The Best Way to Master Trigonometric Identities

In this video I answer a question I received from a viewer. He is asking about learning trigonometric identities. Do you have any advice? If so, please leave a comment in the comment section below.

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[tjalferes]

Focus. Look at the trig identities all day every day. No effort. No struggle. Don't try to memorize them. Just set your eyes on them. Keep looking at them. Make them your whole world, for now. Understand exactly what they mean. Say them out loud. Write the functions and ratios. Draw the graphs and triangles. Keep your eyes on them. Then the clear memory will come.

[garrettrussell]

I've been a private math tutor for 4 years now and whenever my students get caught up with trig identities I always tell them to write the identity off to the side when they use them. After enough practice problems and re-writing the identities, your brain will eventually move faster than your hands!

[sniperwolf]

The holy trinity of trig identities is {sin² x + cos² x = 1; sin (x ± y) = sin x * cos y ± sin y * cos x; cos (x ± y) = cos x * cos y ∓ sin x * sin y}. Everything else is derivable from those. You can even use them to find exact values of sine and cosine of some particular angles if you're so inclined

[chriswilliams]

I used flashcards and memorized them a year ago then stopped using them. I am now studying calculus and have forgotten them so now I'm trying to think of new ways to not forget them.

I enjoy solving trig identities a lot as well. My favorite parts of math are the parts that rely more on memorizing rules and formulas as opposed to the more calculation-intensive math problems that are easy to mess up once you miscalculate one thing in a big equation.
Thanks for the video.

[danielmrosser]

Euler’s Formula e^(i*t) = cos t + i*sin t
is the key to unlock your understanding of the trigonometric identities

[dhickey]

Trig was the first math course I took in college where more than half the class failed. Math Sorcerer hits on one of the key things most of us didn't have when taking trig - get comfortable with proofs and proof notation. Proofs will help with all your math including calculus studies as well.
Also, my 'A-HAH' moment in calculus happened during our study of arc length problems with trig. You learn to simplify complex problems where cos^2 + sin^2 = 1. Without understanding identities, these problems would take a long time to solve on an exam.
Lastly, trig is a major tool used in physics with calculus 1. Try a few basic physics problems involving static friction. Using trig in physics may connect a few ideas together for you. Best of luck.

[alexlilano]

I’ll legitimately deriving most of identities almost every time. Do some Cal II math. That will make you an expert on them.

[sakyijnrsakyijnr]

Two advice for Kyle:
1. Practice is the sure way.

2. Listen to all the helpful advice given by everyone.

[LaithMMA]

My advice is to always convert and change identities to sines and cosines then simplify 👍😊

Best of luck and remember to do a metric ton of practice problems 👍

[Scobson]

100% just do a tonne of problems. My issue was memorising all the identities. I learned for all the reciprocals there always existed the prefix "co" exactly once. I.e.
Sin=1/COsec & COs = 1/sec & tan = 1/COt

tan = sin/cos & Sin^2 + cos^2 = 1 I memorise from its derivation. Then the others you derive from this by dividing each term by sin^2 or cos^2 or, the way I remember it's

1 + tan^2 = sec^2 (a man with a tan is sexy)
1 + cot^2 = cosec^2 (a baby in a cot is cozy)

Having these memorised through these "tricks" made solving problems significantly less challenging as my entire focus was on the problem at hand and not half of my brain power focussing on remembering the identities.

Hope this helps

[MrCliverlong]

I will restrict my suggestions and examples to the circular trig functions: sin, cos, tan and the addition or double-angle formulae. However, the ideas can be extended to the hyperbolic trig functions and multiple-angle identities such as cos 3x, sin 4x . I will also suggest some extension activities.

1) For an example like: cos(A+B) - cos(A-B) = -2sin(A)sin(B) try changing the complicated expression to the simpler expression. Usually the complicated expression has more symbols, in this case the left-hand side with the addition terms.
2) Translate the terms with double-angles into single angles.
3) If you have a mixture of squared terms and linear terms, convert the trig function that is squared to the trig function that is linear using variations of sin^2 x + cos^2 x = 1. More useful in solving trig equations.
4) Look for opportunities to apply any identity on either side and see if the expression simplifies. Sometimes starting at both ends leads to the same expression in the middle of the working. Then join the two workings together writing one backwards.
5) Learn the basic addition, double-angle and sum-product identities. This will make you more aware of ones that are available to use when you are posed with a trig question.

Extensions which are a good, geometric introduction to Complex Number: definition of trig functions as infinite series, Euler’s identity and roots of unity, de Moivre’s Theorem, relationship between circular and hyperbolic trig functions.

[davidhill]

i HAD AN IDENTITY CRISIS DOING THESE !

[dreed]

Great advice from many I see. MIne is pretty much the same for any chapter in almost any math course:
Your brain synapses lack the memory muscle and like most other subjects, you think you'll memorize it like you did some other class, or with a mnemonic, and many do that I suppose. I couldn't really. I didn't start to sight read that and think that way until I worked a zillion problems. Get intimate with the unit circle, seek out every problem you can find. You need to see through and past the identity and how to use them. You will see them like you should see commutative and associative in the language of life, not just on a math problem where you are told "identify which property of . . . blah blah with your answer" :) good luck!

[eymenu]

I think Euler identity is really useful to derive and understand what really the trig functions mean

[EvilSandwich]

I keep telling people again and again and again. If you want to master trig identities, MASTER THE UNIT CIRCLE. Study it, understand it, watch how everything changes as you move a point around the circumference. Grok the crap out of it. As you do that patterns start to emerge.

For example:
1) Remembering the Pythagorean Identities becomes trivial when you realize that it's like trying to find all the hidden Right Triangles on the Unit Circle.
2) Knowing that Secant, Cosecant and Cotangent are reciprocals of Cosine, Sine and Tangent become obvious when you can see that they grow and shrink in proportion to each other as you move to different points on the Unit Circle.
3) Sine = pi/2 - Cosine and Cosine = pi/2 - Sine also become self explanatory.
4) When Sine, Cosine are negative and positive also make sense when you realize which quadrant that takes you in.

I could go on for hours.

[videgameCaster]

Doing integrals where you had to use trigonometric identities was where I truly had to understand using them to their potential.
So my suggestion would be to aim for integrals exercises that focus on trigonometric identities.

[Leonar]

I remember well that I, too, was confused as you mentioned about how the little proofs that consecutive equalities create worked, some years ago xD it is often overlooked as a thing so simple that almost no one bothers explaining... because it is indeed easy, but you have to know why first and a student initially don't if not after a while lol.

[seanhunter]

One thing I noticed is that Chris McMullen now has a "Trig Identities" workbook as well as his other Trig workbook. That might be a good resource if you want tons of practise problems.

[nadeking]

Hey math sorcerer what advice do you have high school students graduating and going into Math Major. Like preparation tips. Thanks for your work. You’ve been a big inspiration man. I went from failing math in school about 1.5 years ago I knew nothing. To learning elementary analysis within roughly 6 months. Math has been literally live saving for me as you helped show me it’s beauty and order in this at times very dark and seemingly chaotic world. There is definitely something calming about understanding a problem and solving it, it is meditative at times. Also have you thought about getting a discord server?

[Quadra]

I am currently in class 12 in India and I am preparing for JEE ADVANCED exam but having a hard time solving the questions of trigonometry. Simplifying the equations by using identities, is the most difficult task which drags my marks to a great extent.