What is the solution to the equation cosx=tanx for -180<= x <= 180?

Question

I can only use the identities, sin^2x + cos^2x =1 and tanx = sinx/cosx, and of course factorisation. I am getting approx. values of 38 and 142 degrees.

Answer 1

cosx=tanx
cosx=sinx/cosx
cos^2(x)=sinx
1-sin^2(x)=sinx {pythagoras}
sin^2(x)+sinx-1=0
use quadratic formula
sinx=(-1+or-sqrt(5))/2
=0.6180339888
or -1.6180339888
sine has to lie between +/-1
sinx= 0.6180339888
x=arcsin(0.6180339888)
=38.17270763
+ve value for sine lies in 1st and 2nd
quadrants
therefore,x=38.17270763 degrees or
141.8272924 degrees

these two values were checked in the
original equation and were found
to be correct

i hope that this helps

Answer 2

3 sin2x = cos2x => tan2x = a million/3 ; x =9d thirteen' 3" {If i'm flawed sin2x=> sq. of sinx, then x = 60d} a million. 3/5 = tanx => x = 30d fifty seven' 50" 2. (a)sin 30=a million/2; cos 30=sqrt3/2 (b)tan 30= sin 30/cos 30 = a million/sqrt3 3. enable a = BC, b = AC=sqrt3/2, c = AB=6 b^2 = c^2 + a^2 - 2ac cos(B) 3/4 = 36 + a^2 -12 a(sqrt3/2) a^2 -6 sqrt3 a + 35.25 =0 a = 3 sqrt3 [+/-]sqrt[27-35.25] because of the fact the radical supplies an imaginary quantity the answer isn't available 4. (a)2 cos^2 x + sin x =2[a million-sin^2 x] + sin x =2 + sin x - 2sin^2 x (b)If the above expression is =2 =>sin x - 2 sin^2 x =0 => sin x(a million -2 sin x)= 0 =>(a million/2 -sin x) =0 => sin x =a million/2 & x=30. 5. (a)3 sin^2 x + 4 cos x =3[a million - cos^2 x] + 4 cos x = -3 cos^2 x + 4 cos x + a million (b) 3[a million - cos^2 x] + 4 cos x - 4 =0 -3 cos^2 x + 4 cos x - a million =0; cos x = 2/3 [+/-]sqrt[4/9 -a million/3] =2/3[+/-]sqrt[a million/9] = a million, a million/3 x = 90d, 70d 31'40 3.sixty one"

Answer 3

cosx = sinx/cosx --->( cosx) ^2 = sinx = 1-sin(x)^2

this is equivalent to (sinx)^2 +sinx -1 =0

sinx =( -1+ (1+4)^0.5)/2 = 0.618 2solutions 38° or 142 °

The negative solution gives sinx <-1 impossible

Answer 4

cos(x)=sin(x)/cos(x)
(cos(x))^2- sin(x) =0
1 - (sin(x))^2 - sin(x) = 0
(sin(x))^2 + sin(x) - 1 =0

Sin(x)=0.61,-1.61

Sin value ranges between -1 to +1 so -1.61 is not valid

x= 38.17 Degrees

Answer 5

You are absolutely right.
sin x =-0.5 +/- (root5)/2
So x =38.2deg. or 141.8deg. (to 2dp)

Answer 6

X= -1/2

OR

X= -0.5