Understanding Engineering Mathematics

6.2.8 Trigonometric equations

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A trigonometric equation is any equation containing ratios of an ‘unknown’ angleθ,tobe
determined from the equation. One attempts to solve such an equation by manipulating it
(possibly using trig identities) until it can be solved by solutions of one or more equations
of the form

sinθ=a

cosθ=b

tanθ=c

As noted in Section 6.2.5, in general the solution to such equations will not be unique,
but will lead to multiple values ofθ.Thegeneral solutionis that which specifiesall
possible solutions. Such general solutions can always be expressed as simple extensions
ofprincipal value solutions, which are those for whichθis confined to a specific range
in which the above basic equations have unique solutions. Thus, from Section 6.2.5 the
principal values for the elementary trigonometric ratios are:

sinθ, θ∈

[

−

π

2

,

π

2

]

which means −

π

2

≤θ≤

π

2

cosθ, θ∈[0,π]or0≤θ≤π

tanθ, θ∈

[

−

π

2

,

π

2

]

or −

π

2

≤θ≤

π

2

In each case there is onlyoneprincipal value solution to each of the equations given
above. If we are asked for a solution in a specific range ofθthen we have to examine
such equations as sinθ=ato determine which solutions fall in this range.
An important type of equation is

f(A)=f(B)

wheref is an elementary trig function. The general solutions for such equations are

For cosA=cosB,we haveA= 2 nπ±B

For sinA=sinB,we haveA= 2 nπ+Bor( 2 k+ 1 )π−B

For tanA=tanB,we haveA=nπ+B

where in each case,nis an arbitrary integer. These results can be confirmed by inspecting
the graphs in Figures 6.8 – 10. The solution to the review question provides an example.

Solution to review question 6.1.8

(i) sinθ+2sinθcosθ=sinθ( 1 +2cosθ)= 0

implies either sinθ=0or1+2cosθ= 0

sinθ=0 has solutionsθ=kπwherekis an integer

The equation 1+2cosθ=0or cosθ=−^12 has general solution

θ=

2 π

3

+ 2 mπor −

2 π

3

+ 2 nπ