Advertisements
Advertisements
Question
If A + B + C = 180°, prove that sin2A + sin2B + sin2C = 2 + 2 cos A cos B cos C
Advertisements
Solution
L.H.S = `(1 - cos2"A")/2 + (1 - cos2"B")/2 + (1 - cos 2"C")/2`
Hint: `[sin^2"A" = (1 - cos2"A")/2]`
= `3/2 - 1/2[cos2"A" + cos2"B" + cos2"C"]`
= `3/2 - 1/2 [2cos("A" + "B") cos("A" - "B") + 2cos^2"C" - 1]`
= `3/2 - cos("A" + "B") cos("A" - "B") - cos^2"C" + 1/2`
= 2 + cos C cos(A – B) – cos2
= 2 + cosC[cos(A – B)(cos(A + B)]
[cos(180° – C) – cos C – cos C]
= 2 + cos C [cos(A – B) + cos(A + B)]
= 2+ cos C[2 cos A cos B]
= 2 + 2 cos A cos B cos C
= R.H.S
APPEARS IN
RELATED QUESTIONS
Find the values of sin(480°)
`(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
Find the value of the trigonometric functions for the following:
cos θ = `2/3`, θ lies in the I quadrant
Show that `sin^2 pi/18 + sin^2 pi/9 + sin^2 (7pi)/18 + sin^2 (4pi)/9` = 2
Find the value of tan `(7pi)/12`
Prove that sin(π + θ) = − sin θ.
Prove that cos(A + B) cos(A – B) = cos2A – sin2B = cos2B – sin2A
Prove that cos 8θ cos 2θ = cos25θ – sin23θ
Prove that `tan(pi/4 + theta) tan((3pi)/4 + theta)` = – 1
Find the value of cos 2A, A lies in the first quadrant, when cos A = `15/17`
Find the value of cos 2A, A lies in the first quadrant, when sin A = `4/5`
Prove that cos 5θ = 16 cos5θ – 20 cos3θ + 5 cos θ
Express the following as a product
sin 75° sin 35°
If A + B + C = 180◦, prove that sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
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
If A + B + C = 180°, prove that sin A + sin B + sin C = `4 cos "A"/2 cos "B"/2 cos "C"/2`
If x + y + z = xyz, then prove that `(2x)/(1 - x^2) + (2y)/(1 - y^2) + (2z)/(1 - z^2) = (2x)/(1 - x^2) (2y)/(1 - y^2) (2z)/(1 - z^2)`
Choose the correct alternative:
`1/(cos 80^circ) - sqrt(3)/(sin 80^circ)` =
Choose the correct alternative:
If `pi < 2theta < (3pi)/2`, then `sqrt(2 + sqrt(2 + 2cos4theta)` equals to
Choose the correct alternative:
`(sin("A" - "B"))/(cos"A" cos"B") + (sin("B" - "C"))/(cos"B" cos"C") + (sin("C" - "A"))/(cos"C" cos"A")` is
