Double, Triple, and Half-Angle Formulae: Which One to Use?

You recognise every formula on the revision sheet. Then a question gives you cos x and asks for sin(x/2), and suddenly the sheet feels much less helpful.

The difficulty is usually choosing a route. Should you find another ratio first? Should you take a square root? Which sign belongs in the answer?

Double, triple, and half-angle formulae become easier when you read them as instructions for changing an angle. Start with what you know, look at what you need, and choose the shortest valid connection between the two.

This guide uses original worked examples to help you make those decisions. In the notation below, sin²x means (sin x)², sin³x means (sin x)³, and sqrt means the nonnegative square root. Angles marked with ° are in degrees; the identities also work in radians when units are used consistently.

Start with this decision table

What the question gives or shows	What to try first
sin x, and you need cos 2x	1 - 2sin²x
cos x, and you need cos 2x	2cos²x - 1
Both sin x and cos x	sin 2x = 2sin x cos x
tan x, and you need a double-angle ratio	A formula written in tan x
Cubes such as 3sin x - 4sin³x	Recognise sin 3x
cos x, and you need a ratio of x/2	Half-angle formula, then a sign check
1 - cos x or 1 + cos x	Rewrite as 2sin²(x/2) or 2cos²(x/2)
sin²x or cos²x in an expression	Consider a power-reduction formula

The table is a starting point, not a command to expand everything. Sometimes recognising a formula backwards saves several lines.

For example, 2sin 17° cos 17° is already sin 34°. There is no reason to find sin 17° and cos 17° separately.

Double-angle formulae: choose the version that matches your information

The main identities are:

• sin 2x = 2sin x cos x
• cos 2x = cos²x - sin²x
• cos 2x = 2cos²x - 1
• cos 2x = 1 - 2sin²x
• tan 2x = 2tan x / (1 - tan²x), where tan x exists and the denominator is nonzero.

The three cosine versions say the same thing. The identity sin²x + cos²x = 1 lets you move between them. Choose the form that uses the ratio you already have. These identities and their derivations appear in NCERT’s Trigonometric Functions chapter, Section 3.4.

Example 1: one given ratio, two different routes

Suppose sin x = 7/25 and 90° < x < 180°. Find cos 2x and sin 2x.

For cos 2x, the given sine is enough:

cos 2x = 1 - 2(7/25)² = 1 - 98/625 = 527/625.

For sin 2x, we need cos x too. Since x lies in the second quadrant, cosine is negative:

cos x = -sqrt(1 - 49/625) = -24/25.

Therefore:

sin 2x = 2(7/25)(-24/25) = -336/625.

Notice the difference. Finding cos x was necessary for the second calculation, but unnecessary for the first. Choosing the right cosine form saves time and avoids introducing an extra sign decision.

As a check, the two answers obey sin²2x + cos²2x = 1 because 336² + 527² = 625².

Example 2: when tangent is already given

If tan x = 2/5, then:

tan 2x = (4/5)/(1 - 4/25) = (4/5)/(21/25) = 20/21.

You can also obtain sine and cosine of 2x directly:

• sin 2x = 2tan x / (1 + tan²x) = 20/29.
• cos 2x = (1 - tan²x) / (1 + tan²x) = 21/29.

These two forms require tan x to be defined. Their denominator, 1 + tan²x, is positive for every real value of tan x.

Why do we not need x’s quadrant here? Angles with the same tangent differ by a multiple of 180°. Doubling that difference gives a multiple of 360°, so both double-angle sine and cosine stay the same.

That reasoning does not apply automatically to half angles.

A zero denominator is a result you must interpret

If tan x = 1, the denominator in the tangent double-angle formula becomes zero. You cannot assign an ordinary number to the fraction.

Here x = 45° + 180°n for an integer n. Hence 2x = 90° + 360°n, where tangent is undefined.

There is a different problem at x = 90°. The value tan 180° exists and equals zero, but tan x itself is undefined. So a formula written in tan x cannot be used at that input. Evaluate the target directly or use sine and cosine.

Triple-angle formulae: recognise the cubic pattern

The three standard forms are:

• sin 3x = 3sin x - 4sin³x
• cos 3x = 4cos³x - 3cos x
• tan 3x = (3tan x - tan³x)/(1 - 3tan²x), where tan x exists and the denominator is nonzero.

Look carefully at the order of the sine and cosine terms. Sine starts with the linear term. Cosine starts with the cubic term.

A quick memory check is x = 0°. The cosine formula must produce 1, so 4(1)³ - 3(1) works. Reversing the terms would give -1.

Where the sine formula comes from

You can rebuild it using the angle-sum identity:

sin 3x = sin(2x + x)

= sin 2x cos x + cos 2x sin x

= 2sin x cos²x + (1 - 2sin²x)sin x.

Now replace cos²x by 1 - sin²x:

= 2sin x(1 - sin²x) + sin x - 2sin³x

= 3sin x - 4sin³x.

The useful lesson is the substitution choice. To finish with sine only, eliminate cosine squared. If you were deriving the cosine triple-angle identity, you would eliminate sine squared instead.

Example 3: triple angles do not always need a quadrant

Given sin x = 2/7, find sin 3x.

sin 3x = 3(2/7) - 4(2/7)³

= 6/7 - 32/343

= 262/343.

No cosine appears in the formula, so there is no missing cosine sign to resolve. The given sine determines sin 3x even though it does not determine a unique x.

Similarly, if cos y = -3/5:

cos 3y = 4(-3/5)³ - 3(-3/5)

= -108/125 + 225/125

= 117/125.

