Solving Trigonometric Equations

Summary

To solve trigonometric equations, several identities and formulas are used i.e:

  • tan\theta \quad =\quad \frac { sin\theta }{ cos\theta }
  • { sin }^{ 2 }\theta \quad +\quad { cos }^{ 2 }\theta \quad =\quad 1
  • { cosec }^{ 2 }\theta \quad =\quad 1\quad +\quad { cot }^{ 2 }\theta
  • { sec }^{ 2 }\theta \quad =\quad 1\quad +\quad { tan }^{ 2 }\theta
  • sin2x\quad =\quad 2sin(x)cos(x)
  • cos2x\quad =\quad { 2cos }^{ 2 }x\quad -\quad 1\quad =\quad { 1\quad -\quad 2sin }^{ 2 }x\quad

For basic trigonometric equations, we follow the following steps to solve them:
1.    Make sine, cosine or tangent the subject.
2.    Use any method including a calculator to find basic angles.
3.    Using quadrants, find all solutions in the given range.

Example #1

Q. Solve the equation  cos\theta \quad =\quad \frac { 1 }{ 2 }   for the range  0\quad \le \quad \theta \quad \le \quad 2\pi

Solution:

Using a calculator we can find that the basic angle  \theta \quad =\quad \frac { \pi }{ 3 }

Now we know that cos is positive in the first quadrant and in the fourth quadrant hence another value of \theta would be:

\theta \quad =\quad 2\pi \quad -\quad \frac { \pi }{ 3 } \quad =\quad \frac { 5\pi }{ 3 }

Ans:     \theta \quad =\quad \frac { \pi }{ 3 } \quad \quad \quad \quad and\quad \quad \quad \theta \quad =\quad \frac { 5\pi }{ 3 }

Trigonometric Identities

Refer to Fig 1.

We know that:

sin\theta \quad =\quad \frac { y }{ r } \quad \quad \Rightarrow \quad 1\quad

cos\theta \quad =\quad \frac { x }{ r } \quad \quad \Rightarrow \quad 2

tan\theta \quad =\quad \frac { y }{ x } \quad \quad \Rightarrow \quad 3

Dividing 1 by 2 we get:

tan\theta \quad =\quad \frac { sin\theta }{ cos\theta }

This is called an ”Identity”.

Using pythagoras theorem:   { x }^{ 2 }\quad +\quad { y }^{ 2 }\quad =\quad { r }^{ 2 }

Dividing the whole equation by  { r }^{ 2 }:

{ (\frac { x }{ r } ) }^{ 2 }\quad +\quad { (\frac { y }{ r } ) }^{ 2 }\quad =\quad 1

Substituting the values:   { (cos\theta ) }^{ 2 }\quad +\quad { (sin\theta ) }^{ 2 }\quad =\quad 1

We get another identity:   { sin }^{ 2 }\theta \quad +\quad { cos }^{ 2 }\theta \quad \equiv \quad 1

where  \equiv   means ”equivalent to” or ”identical to”.

These identities are very useful when solving many trigonometric equations.

Example #2

Q. Prove the identity   tan^{ 2 }\theta \quad -\quad { sin }^{ 2 }\theta \quad =\quad \frac { { sin }^{ 4 }\theta }{ { cos }^{ 2 }\theta }

Solution:

Taking left hand side only:

=\quad tan^{ 2 }\theta \quad -\quad { sin }^{ 2 }\theta

=\quad (\frac { sin\theta }{ cos\theta } )^{ 2 }\quad -\quad { sin }^{ 2 }\theta

=\quad \frac { { sin }^{ 2 }\theta }{ { cos }^{ 2 }\theta } \quad -\quad { sin }^{ 2 }\theta

Take  { sin }^{ 2 }\theta   common:

=\quad { sin }^{ 2 }\theta (\frac { 1 }{ { cos }^{ 2 }\theta } \quad -\quad 1)

=\quad { sin }^{ 2 }\theta (\frac { 1\quad -\quad { cos }^{ 2 }\theta }{ { cos }^{ 2 }\theta } )

=\quad \frac { { sin }^{ 2 }\theta }{ { cos }^{ 2 }\theta } ({ sin }^{ 2 }\theta \quad +\quad { cos }^{ 2 }\theta \quad -\quad cos^{ 2 }\theta )

=\quad \frac { { sin }^{ 4 }\theta }{ { cos }^{ 2 }\theta }      Hence proven!

Similarly, there are a few other formulas that help solve trigonometric equation:

  • { cosec }^{ 2 }\theta \quad =\quad 1\quad +\quad { cot }^{ 2 }\theta
  • { sec }^{ 2 }\theta \quad =\quad 1\quad +\quad { tan }^{ 2 }\theta
  • sin2x\quad =\quad 2sin(x)cos(x)
  • cos2x\quad =\quad { 2cos }^{ 2 }x\quad -\quad 1\quad =\quad { 1\quad -\quad 2sin }^{ 2 }x\quad