15 Trigonometry Formulas for Grade 11 — Complete Cheat Sheet with Memory Tips

Why do grade 11 students struggle with trigonometry?

Trigonometry is one of the topics that worries grade 11 students the most. Not because it's inherently difficult — but because the number of trigonometry formulas is overwhelming and they look similar. When students see pages filled with dozens of sin, cos, tan, cot formulas for a+b, a−b, 2a, half-angle... many feel lost from the start.

The fear of trigonometry usually starts with the first mistake: trying to memorize everything at once, without categorizing, without understanding the logic. The result is vague memory, and in exams students confuse sin(a+b) with cos(a+b).

In reality, the grade 11 trigonometry curriculum only requires 15 core formulas. If you master these 15 formulas and understand how to apply each type, you can solve 90% of trigonometry problems in exams. This article organizes the 15 grade 11 trigonometry formulas into 5 logical groups, with memory tips and practice problems.

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6 Basic Trigonometry Identities (Essential Foundation)

Before moving to more complex formulas, you need to master these 6 — they are the foundation from which all other formulas can be derived.

Definitions on the Unit Circle

For point M on the unit circle (radius R = 1) corresponding to an arc of length x (in radians):

• sin(x) = y-coordinate of point M.
• cos(x) = x-coordinate of point M.
• tan(x) = sin(x)/cos(x) — where cos(x) ≠ 0.
• cot(x) = cos(x)/sin(x) — where sin(x) ≠ 0.

Two Fundamental Identities

The Pythagorean identity in trigonometry:

${\sin }^{2}x+{\cos }^{2}x=1$

$1+{\tan }^{2}x=\frac{1}{{\cos }^{2}x}$

These two identities are critically important — used to:

• Find cos(x) when sin(x) is known (and vice versa).
• Simplify trigonometric expressions.
• Prove trigonometric identities.

Important corollary:

$1+{\cot }^{2}x=\frac{1}{{\sin }^{2}x}$

Table of Special Trigonometric Values

Angle	0°	30° (π/6)	45° (π/4)	60° (π/3)	90° (π/2)
sin	0	1/2	√2/2	√3/2	1
cos	1	√3/2	√2/2	1/2	0
tan	0	√3/3	1	√3	Undefined
cot	Undefined	√3	1	√3/3	0

Memory tip for sin values: Reading from 0° to 90° — sin values are √0/2, √1/2, √2/2, √3/2, √4/2. The cos table is read in reverse order.

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4 Addition Formulas (Most Important in Exams)

Addition formulas are the most important group — they are the foundation from which double-angle formulas and many others are derived.

Sin and Cos of Sum/Difference

$\sin (a+b)=\sin a\cos b+\cos a\sin b$

$\sin (a-b)=\sin a\cos b-\cos a\sin b$

$\cos (a+b)=\cos a\cos b-\sin a\sin b$

$\cos (a-b)=\cos a\cos b+\sin a\sin b$

Memory tip:

• sin of sum/difference: signs inside the formula match the sign outside (+ matches +, − matches −).
• cos of sum/difference: signs inside the formula are opposite to the sign outside.

Tan of Sum/Difference

$\tan (a+b)=\frac{\tan a+\tan b}{1-\tan a\tan b}$

$\tan (a-b)=\frac{\tan a-\tan b}{1+\tan a\tan b}$

Practical application: Calculate tan(75°) = tan(45° + 30°):

$\tan 75°=\frac{1+\sqrt{3}/3}{1-\sqrt{3}/3}=2+\sqrt{3}$

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3 Double Angle Formulas

Double angle formulas are derived from the addition formulas by setting b = a.

Sin Double Angle

$\sin 2a=2\sin a\cos a$

Derived from sin(a + a) = sin(a)·cos(a) + cos(a)·sin(a) = 2·sin(a)·cos(a).

