Advertisements
Advertisements
प्रश्न
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)
Advertisements
उत्तर
Given sin x = `15/17, 0 < x < pi/2`
We have cos2x + sin2x = 1
∴ cos2x = 1 – sin2x
= `1 - (15/17)^2`
= `1 - 225/289`
cos2x = `(289 - 225)/289 = 64/289`
cos x = `+- sqrt(64/289)`
= `+- 8/17`
Given that `0 < x < pi/2`, that is x lies in the first quadrant
∴ cos x is positive.
cos x = `8/17`
Also given cos y = `12/13, 0 < x < pi/2`
We have cos2y + sin2y = 1
sin2y = 1 – cos2y
sin2y = `1 - (12/13)^2 = 1 - 14/169`
sin2y = `(169 - 144)/169 = 25/169`
sin y = `+- sqrt(25/169) = +- 5/13`
Since `0 < y < pi/2, y lies in the first quadrant sin y is positive.
∴ sin y = `5/13`
sin x = `15/17`
sin y = `5/13`
cos x = `8/17`
cos y = `12/13`
sin(x + y) = sin x cos y + cos x sin y
= `15/17 * 12/13 + 8/17 * 5/13`
sin(x + y) = `180/221 + 40/221`
= `220/221`
APPEARS IN
संबंधित प्रश्न
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
If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2`, find the value of cos(x − y)
If sin A = `3/5` and cos B = `9/41 0 < "A" < pi/2, 0 < "B" < pi/2`, find the value of sin(A + B)
Prove that cos 8θ cos 2θ = cos25θ – sin23θ
Show that cos2 A + cos2 B – 2 cos A cos B cos(A + B) = sin2(A + B)
If tan x = `"n"/("n" + 1)` and tan y = `1/(2"n" + 1)`, find tan(x + y)
If θ + Φ = α and tan θ = k tan Φ, then prove that sin(θ – Φ) = `("k" - 1)/("k" + 1)` sin α
Find the value of cos 2A, A lies in the first quadrant, when tan A `16/63`
If θ is an acute angle, then find `cos (pi/4 + theta/2)`, when sin θ = `8/9`
If cos θ = `1/2 ("a" + 1/"a")`, show that cos 3θ = `1/2 ("a"^3 + 1/"a"^3)`
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 sum or difference
sin 35° cos 28°
Express the following as a sum or difference
cos 5θ cos 2θ
Express the following as a product
sin 50° + sin 40°
Show that `(sin 8x cos x - sin 6x cos 3x)/(cos 2x cos x - sin 3x sin 4x)` = tan 2x
Prove that sin x + sin 2x + sin 3x = sin 2x (1 + 2 cos x)
Prove that `(sin(4"A" - 2"B") + sin(4"B" - 2"A"))/(cos(4"A" - 2"B") + cos(4"B" - 2"A"))` = tan(A + B)
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 = `pi/2`, prove the following cos 2A + cos 2B + cos 2C = 1 + 4 sin A sin B sin C
