3cosx+4sinx=5

what is x is too difficulat form me to solve?
Shw working please

3 cos x = 5-4 sin x

9 cos^2 x = 25 - 40 sin x + 16 sin^2 x

9 (1 -sin^2x) = 16 sin^2 x -40 sin x + 25

9 - 9 sin^2 x = 16 sin^2 x -40 sin x + 25

0 = 25 sin^2 x - 40 sin x + 16
let z = sin x

25 z^2 - 40 z + 16 = 0

(5 z - 4)(5 z - 4) = 0 wow, that was lucky

sin x = 4/5 :)

Im not in Trig yet but i did look up online trigonometry calculators... see if that help?

PS, got this from 1 of those sites:
3cos(x) + 4sin(x) = 5 : x = 2πn + arcsin(4/5)
Hope this helps.

let's change 3cosx+4sinx into a single sine curve of the form y = a sin (x+k)

we know asin(x+k)
= asinxcosk + acosxsink
comparing our terms

asinxcosk = 4sinx
acosk = 4
a = 4/cosk

acosxsink = 3cosx
a sink = 3
a = 3/sink

then 3/sink = 4/cosk
4sink = 3cosk
sink/cosk = 3/4
tank = 3/4
k = 36.87°
also a^2 = 3^2 + 4^2
a = 5

so 3cosx+4sinx=5 becomes
5sin(x+36.87°) = 5
sin(x+36.87°) = 1
x+36.87° = 90°
x = 53.13°

check:
LS = 3cos53.13 + 4sin53.12
= 4.99958
not bad, we could carry more decimals for more accuracy

of course we could get more solutions by simply adding multiples of 360° to 53.12°

e.g x = 413.13 or x = -306.87° etc

thnx Damon. im not in trig so i just looked it up :)

To solve the equation 3cos(x) + 4sin(x) = 5, you can use a trigonometric identity to simplify it before solving for x. The identity we'll use is:

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

In our equation, we can rewrite 3cos(x) + 4sin(x) as 5cos(x - ?), where ? is a constant that we need to determine.

Comparing the equations, we have:
5cos(x - ?) = 5

From this, we can conclude that cos(x - ?) = 1, which means x - ? = 0. Solving for x, we get:

x = ?

Since we don't know the exact value of ?, we cannot determine the exact value of x.

However, we can find the general solutions for x, using the inverse trigonometric functions:

1. First, solve for x when cos(x - ?) = 1:
x - ? = 0
x = ?

2. Find the general solutions for x:
x = ? + 2?n, where n is an integer representing the number of complete cycles.

In conclusion, the exact value of x cannot be determined without knowing the value of ?. However, the general solutions for x are given by x = ? + 2?n, where n is an integer.

Go with Damon's , he used an easier method

(but I thought mine was quite elegant)

well I agree with anonymous :)