Given that sin x + sin y = a and cos x + cos y =a, where a not equal to 0, express sin x + cos x in terms of a.

attemp:
sin x = a - sin y
cos x = a - cos y
sin x + cos x = 2A - (sin y + cos y)

so sinx + siny = cosx + cosy

the trivial and obvious solution is x = y = 45° or π/4

another obvious one is 225° or 5π/4, because of the CAST rule

are there others , not multiples of 45°

recall:

sin a + sin b = 2 sin (1/2)(a+b) cos (1/2)(a-b)
cos a + cos b = 2 cos (1/2)(a+b) cos (1/2)(a-b)

so:
sinx + siny = 2 sin (1/2)(x+y) cos (1/2)(x-y)
cosx + cosy = 2 cos (1/2)(x+y) cos (1/2)(x-y)

2 sin (1/2)(x+y) cos (1/2)(x-y) = 2 cos (1/2)(x+y) cos (1/2)(x-y)
divide both sides by 2cos (1/2)(x-y)
sin (1/2)(x+y) = cos (1/2)(x+y)
divide by cos (1/2)(x+y)

tan (1/2)(x+y) = 1
we know tan 45° = 1
(1/2)(x+y) = 45
x+y = 90

so any pair of complimentary angles will work
e.g let x = 10°, y = 80°
LS = sin10° + sin80°
= cos80° + sin10
= RS

of course, I should have seen the property that
cosØ = sin(90°- Ø)

then sinx + siny = sin(90-x) + sin(90-y)
sinx + siny = sin90cosx - cos90sinx + sin90cosy - cos90siny
sinx + siny = (1)cosx - (0)sinx + (1)cosy - (0)siny
sinx + siny = cosx + cosx

back to the beginning

not answered.

Question asked: express sinx + cosx in term of a.

To express sin x + cos x in terms of a, we can use the given equations:

sin x + sin y = a

cos x + cos y = a

We can solve for sin x and cos x individually in terms of a:

sin x = a - sin y

cos x = a - cos y

Now, let's substitute these values back into the expression sin x + cos x:

sin x + cos x = (a - sin y) + (a - cos y)

Using the distributive property, we get:

sin x + cos x = 2a - sin y - cos y

Since we are asked to express sin x + cos x in terms of a, we can rewrite the equation as:

sin x + cos x = 2a - (sin y + cos y)

Therefore, sin x + cos x in terms of a is given by 2a - (sin y + cos y).

To express sin x + cos x in terms of a, we can substitute the values of sin x and cos x into the equation:

sin x + cos x = (a - sin y) + (a - cos y)
= 2a - (sin y + cos y)

Now, we need to find the value of sin y + cos y. We can use the given information:

sin x + sin y = a
cos x + cos y = a

Since a is not equal to 0, we can multiply the first equation by cos y and the second equation by sin y:

sin x * cos y + sin y * cos y = a * cos y
cos x * sin y + cos y * sin y = a * sin y

Using the trigonometric identity sin a * cos b + cos a * sin b = sin(a + b), we can simplify the equations:

sin(x + y) = a * cos y
sin(x + y) = a * sin y

Since these equations are true for all values of x and y, we can conclude that cos y = sin y. Now, substituting this into our expression for sin x + cos x:

sin x + cos x = 2a - (sin y + cos y) = 2a - 2sin y

Thus, sin x + cos x is equal to 2a - 2sin y in terms of a.