please solve these if possible
Q1. If sinx +siny=3(cosy-cosx) then the value of sin3x/sin3y.
Q2. If sina ,cosa,and tan a are in g.p.then cos cubea+cos square a is equal to
Q1.
From the given relation, we can write, 3 cos x + sin x = 3 cos y - sin y
put r cos ƒ¿=3 and r sin ƒ¿ = 1 , then we get , r = �ã 10, tanƒ¿= 1/3
so r cos ( x - ƒ¿) = r cos (x+ƒ¿)
x = - y or x = �}(y+ƒ¿)
clearly x = - y satisfies the equation
therefore, x = - y
so 3x = -3y
sin 3x = sin (-3y) = - sin 3y
sin 3x / sin 3y = -1
ƒ¿= Greek letter Alpha
Q2)Because sina, cosa, tana - g.p.
cos^2(a)=sina*tana=sin^2(a)/cosa
cos^2(a)=(1-cos^2(a))/cosa
cos^3(a)=1-cos^2(a)
Sure! Let's solve each problem step by step.
Q1. If sinx + siny = 3(cosy - cosx), find the value of sin3x/sin3y.
To find the value of sin3x/sin3y, we can rewrite sin3x and sin3y using the multiple angle formula for sine.
sin3x = 3sinx - 4sin^3x
sin3y = 3siny - 4sin^3y
Now, we can substitute these expressions into the ratio sin3x/sin3y.
sin3x/sin3y = (3sinx - 4sin^3x)/(3siny - 4sin^3y)
Now, let's manipulate the given equation sinx + siny = 3(cosy - cosx) to express sinx and siny in terms of cosx and cosy.
sinx = √(1 - cos^2x) (using the identity sin^2x + cos^2x = 1)
siny = √(1 - cos^2y) (using the identity sin^2y + cos^2y = 1)
Substituting these expressions into the ratio sin3x/sin3y:
sin3x/sin3y = (3√(1 - cos^2x) - 4(√(1 - cos^2x))^3) / (3√(1 - cos^2y) - 4(√(1 - cos^2y))^3)
Simplifying the expression might require some computational work, including some calculations.
Q2. If sina, cosa, and tana are in geometric progression (G.P.), find the value of cos^3a + cos^2a.
For a geometric progression, we have:
cosa, sina, tana
We can relate these terms using the trigonometric identity tan^2a = sin^2a / cos^2a.
Since tan a = sina / cosa, we can substitute this into the identity:
tan^2a = (sina / cosa)^2
Simplifying, we get:
sin^2a = tan^2a * cos^2a
sin^2a = (sina / cosa)^2 * cos^2a
sin^2a = sin^2a
This identity confirms that sina, cosa, and tana are indeed in geometric progression.
Now, let's use the relation sina^2 = (sina / tana)^2 to obtain an expression for sina^2 in terms of tana:
sina^2 = (sina / tana)^2 * tana^2
sina^2 = (sina^2 / tana^2) * tana^2
sina^2 = sina^2
This also confirms that sina, cosa, and tana are in geometric progression.
To find the value of cos^3a + cos^2a, we can use the properties of a geometric progression.
Let's assume the common ratio is r.
Since sina, cosa, and tana are in G.P., we have:
cosa = sina * r
Using the identity cos^2a + sin^2a = 1, we can substitute these values:
(sina * r)^2 + sina^2 = 1
sina^2 * r^2 + sina^2 - 1 = 0
We now have a quadratic equation in terms of sina^2. Solving this equation will give us the values of sina^2 and allow us to calculate cos^3a + cos^2a.
Once we find sina^2, we can substitute it into cos^3a + cos^2a and calculate the value.