 # Solving Trig Equations

 From: Nicholas Lawton Date: 9 July 1999 Subject: Solve 2cos(x+50)=sin(x+40) How do I solve the equation   2cos(x+50) = sin(x+40)   for  0<=x<=360 degrees? ### Maths Help suggests:

First, let us sketch the graphs of
y = 2cos(x+50)   (in red)
and
y = sin(x+40)   (in blue).

The solutions to your problem are the x-values of the points where the curves intersect.

There appear to be two solutions in the required range.

To solve the equation, use the trig. identities:

• sin(A + B) = sinA cosB + cosA sinB
• cos(A + B) = cosA cosB - sinA sinB

So:   cos (x + 50) = cos x cos 50 - sin x sin 50
and   sin (x + 40) = sin x cos 40 + cos x sin 40

You must solve
2(cos x cos 50 - sin x sin 50) = sin x cos 40 + cos x sin 40

Re-arranging,
cos x(2cos 50 - sin 40) = sin x(cos 40 + 2sin 50)

Dividing through by cos x  and  (cos 40 + 2sin 50) This simplifies to:
tan x = 0.2796...

And the two solutions in the range  0 <= x <= 360  are:
x = 15.6°, 195.6°   (1 d.p.)

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