5 sin x + 3 cos x = 5?

Question

Solve for 0

@ironduke8159

If you divide the whole equation by cos x
Then it will be
5 tan x + 3 = 5/cos x

Answer 1

5sin(x) + 3cos(x) = 5

Note that sin^2(x) = 1 - cos^2(x), so it follows that
sin(x) = +/- sqrt(1 - cos^2(x))

But, since 0 <= x <= 180, we know that sin(x) is positive in that region, so we take the positive result, and

sin(x) = sqrt(1 - cos^2(x))

Plugging this into our equation,

5sqrt(1 - cos^2(x)) + 3cos(x) = 5

Moving 3cos(x) to the other side,

5sqrt(1 - cos^2(x)) = 5 - 3cos(x)

Square both sides,

25 (1 - cos^2(x)) = 25 - 30cos(x) + 9cos^2(x)

Expand both sides,

25 - 25cos^2(x) = 25 - 30cos(x) + 9cos^2(x)

Move everything to the right hand side.

0 = -30cos(x) + 34cos^2(x)

Solve normally by factoring.

0 = -2cos(x)(15 - 17cos(x))

Equate each factor to 0

-2cos(x) = 0
15 - 17cos(x) = 0

Therefore,

cos(x) = 0
cos(x) = 15/17

cos(x) = 0 when x = 90 degrees.
To solve cos(x) = 15/17, take the arccos of both sides, toget

x = arccos(15/17), which is approximately equal to
x ~= 28.072

{fortunately for us, arccos(15/17) falls within the interval of 0 < x < 180).

Therefore, our solutions are

x = {pi/2, arccos(15/17)}

Answer 2

5 sin x + 3 cos x = 5

Answer 3

5 sin x + 3 cos x =5
5 tanx +3 =5
tanx = (5-3)/2 = 1
x = arctan(1) = 45 degrees [0

Answer 4

5 sin x + 3 cos x
= √34 sin[x+arctan(3/5)] = 5......(1)
x1 = arcsin(5/√34)-arctan(3/5) = 28.07248694 degrees

or
5 sin x + 3 cos x
= √34 sin[180 - x - arctan(3/5)] = 5......(2)
x2 = 180 - arctan(3/5) - arcsin(5/√34) = 90.0000000 degrees
------------
Reason:
a sinx + b cosx = √(a^2+b^2)sin[x+arctan(b/a)]

Answer 5

5sinx+3cosx=5
hepl me

Answer 6

I tried to figure it out, but I haven't had any caffiene today so it's probably wrong.

x = -3

Your answer seems more reasonable.