Let’s be real—memorizing trig identities isn’t exactly anyone’s idea of fun. But having a trig identities sheet in your back pocket (or backpack) can be a total game-changer when you’re cramming for exams or just trying to make sense of a tough problem.
This article will walk you through all the essential trig identities you need, tips on how to memorize them, and even where to grab a free printable version.
What Are Trigonometric Identities?
Trig identities are equations that relate trigonometric functions like sine, cosine, and tangent to each other. They’re always true, no matter what angle you plug in. Think of them as shortcuts to simplify expressions, solve equations, and understand the geometry of triangles and circles.
Why Use a Trig Identities Sheet?
Because who has time to memorize 20+ formulas by heart?
A trig identities sheet gives you a fast reference while doing homework, taking tests, or prepping for exams. It’s like having the answers whispered to you—except totally allowed (outside of exams, of course!).
Fundamental Trig Identities
These are the foundation of everything else. If you know these, the rest becomes much easier.
Reciprocal Identities
| Function | Reciprocal |
|---|---|
| sin(θ) = 1/csc(θ) | csc(θ) = 1/sin(θ) |
| cos(θ) = 1/sec(θ) | sec(θ) = 1/cos(θ) |
| tan(θ) = 1/cot(θ) | cot(θ) = 1/tan(θ) |
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tan(θ) = sin(θ)/cos(θ)
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cot(θ) = cos(θ)/sin(θ)
Pythagorean Identities
These are arguably the most important trig identities. They’re derived straight from the Pythagorean theorem.
The Core Three
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sin²(θ) + cos²(θ) = 1
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1 + tan²(θ) = sec²(θ)
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1 + cot²(θ) = csc²(θ)
How to Derive Them
Start with sin²(θ) + cos²(θ) = 1 and divide by either sin²(θ) or cos²(θ) to derive the others.
Co-Function Identities
These are great for working with complementary angles.
Understanding Co-Functions
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sin(90° − θ) = cos(θ)
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cos(90° − θ) = sin(θ)
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tan(90° − θ) = cot(θ)
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cot(90° − θ) = tan(θ)
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sec(90° − θ) = csc(θ)
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csc(90° − θ) = sec(θ)
Basically, co-functions of complementary angles are equal.
Even-Odd Trigonometric Identities
These show how trig functions behave with negative angles.
Positive vs Negative Angles
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sin(−θ) = −sin(θ)
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cos(−θ) = cos(θ)
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tan(−θ) = −tan(θ)
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csc(−θ) = −csc(θ)
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sec(−θ) = sec(θ)
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cot(−θ) = −cot(θ)
Sine, tangent, and cotangent are odd functions, while cosine, secant are even.
Sum and Difference Formulas
Use these when you’re adding or subtracting angles in a trig function.
Sine, Cosine, and Tangent
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sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
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cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)
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tan(A ± B) = (tan(A) ± tan(B)) / (1 ∓ tan(A)tan(B))
Double Angle Identities
These are perfect when working with 2θ situations.
Sine, Cosine, and Tangent Forms
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sin(2θ) = 2sin(θ)cos(θ)
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cos(2θ) = cos²(θ) − sin²(θ)
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Alternative: cos(2θ) = 2cos²(θ) − 1 or 1 − 2sin²(θ)
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tan(2θ) = 2tan(θ) / (1 − tan²(θ))
Half-Angle Identities
For when you need to simplify expressions like sin(θ/2).
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sin²(θ/2) = (1 − cos(θ)) / 2
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cos²(θ/2) = (1 + cos(θ)) / 2
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tan(θ/2) = ±√[(1 − cos(θ)) / (1 + cos(θ))]
or tan(θ/2) = sin(θ)/(1 + cos(θ))
Product-to-Sum and Sum-to-Product Identities
These come in handy for simplifying multiple-angle expressions.
Product-to-Sum:
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sin(A)sin(B) = ½[cos(A − B) − cos(A + B)]
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cos(A)cos(B) = ½[cos(A − B) + cos(A + B)]
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sin(A)cos(B) = ½[sin(A + B) + sin(A − B)]
Sum-to-Product:
These are the reverse—turning sums into products.
Tips for Memorizing Trig Identities
Let’s be honest—it’s a lot to remember. But here are a few tricks:
Use Mnemonics
“Sally Can Tell Cold Secrets” = Sin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj
Practice with Flashcards
Write the name on one side and the identity on the other. Quiz yourself regularly.
Make Your Own Cheat Sheet
Rewrite your trig trig identities sheet by hand. It’ll reinforce memory and give you a custom resource.
Printable Trig Identities Sheet
What to Include
A solid trig sheet should have:
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Reciprocal, Quotient, and Pythagorean identities
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Co-function and Even-Odd formulas
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Sum/Difference, Double, and Half-Angle identities
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Product-to-Sum and Sum-to-Product conversions
Where to Find Free Printables
Websites like MathIsFun, KutaSoftware, and TeachersPayTeachers often offer free downloadable PDFs. You can also make your own in Google Docs or Canva!
How to Use Your Trig Identities Sheet Effectively
During Homework
Keep the sheet next to you to save time. It’ll help you recognize which identity fits the problem.
For Test Prep
Highlight the identities you forget the most and focus your practice there.
Final Thoughts
Trigonometry can feel like learning a new language—but your trig identities sheet is your phrasebook. With the right sheet in hand, solving trig problems becomes a whole lot less intimidating. Use it smartly, practice consistently, and before you know it, those once-confusing formulas will feel second nature.
FAQs
Q1: Can I use a trig identities sheet during exams?
A1: That depends on your teacher or test guidelines. Some allow formula sheets—some don’t.
Q2: What’s the best way to memorize trig identities?
A2: Practice writing them out, use flashcards, and quiz yourself regularly.
Q3: Are all trig identities necessary to learn?
A3: Not all at once. Start with the basics like Pythagorean and Reciprocal, then build up.
Q4: Where can I find visual versions of a trig sheet?
A4: Pinterest and Google Images often have colorful, easy-to-read versions.
Q5: What if I still can’t remember them?
A5: Don’t stress—understanding how they’re derived is often more helpful than raw memorization.
