Trigonometric identities are, maybe, the most difficult trig topic to master.
It is, on the surface, an easy task – all you need to show is that the expression on one side of the equation is equal to the expression written on the other side.
You don’t need to solve anything – just rewrite !
However, due to the multitude of options, formulas, and sometimes lack of time or will to explain this topic in an adequate way, many students find themselves reluctant to solve problems involving trigonometric identities.
The way I’m describing here is by no means the fastest, most elegant, or the simplest – however the principles to follow are straightforward and comprehensive. They are general enough to enable you to use them each time, and with practice, these guidelines will help you overcome even the most daunting trig identities problem.
1. REMEMBER YOUR ALGEBRA
Many students assume that since they have taken the Algebra HSA (Montgomery county) and are done with Algebra I and Algebra II the material covered during the study of these topics is no longer needed – as you will see they are wrong.
An important first step in mastering trig identities is mastering FOILing and unFOILing as well as memorizing and being able to identify on sight several identities.
The most common ones are:
and the most important to identify is the third one (when A=X and B=1) which take the form of:
which can and should be expanded to
2. Change the trigonometric expressions appearing on both left and right to expressions containing SIN and COS only – second from the left column in the table below.
If all terms are single multiples of the variable (x, t, alpha, etc) don’t bother to copy the variable throughout the steps – saves time and less messy.
3. Remember to check after each step what is the expression you are trying to obtain (usually the one on the right side).
This will help you determine which additional steps are needed to expand, rewrite, or manipulate the intermediate form you have.
4. Trig identities expressions are separated by an equal sign – that implies you can transition from left to right as well as from right to left. If you get stuck – start with the other side. All the time the expressions you write are identical somewhere in the middle you demonstrated they are equivalent.
Left side expression = Right side equation
Left side expression –>step 1 —> step 2 —> …..last step —> Right side equation
Left side expression <– last step…. <— step 2 <— step 1 <— Right side equation
Left side expression –> step 2 —> step 3—-> last step <—- step 4 <— step 1 <— Right side equation
5. If one of the sides has an expression with dividend or numerator and a divisor or denominator and somewhere along the way you obtained either one – don’t change it. Work on the one you don’t have and try to make it the same as in the final expression.
6. If worse comes to worse and you are stuck with a numerator and a denominator – you can always on one side and the expression on the other side has a different numerator and a different denominator and you can’t see a way out – you can expand one side by the other expression denominator (multiply and divide). The numerator multiplied by the other side denominator and divided by the original denominator should then be simplified to give the desired numerator.
7. Most important of all :