B) = cos2A – sin2B = cos2B – sin2A
Prove that sin2(A + B) – sin2(A – B) = sin2A sin2B
Show that cos2 A + cos2 B – 2 cos A cos B cos(A + B) = sin2(A + B)
Prove that cot(A + B) = `(cot "A" cot "B" - 1)/(cot "A" + cot "B")`
Find the value of tan(α + β), given that cot α = `1/2`, α ∈ `(pi, (3pi)/2)` and sec β = `- 5/3` β ∈ `(pi/2, pi)`
Prove that `tan (pi/4 + theta) - tan(pi/4 - theta)` = 2 tan 2θ
Prove that `32(sqrt(3)) sin pi/48 cos pi/48 cos pi/24 cos pi/12 cos pi/6` = 3
Express the following as a product
cos 65° + cos 15°
Express the following as a product
cos 35° – cos 75°
Prove that 1 + cos 2x + cos 4x + cos 6x = 4 cos x cos 2x cos 3x
Show that cot(A + 15°) – tan(A – 15°) = `(4cos2"A")/(1 + 2 sin2"A")`
If A + B + C = 180°, prove that `tan "A"/2 tan "B"/2 + tan "B"/2 tan "C"/2 + tan "C"/2 tan "A"/2` = 1
Choose the correct alternative:
cos 1° + cos 2° + cos 3° + ... + cos 179° =
