Advertisements
Advertisements
Question
If A + B + C = 180◦, prove that sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
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°)
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)
If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2`, find the value of tan(x + y)
Find the value of sin 105°
Prove that sin 75° – sin 15° = cos 105° + cos 15°
Prove that cos(A + B) cos C – cos(B + C) cos A = sin B sin(C – A)
Prove that sin(n + 1) θ sin(n – 1) θ + cos(n + 1) θ cos(n – 1)θ = cos 2θ, n ∈ Z
Show that tan(45° − A) = `(1 - tan "A")/(1 + tan "A")`
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)`
If θ is an acute angle, then find `sin (pi/4 - theta/2)`, when sin θ = `1/25`
If cos θ = `1/2 ("a" + 1/"a")`, show that cos 3θ = `1/2 ("a"^3 + 1/"a"^3)`
Prove that sin 4α = `4 tan alpha (1 - tan^2alpha)/(1 + tan^2 alpha)^2`
Show that `cot(7 1^circ/2) = sqrt(2) + sqrt(3) + sqrt(4) + sqrt(6)`
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 35° – cos 75°
Prove that `(sin 4x + sin 2x)/(cos 4x + cos 2x)` = tan 3x
If ∆ABC is a right triangle and if ∠A = `pi/2` then prove that cos B – cos C = `- 1 + 2sqrt(2) cos "B"/2 sin "C"/2`
Choose the correct alternative:
If cos 28° + sin 28° = k3, then cos 17° is equal to
Choose the correct alternative:
`(1 + cos pi/8) (1 + cos (3pi)/8) (1 + cos (5pi)/8) (1 + cos (7pi)/8)` =
