Cosine x= sine(x+20)

Question

The question is asking to find the value of x if Cosine x = sine (x+20) 

Answer 1

sin and cos  have the same sign in the 1st and 3rd quadrants

cos(x)=sin(90-x)

cos(x)=sin(x+20)

90-x=x+20

90-20=x+x

70=2x

x=70/2

x=35º

add 180º to 35º to get 215º (3rd quadrant)

x=215º also

cos(215º)=sin(215º+20º)=sin(235º)=-0.81915204428

cos(35º)=sin(35º+20º)=sin(55º)=0.81915204428

Answer 2

1.  If degrees is intended, that symbol should have been appended to the 20, i.e. 20^o.

 

2.  In my opinion this problem should be treated as an empirical problem, not theoretical.  So just use a calculator to graph the function on the left side as Y1 and the right side function as Y2.  Then ask the calculator to find the intersection!

Answer 3

There are a couple of approaches to solving this.

 

1) Expand the right side using a sum identity for sin.

2) Square both sides and simplify using trig identities.

 

My judgement is that the first method will be more effective.

 

sin (x + 20°) = sin x cos 20° + cos x sin 20° .

 

Set this equal to cos x:

 

cos x = sin x cos 20° + cos x sin 20° .

 

Move the last term to the left and factor out cos x:

 

cos x - cos x sin 20° = sin x cos 20°

 

cos x (1 - sin 20°) = sin x cos 20° .

 

Now, divide both sides by cos x and cos 20°:

 

(1 - sin 20°)/(cos 20°) = (sin x)/(cos x) = tan x (the last by a trig identity)

 

then you can finish the problem by computing the left side with a calculator and using the inverse tangent to locate the solutions. You did not specify any interval for the solutions, so you must find ALL of them. Start by looking in one period of tangent, from 0 to 180 degrees, find all of the solutions (there should be just one), then add 180°n to get the others.