Cos Double Angle (3 Forms)

$\cos 2a={\cos }^{2}a-{\sin }^{2}a$

From the fundamental identity, two equivalent forms:

$\cos 2a=2{\cos }^{2}a-1$

$\cos 2a=1-2{\sin }^{2}a$

These three forms are used for half-angle reduction:

• cos²(a) = (1 + cos(2a)) / 2
• sin²(a) = (1 − cos(2a)) / 2

Tan Double Angle

$\tan 2a=\frac{2\tan a}{1-{\tan }^{2}a}$

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2 Groups of Sum-to-Product and Product-to-Sum Formulas

This is the group most often overlooked by students but appears frequently in simplification and trigonometric equation problems.

Product-to-Sum

Product	Corresponding Sum/Difference
cos(a)·cos(b)	(1/2)·[cos(a−b) + cos(a+b)]
sin(a)·sin(b)	(1/2)·[cos(a−b) − cos(a+b)]
sin(a)·cos(b)	(1/2)·[sin(a+b) + sin(a−b)]

Memory tip: cos·cos and sin·sin products → cos sum/difference. sin·cos product → sin sum/difference.

Sum-to-Product

Sum/Difference	Corresponding Product
sin(a) + sin(b)	2·sin((a+b)/2)·cos((a−b)/2)
sin(a) − sin(b)	2·cos((a+b)/2)·sin((a−b)/2)
cos(a) + cos(b)	2·cos((a+b)/2)·cos((a−b)/2)
cos(a) − cos(b)	−2·sin((a+b)/2)·sin((a−b)/2)

Memory tip: "sin + sin → sin times cos"; "cos + cos → cos times cos"; "cos − cos → negative sin times sin".

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5 Effective Memory Tips for Trigonometry Formulas

Tip 1: Group by logic, not individually

Don't try to memorize each formula independently. Group them by theme (addition formulas, double angle formulas, sum-to-product) and understand the relationships between them. Double angle formulas are a special case of addition formulas when a = b.

Tip 2: Derive from scratch instead of rote memorization

If you forget the double angle formula, don't panic — derive it again from the addition formula. This takes about 30 seconds in an exam and guarantees 100% accuracy. Being able to derive formulas from fundamentals is more valuable than mechanical memorization.

Tip 3: Special value table — learn through visuals

Draw the unit circle with 5 special points (0°, 30°, 45°, 60°, 90°) and fill in the sin/cos values. Visual images help the brain retain information much longer than text.

Tip 4: Practice problems immediately after studying

Right after reading a group of formulas, solve at least 5 direct application problems. The brain learns most effectively through practice with immediate feedback. The Witza app lets you practice trigonometry problems and receive detailed solutions instantly — perfect for quick practice after each theory session.

Tip 5: Understand the sign pattern

For addition formulas: sin follows the "same sign" rule, cos follows the "opposite sign" rule. This simple pattern eliminates half the memorization effort.

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10 Practice Problems with Solutions

Problem 1: Simplify the expression

Simplify A = sin²(3x) + cos²(3x).

Solution: This is the fundamental identity sin²(u) + cos²(u) = 1 with u = 3x.

$A=1$

Problem 2: Find trigonometric values

Given cos(x) = 3/5 and x is in the first quadrant. Find sin(x), tan(x), cot(x).

Solution:

${\sin }^{2}x=1-{\cos }^{2}x=1-\frac{9}{25}=\frac{16}{25}$

Since x is in quadrant I, sin(x) is positive: sin(x) = 4/5.

tan(x) = sin(x)/cos(x) = (4/5)/(3/5) = 4/3, cot(x) = 3/4.

Problem 3: Using the addition formula

Calculate sin(75°).

Solution: 75° = 45° + 30°

$\sin 75°=\sin 45°\cos 30°+\cos 45°\sin 30°=\frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot \frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}$

Problem 4: Prove the identity

Prove: sin(x + y)·sin(x − y) = sin²(x) − sin²(y).

Solution:

Left side = (sin(x)·cos(y) + cos(x)·sin(y))(sin(x)·cos(y) − cos(x)·sin(y))

$={\sin }^{2}x{\cos }^{2}y-{\cos }^{2}x{\sin }^{2}y$

$={\sin }^{2}x(1-{\sin }^{2}y)-(1-{\sin }^{2}x){\sin }^{2}y={\sin }^{2}x-{\sin }^{2}y✓$

Problem 5: Using the double angle formula

Given sin(x) = 1/3. Calculate sin(2x) and cos(2x).

Solution:

cos²(x) = 1 − 1/9 = 8/9, so cos(x) = ±2√2/3.

With cos(x) = 2√2/3 (acute angle):

$\sin 2x=2\sin x\cos x=2\cdot \frac{1}{3}\cdot \frac{2\sqrt{2}}{3}=\frac{4\sqrt{2}}{9}$

$\cos 2x=1-2{\sin }^{2}x=1-\frac{2}{9}=\frac{7}{9}$

Problem 6: Sum-to-product

Simplify B = cos(40°) + cos(80°).

Solution:

$B=2\cos \frac{40°+80°}{2}\cos \frac{40°-80°}{2}=2\cos 60°\cdot \cos 20°=2\cdot \frac{1}{2}\cdot \cos 20°=\cos 20°$

Problem 7: Half-angle reduction

Simplify C = cos²(x) − sin²(x).

Solution: This is one form of the double angle formula:

$C={\cos }^{2}x-{\sin }^{2}x=\cos 2x$

Problem 8: Basic trigonometric equation

Solve 2cos(2x) − 1 = 0.

Solution:

$\cos 2x=\frac{1}{2}=\cos 60°$

$2x=\pm 60°+k\cdot 360°(k\in \mathbb{Z})$

$x=\pm 30°+k\cdot 180°$

Problem 9: Product-to-sum

Calculate 2sin(75°)·cos(15°).

Solution: Using 2sin(a)·cos(b) = sin(a+b) + sin(a−b):

$2\sin 75°\cos 15°=\sin 90°+\sin 60°=1+\frac{\sqrt{3}}{2}=\frac{2+\sqrt{3}}{2}$

Problem 10: Comprehensive problem

Prove: (sin(3x) − sin(x)) / (cos(x) − cos(3x)) = cot(2x).

Solution:

Numerator: sin(3x) − sin(x) = 2cos(2x)·sin(x)

Denominator: cos(x) − cos(3x) = 2sin(2x)·sin(x)

$\frac{\sin 3x-\sin x}{\cos x-\cos 3x}=\frac{2\cos 2x\sin x}{2\sin 2x\sin x}=\frac{\cos 2x}{\sin 2x}=\cot 2x✓$

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Frequently Asked Questions

Q1: Why is sin²(x) different from sin(x²)?

This is the most common mistake in trigonometry. sin²(x) means (sin(x))² — the square of the sine value at x. While sin(x²) is the sine value of x² — sin applied to the number x². These are completely different. For example: with x = π/2, sin²(π/2) = 1² = 1, but sin((π/2)²) = sin(π²/4) ≈ 0.61.

The notation sin^n(x) in mathematics always means (sin(x))^n, not sin(x^n).

Q2: How to quickly memorize the special trigonometric values table?

Use the "staircase of √n divided by 2" trick for sin: values of sin at 0°, 30°, 45°, 60°, 90° are √0/2, √1/2, √2/2, √3/2, √4/2. The cos table is read in reverse (from 90° back to 0°). tan = sin/cos, cot = cos/sin.

Q3: Why must you memorize sin, cos, tan of 30°, 45°, 60°?

Because these are the angles that appear most frequently in problems and exams — especially in trigonometry and geometry. Knowing the special values table by heart lets you calculate quickly without a calculator, and this is essential in national exam conditions (since many multiple-choice questions require exact calculations with these special angles).

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Studying grade 11 trigonometry formulas systematically with memory tips will save you a lot of exam preparation time. The 15 formulas in this article are sufficient to solve the vast majority of trigonometry problems in the high school curriculum. If you want to test your mastery right now, try the trigonometry practice set on Witza — the AI math solving app for Vietnamese students, where you can snap a photo of any problem and receive a detailed step-by-step solution in seconds.