6.5. Trigonometric Equations http://www.ck12.org
When solving trigonometric equations, try to give exact (non-rounded) answers. If you are working with a calculator,
keep in mind that while some newer calculators can provide exact answers like
√ 3
2 , most calculators will produce
a decimal of 0.866... If you see a decimal like 0.866..., try squaring it. The result might be a nice fraction like
(^34). Then you can logically conclude that the original decimal must be the square root of (^34) or√ 23.
When solving, if the two sides of the equation are always equal, then the equation is an identity. If the two sides of
an equation are never equal, as with sinx=3, then the equation has no solution.
Example A
Solve the following equation algebraically and confirm graphically on the interval[− 2 π, 2 π].
cos 2x=sinx
Solution:
cos 2x=sinx
1 −2 sin^2 x=sinx
0 =2 sin^2 x+sinx− 1
0 = (2 sinx− 1 )(sinx+ 1 )
Solving the first part set equal to zero within the interval yields:
0 =2 sinx− 1
1
2 =sinx
x=π 6 ,^56 π,−^116 π,−^76 π
Solving the second part set equal to zero yields:
0 =sinx+ 1
− 1 =sinx
x=−π 2 ,^32 π
These are the six solutions that will appear as intersections of the two graphsf(x) =cos 2xandg(x) =sinx.