Trigonometric Identities Summary Notes

I understand that learning trigonometric identities can be tricky, as I’ve experienced the same challenge myself. However, I’m here to make them simple and easy for you. The methods I used to learn these formulas have stayed with me for more than a decade. So, I want to help you learn them the same way.

After learning about the domain and range of trigonometric functions in the previous article, this one focuses on key Trigonometric Formulas, including sum and difference formulas, sum to product, product to sum, and the 2A and 3A formulas.

Understanding these identities is not only important for your exams but also for solving complex problems in higher Mathematics.

Let’s go through it step by step, and you’ll see how easy it can be. We will also explore tips to prove and remember them easily.

1. Sum and Difference Formulas

The sum and difference formulas allow us to express the sine (sin) cosine (cos), and tangent (tan) of sums and differences of angles.

They are:

These formulas are incredibly useful in simplifying trigonometric expressions and proving more identities. For example, using the sum/difference formula for sine, you can easily find the values of sin 15°, sin 75° etc. by breaking these angles into a difference (45° – 30°) or a sum (45° + 30°) of special angles respectively.

Observations:

• Learn them in the same order as mentioned above.
• (First 4 Formulas) Let’s remember them as sum or difference of two parts (on the RHS) – the first and second parts, so we can use them to prove more identities.

2. Product to Sum Formulas

Product to sum or difference formulas transform product of trigonometric functions into sums or differences.

This set of formulas is particularly helpful in integration wherein it’s simpler to integrate a function – sin x sin 2x sin 4x by converting it into sum or difference.

Observations:

They can be easily verified and learnt as mentioned in blue in the image above.

• When you add the first two formulas (F1 and F2) or the next two (F3 and F4) above, you get twice of first part of sum or difference formulas.
• Likewise, when you subtract them (F2 from F1 or F4 from F3), you get twice of second part with an exception of minus sign in the last one.

3. Sum to Product Formulas

Conversely, sum to product formulas convert sums or differences into products of trigonometric functions. These trigonometric identities include the following:

As per the requirement, you can easily use the above formulas wherein you want to get a product (factors) instead of sum or difference. For example, this is helpful in solving equations of the types sin 2x – sin 4x + sin 6x = 0 and likewise.

Once you understand these formulas, it will help you tackle more complex trigonometric equations.

Observations:

• Link them up with the first two sets of formulas above, so you can easily remember.
• I suggest you to derive and also verify them for different known angles.

4. 2A and 3A Formulas

These formulas simplify expressions involving angles that are multiples of a certain angle. As you can see the image below:

These formulas are frequently used in both integration and differentiation problems involving trigonometric functions.

Observations:

• You can easily derive these trigonometric identities from the first set of sum of difference formulas, so do try them.
• Connect them with each other and previously learnt formulas for easy learning.
• The 3A formulas are just extensions of 2A identities, do check them by replacing B with 2A in sum and difference formulas in the first set above.

Tips to Remember:

1. Visual Aids: By using the images provided here, you must try to visualize and connect the relationships between the formulas.

2. Mnemonics: Create your own simple phrases or stories, so you can remember the order of terms in formulas.

3. Practice Regularly: Frequent practice will help solidify these formulas in your memory

All 26 Trigonometric Identities Together! (Light Mode)

Here is a list of all the above important formulas for your ready reference:

Practice Questions for You:

One effective way to master these formulas is through practice and proving related identities. For instance, try deducing the following using the above formulas:

Final Thoughts:

By now, am sure you must have gained a clearer understanding of these trigonometric identities. Remember, with a little practice, they’ll start to feel more natural, just like they did for me. Don’t rush—take it one step at a time, and you’ll find it easier to remember and apply them. Keep practicing, and soon enough, you’ll master these formulas! If you ever get stuck, come back to these steps, and you’ll be on the right track. Happy studying!

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