Everything Maths Grade 11

CHAPTER 17. TRIGONOMETRY 17.4

And then we are left with two linear trigonometric equations. Be careful: sometimes one of the two
solutions will be outsidethe range of the trigonometric function. In that case you need to discard that
solution. For example consider the same equation with cosines insteadof tangents

cos^2 (2x + 1) + 3cos(2x + 1) + 2 = 0

Using the same methodwe find that

cos(2x + 1) =− 1 or cos(2x + 1) =− 2

The second solution cannot be valid as cosine must lie between− 1 and 1. We must, therefore, reject
the second equation. Only solutions to the firstequation will be valid.

More Complex Trigonometric Equations EMBDL

Here are two exampleson the level of the hardest trigonometric equations you are likely to encounter.
They require using everything that you have learnt in this chapter. If you can solve these, you should
be able to solve anything!

Example 12:

QUESTION

Solve 2cos^2 x− cos x− 1 = 0 for x∈ [− 180 ◦;360◦]

SOLUTION

Step 1 : Use a temporary variable

We note that cos x occurs twice in the equation. So, let y = cos x. Then we have

2 y^2 − y− 1 = 0 Note that with practiseyou may be able to leave out this step.

Step 2 : Solve the quadratic equation

Factorising yields

(2y + 1)(y− 1) = 0

∴ y =− 0 , 5 or y = 1

Step 3 : Substitute back and solve the two resulting equations

We thus get

cos x =− 0 , 5 or cos x = 1

Both equations are valid(i.e. lie in the range of cosine).

General solution: