Trigonometric Identity Transformation Calculator & Formula Lookup

The tool above brings every trigonometric transformation formula from the high-school curriculum into one place. The Formula table tab is a quick reference with copyable LaTeX. The Evaluate tab plugs concrete angles into each formula and computes both sides as exact surds so you can verify the identity yourself. The Transform tab gives step-by-step solutions for sum ↔ product, power reduction, multiple angles, a·sin x + b·cos x and expansions. Below is a summary of the theory behind each formula group: where it comes from, when to use it and where students usually slip. The article ends with a 5-question quiz, one question per formula group, so you can check how well you understand it.

➕ 1. Addition formulas: the root of every identity

For all angles a and b:

Where does it come from? The points M(cos a; sin a) and N(cos b; sin b) on the unit circle make an angle a − b at the origin O, so the dot product OM·ON = cos(a − b). Computed from coordinates, the same dot product equals cos a cos b + sin a sin b. Replacing b with −b and using sin x = cos(π/2 − x) gives the rest. So if you remember one formula well, you can rebuild the whole group.

Remembering the signs: sine keeps the sign (the sine of a sum has a plus), cosine flips it (the cosine of a sum has a minus). The tangent formula needs a, b, a ± b ≠ π/2 + kπ.

Example. Find the exact value of cos 15°:

Application: combining a·sin x + b·cos x. Read the addition formula backwards and any expression of this form becomes a single sine:

For example √3 sin x + cos x = 2 sin(x + π/6), so the expression always lies in [−2; 2]. This is the key step for solving a sin x + b cos x = c and for finding maximum and minimum values.

✖️ 2. Double- and triple-angle formulas

Set b = a in the addition formulas and the double-angle formulas appear at once:

The three forms of cos 2a come from sin²a + cos²a = 1. Choose the one that fits the problem: if only sin a is known use 1 − 2sin²a, if only cos a is known use 2cos²a − 1, and to factor use cos²a − sin²a = (cos a − sin a)(cos a + sin a).

Applying the addition formula to 3a = 2a + a and substituting the double-angle formulas gives the triple-angle formulas:

Notice the opposite order: for sine the linear term comes first, for cosine the cubic term comes first.

Example. If cos a = 1/3, then cos 2a = 2·(1/9) − 1 = −7/9, whatever quadrant a is in.

⬇️ 3. Power-reduction and half-angle formulas

Solve the last two forms of cos 2a for cos²a and sin²a: the square drops to the first power, and in exchange the angle doubles:

Likewise, from the triple-angle formulas: sin³a = (3sin a − sin 3a)/4 and cos³a = (3cos a + cos 3a)/4.

Replacing a with a/2 gives the half-angle formulas, used to find values at angles such as 22.5° or 15°: cos²(a/2) = (1 + cos a)/2. When taking the square root, pick the sign from the quadrant of a/2. With t = tan(a/2) (for a ≠ π + 2kπ), every trigonometric value of a can be written in terms of t:

When to use them? Power reduction turns powers into sums of first-degree terms, which is exactly what you need for antiderivatives (such as ∫sin²x dx), higher-degree equations and max/min problems.

Example. Reduce twice:

🔀 4. Product-to-sum formulas

Expand cos(a − b) and cos(a + b) with the addition formula, then add (or subtract) the two lines: the sin a sin b terms (or the cos a cos b terms) cancel. Doing the same with sin(a − b) and sin(a + b) gives three formulas:

What they share: a factor of ½, the difference a − b first and the sum a + b second. Only sin a sin b is a difference of cosines. Quick check: with a = b you get sin²a = ½(1 − cos 2a), which must be non-negative, so cos(a − b) has to come before the minus sign.

Example. A product of two awkward values becomes a sum of two special ones:

🔁 5. Sum-to-product formulas

Read the product-to-sum formulas backwards: put u = a + b and v = a − b, that is a = (u + v)/2, b = (u − v)/2, and you get four formulas:

The first angle is always half the sum and the second is half the difference. Note the minus sign in front of cos u − cos v. Two companion formulas you will use often:

When to use them? Sum-to-product turns a sum into a product, which makes it the main tool for “product equals zero” equations and for simplifying trigonometric fractions.

Example. sin 5x + sin 3x = 2 sin 4x cos x, so the equation sin 5x + sin 3x = 0 is equivalent to sin 4x = 0 or cos x = 0.

⚠️ Common mistakes

1. Distributing sine or cosine like multiplication. sin(a + b) ≠ sin a + sin b. Try a = b = 30°: the left side is sin 60° ≈ 0.866, the right side is 1.
2. Wrong sign in cos(a ± b). The cosine of a sum has a minus, the cosine of a difference has a plus.
3. Dropping a factor. cos²a = (1 + cos 2a)/2, not 1 + cos 2a; product-to-sum always has ½ and sum-to-product always has 2.
4. Swapping the order in sin a sin b. The correct form is ½[cos(a − b) − cos(a + b)]. The reversed version would make sin²a negative.
5. Using a tangent formula where it is undefined. tan 2a needs cos 2a ≠ 0, and tan(a + b) needs a + b ≠ π/2 + kπ. The tool reports “undefined” instead of returning a number.
6. Wrong sign in a half-angle formula. The sign of cos(a/2) or sin(a/2) depends on the quadrant of a/2, not of a.

🎯 Practice quiz: one question per formula group

Pick one answer for each question. The correct answer turns green and the worked solution appears right below, along with why the other options are wrong. You can use the Evaluate tab of the tool to double-check.

❓ Frequently asked questions (FAQ)

Which groups of trigonometric identities should I know?

There are five main groups: the addition formulas, the double- and triple-angle formulas, the power-reduction (and half-angle) formulas, product-to-sum and sum-to-product. All of them follow from the addition formulas, so mastering those lets you rebuild the rest.

How do I remember the signs in the addition formulas?

Sine keeps the sign: sin(a + b) = sin a cos b + cos a sin b. Cosine flips it: cos(a + b) = cos a cos b − sin a sin b.

When should I use the power-reduction formulas?

Whenever an expression contains sin²x, cos²x, sin³x, cos³x or sin⁴x and you need first-degree terms: antiderivatives, higher-degree equations and max/min problems. In exchange the angle doubles, for example cos²x = (1 + cos 2x)/2.

What is the difference between sum-to-product and product-to-sum?

Product-to-sum turns a product such as cos a cos b into a sum with a factor of ½, which helps with simplifying and integrating. Sum-to-product goes the other way, turning cos u + cos v into 2cos((u + v)/2)cos((u − v)/2), which helps with solving equations because it produces a product equal to zero.

Why is the result exact for 15° or 75° but approximate for 20°?

Multiples of 15° have sines and cosines that can be written with square roots of 2, 3 and 6, so the tool computes them exactly. An angle like 20° has no simple square-root form, so the tool gives a decimal approximation labelled “Approximate” with the ≈ sign.

Is the tool available in Vietnamese?

Yes. Click VI in the corner of the tool to switch the interface, the formula table and the solutions to Vietnamese instantly, or open the Vietnamese article with the site language switcher.

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