Solve 4sinx=3cosx 0<x<360?

Question

Have used sin squared x +cos squared x =1 and solved to make cos x = plus or minus 0.8.
This gives two answers for each value which fall between 0 and 360.
A level answer paper just gives 36.9 and 216.9.
Am I missing something?

Answer 1

It's not wrong to square both sides and substitute (sinx)^2 with 1-(cosx)^2 and then solve for cosx to get cosx = +-0.8 but it's more easily solved using tangent like a few people did. Doing it with cosine does produce 4 answers: x=arccos(0.8)=36.9 and then you would also have x=360-36.9=323.1.
Then x=arccos(-0.8)=143.1 and also x=360-143.1=216.9. But, solving it this way by squaring both sides you DO have to check those 4 answers. Only 36.9 and 216.9 work.

Answer 2

4

Answer 3

No, there's sufficient information to calculate the answers.

4sinx=3cosx
4sinx/cosx =3
sinx/cosx =3/4
tan x = 3/4
x = tan^(-1) 3/4
x = 36.869...°

In the first quadrant, Sin and Cos are Positive. → x = 36.869...°
In the third quadrant, Sin and Cos are Negative. → x = 180 + 36.869...° = 216∙869...°

So x ≈ 36∙9°
and x ≈ 216∙9°

Answer 4

4sinx =3cos x
sinx/cosx =3/4
tanx = 3/4 since tan is positive there are two answers, 1 in the first quadrant and 1 in the third quadrant that is 36.9 deg and 216.9 deg you seem to have the correct answer.

Answer 5

4sinx=3cosx
=>4sinx/cosx =3
=>sinx/cosx =3/4
=>tan x =3/4
=>x=tan^(-1) 3/4
=>x= 36.9
tan =+ => 1st & 3rd quad
=>x=180 + 36.9= 216.9 or 36.9

Answer 6

sin(x)/cos(x) = tan(x)

x = arctan(3/4)

3/4 can also be -3/-4

this will fall into quadrants 1 and 3

solve for just quadrant 1 and add 180 (degrees) or pi (radians) to answer

arctan(3/4) = 36.9 (quadrant 1)
arctan(3/4) +180 = 216.9 (quadrant 3)

Answer 7

You have forgotten the negative root cosx =-0.8 works

16 (sin x)^2 = 9 (cos x)^2
16(1-(cos x)^2) = 9 (cos x)^2
16 -16 (cos x)^2= 9(cos x)^2
16= 25 (cos x)^2
4= 5 cos x
first answer : cos x =0.8
x= 36.9 (correct)
and x= -36.9 . this is also equal to 360-36.9 = 323.1

second answer cosx = -0.8 x= -143.1 and x= 216.9

So you have 4 answers

36.9 , 143.1 , 216.9 , and 323.1

Answer 8

4sinx = 3cosx
sinx/cosx = 3/4
tanx = 3/4
x = 36.9, 180+36.9
x = 36.9, 216.9

You cannot use (sinx)^2+(cosx)^2 = 1 because that is for the square of the trigonometrical functions only.

Answer 9

i would not have proceed that way!

divide throughout by cosx.
u'll get:

4tanx=3

therefore, tanx =0.75

then solve for x.

tan inverse of 0.75=36.9

then, use the quadrants where tan is positive, i.e the first and the third quadrant.

therefore x=36.9 and (180+36.9)

Answer 10

cannot use (sin x)^2 + (cos x)^2 = 1 . It would make the equation a lot harder to solve!!!
Also if you square an equation, yuo introduce answers which are not solutions of the original, so if you have to square an equation to solve it, you would need to plug your answers in original equation to see if it is true.
Example
x = -1 is an equation which has one solution, namely -1
however, if you square it, you get x^2 = 1
this equation has two solutions, 1, -1.A simple plug into original equation shows that you have to reject 1.

As for your equation,I am not even sure what you did to get cos x = 0.8 or -0.8.
[okay i see now. you squared. watch out, you will have to reject some solutions, since as i have explained, squaring introduces solutions not solutions of original equation, so one of the people who answered you is not correct]
it is equivalent to tanx = 0.75 (divide by 4cos x both sides) which is easy to solve.