Help me solve for x (ex. tanx=-4sinx)?

Question

Here are the problems I need help solving "x" for -

tanx=-4sinx

csc²x=7tan(90° - x) - 11

5sin(90° - x) = sinx

6cos²x=7-5sin x

If you can help with any, I would be so grateful!

Cheers!

Cooper

Answer 1

tanx=-4sinx
sinx/cosx -4sinx = 0
sinx(1/cosx -4)=0 so sinx=0 or cosx = 1/4

csc²x=7tan(90° - x) - 11
csc²x=-7tan(x) - 11
1+cot^2(x)=-7tan(x) - 11
7tanx + 12 +1/tan^2(x) = 0
7tan^3(x) + 12tan^2(x) + 1 = 0
now try to solve the cubic....

5sin(90° - x) = sinx
-5sinx = sinx
sinx = 0

6cos²x=7-5sin x
6 - 6sin^2(x) = 7 - sinx
simplify and solve quadratic

good luck!

Answer 2

tanx=-4sinx
sinx/cosx = -4sin
sinx = -4 sinx cosx
cos x = sinx/-4sin x = -1/4
x= arccos (-1/4) = 104.48 degrees

csc²x=7tan(90° - x) - 11
Sorry, can't seem to figure this one out


5sin(90° - x) = sinx
5(-cosx) = sinx
-5 = sinx/cosx = tanx
x = arctan(-5) = -78.69 degrees

6cos²x=7-5sin x
6(1-sin^2x) = 7-5sinx
6 - 6sin^x = 7 - 5sinx
6sin^2x -5sinx + 1 = 0
sinx = [5 +/- sqrt(25-24)]12
sinx = 5/12 +/- 1/12 = 1/2 or 1/3
x arcsin (1/2) = 30 degrees, or
x = arcsin (1/3) = 19.47 degrees

Answer 3

tanx=-4sinx
→ sinx/cosx = -4sinx. The sin's cancel out
→ 1/cosx = -4
→ cosx = -1/4
→ x = arccos(-1/4) or about 104.5°

5sin(90° - x) = sinx
→ 5cosx = sinx
→ 5 = tanx
→ x = arctan5 or 78.7° (approx.)

Good luck with the rest. Use your equalities to express everything in terms of sinx and cosx, and go for a simple equality like sinx = ? or tanx=? or cosx=?.

Answer 4

tanx=-4sinx
sinx/cosx=-4sinx
sinx=-4sinxcosx
sinx+4sinxcosx=0
sinx(1+4cosx)=0
sinx=0 or x=0
1+4cosx=0
4cosx=-1
cosx=-1/4
x=arccos(-1/4)

csc²x=7tan(90° - x) - 11
csc^2=7cotx-11
1+cot^2x-7cotx+11=0
cot^2x-7cotx+12=0
(cotx-3)(cotx-4)=0
cotx=3 or cotx=4
x=arccot3 or arc cot4

5sin(90° - x) = sinx
5cosx=sinx
tanx=5
x=arctan5

6cos²x=7-5sin x
6(1-sin^2x)=7-5sinx
6-6sin^2x=7-5sinx
6sin^2x-5sinx+1=0
(6sinx+1)(sinx-1)=0
sinx=1 or x=pi/2
sinx=-1/6
x=arcsin(-1/6)

Answer 5

No, there is adequate ideas to calculate the solutions. 4sinx=3cosx 4sinx/cosx =3 sinx/cosx =3/4 tan x = 3/4 x = tan^(-a million) 3/4 x = 36.869...° interior the first quadrant, Sin and Cos are helpful. ? x = 36.869...° interior the third quadrant, Sin and Cos are adverse. ? x = one hundred eighty + 36.869...° = 216?869...° So x ? 36?9° and x ? 216?9°

Answer 6

tan(x)=-4sin(x)
sin(x)/cos(x)+4sin(x)=0
sin(x)(1/cos(x)+4)=0
sin(X)=0 => x=k*pi and 1/cos(x)+4=0 => (1+4cos(x))/cos(x)=0
=>
1+4cos(x)=0 => cos(x)=-1/4 => x=arc cos(1/4)
____________________________________________________

5sin(90-x)=sin(x) => 5cos(x)=sin(x) => sin(x)/cos(x)=5

x=arc tan(5)
____________________________________________________
6cos^2(x)=7-5sin(x) => 6cos^2(x)+5sin(x)-7=0
6(1-sin^2(x))+5sin(x)-7=0
-6sin^2(x)+5sin(x)-1=0

after solving this equation we have
sin(x)=[-5+1]/-12=1/3 => x=arc sin(1/3)
and
sin(x)=[-5-1]/-12=1/2 => x=2kpi+pi/6 and x=2kpi +pi -pi/6
____________________________________________________