Do not assume that a negative cos y forces cos 3y to be negative. The angle has changed.

Example 4: a tangent calculation with a useful sign check

Let tan x = 1/2 and 0° < x < 90°. Then:

tan 3x = [3/2 - 1/8]/[1 - 3/4]

= (11/8)/(1/4)

= 11/2.

Because tan x = 1/2 is less than tan 30° = 1/sqrt(3), x is between 0° and 30°. Therefore 3x lies between 0° and 90°, so the positive answer makes sense.

If tan x had instead been 1/sqrt(3), the denominator would vanish. For acute x, that means x = 30°, so 3x = 90° and tangent is undefined.

Half-angle formulae: the square root gives magnitude, not the sign

The basic forms are:

• sin(x/2) = ±sqrt[(1 - cos x)/2]
• cos(x/2) = ±sqrt[(1 + cos x)/2]
• tan(x/2) = ±sqrt[(1 - cos x)/(1 + cos x)], where 1 + cos x is nonzero.

Choose each sign from the position of x/2. The sign of sin x or cos x by itself does not decide the sign of the half-angle ratio. OpenStax’s half-angle lesson explains this quadrant condition alongside the formulas.

Why is there a square root? Start with cos x = 1 - 2sin²(x/2). Rearranging gives sin²(x/2) = (1 - cos x)/2. Taking the nonnegative square root produces the absolute value of sin(x/2). The angle’s location supplies the sign.

Halve the actual interval

For angles between 0° and 360°, this table helps:

Interval containing x	Interval containing x/2	Signs of sin, cos, tan at x/2
0° < x < 90°	0° < x/2 < 45°	Positive, positive, positive
90° < x < 180°	45° < x/2 < 90°	Positive, positive, positive
180° < x < 270°	90° < x/2 < 135°	Positive, negative, negative
270° < x < 360°	135° < x/2 < 180°	Positive, negative, negative

Boundary angles need their exact axis values. Outside this range, halve the interval you were actually given rather than forcing the angle into this table.

Example 5: the same data used for all three half-angle ratios

Let cos x = -7/25 and 180° < x < 270°.

First, halve the interval:

90° < x/2 < 135°.

Thus x/2 lies in the second quadrant. Sine is positive; cosine and tangent are negative.

sin(x/2) = +sqrt[(1 + 7/25)/2]

= sqrt(16/25) = 4/5.

cos(x/2) = -sqrt[(1 - 7/25)/2]

= -sqrt(9/25) = -3/5.

Finally:

tan(x/2) = (4/5)/(-3/5) = -4/3.

The sign check belongs before the arithmetic. If you attach signs after several lines of calculation, it is much easier to forget which angle you are considering.

Example 6: a half angle outside the usual first two quadrants

Suppose 360° < x < 540° and cos x = 9/41. Find sin(x/2) and cos(x/2).

Halving gives 180° < x/2 < 270°, so both answers must be negative:

• sin(x/2) = -sqrt[(1 - 9/41)/2] = -4/sqrt(41).
• cos(x/2) = -sqrt[(1 + 9/41)/2] = -5/sqrt(41).

This example explains why you should keep the original angle interval. Reducing x by 360° before halving changes the half angle by 180°, which changes both sine and cosine signs.

Tangent half-angle forms that avoid square roots

Two useful alternatives are:

tan(x/2) = sin x / (1 + cos x), when 1 + cos x is nonzero.

tan(x/2) = (1 - cos x) / sin x, when sin x is nonzero.

Signed sine and cosine values carry the sign through these fractions automatically. You still need to know their signs correctly.

For the data in Example 5, sin x = -24/25 because x is in the third quadrant. Therefore:

tan(x/2) = (-24/25)/(1 - 7/25)

= (-24/25)/(18/25)

= -4/3.

The second version gives the same answer:

(1 + 7/25)/(-24/25) = (32/25)/(-24/25) = -4/3.

But the two fraction forms do not have identical domains. At x = 0°, sin x/(1 + cos x) gives 0/2 = 0, while (1 - cos x)/sin x becomes 0/0 and cannot be used. The target tan 0° still exists.

At x = 180°, the target tan 90° is itself undefined.

These are good reminders to check the actual fraction instead of assuming every displayed version works everywhere.

Finding exact values without a calculator

Example 7: sin 22.5° and cos 22.5°

Since 22.5° is half of 45° and lies in the first quadrant, use positive square roots:

sin 22.5° = sqrt[(1 - cos 45°)/2]

= sqrt[(1 - sqrt(2)/2)/2]

= sqrt(2 - sqrt(2))/2.

Similarly:

cos 22.5° = sqrt(2 + sqrt(2))/2.

The outer division by 2 follows from taking the square root of a denominator of 4. Do not leave a denominator of 4 after removing the square root.

Example 8: tan 7.5°

Here 7.5° is half of 15°. We can use familiar exact values for 15°:

sin 15° = [sqrt(6) - sqrt(2)]/4,

cos 15° = [sqrt(6) + sqrt(2)]/4.

Then:

tan 7.5° = [sqrt(6) - sqrt(2)]/[4 + sqrt(6) + sqrt(2)].

This is already an exact value. Unless the question asks for a particular form or a rationalised denominator, a complicated extra rearrangement is not needed.

To recover the 15° values if you forget them, use 15° = 45° - 30° with the sine and cosine difference identities. Those identities are reviewed in OpenStax’s sum and difference lesson.

Identity proofs: decide which direction makes the expression simpler

In a proof, you are showing equality wherever both sides are defined. Start with one side and transform it. Avoid assuming the required result is already true.

Example 9: recognise a triple angle backwards

Simplify 8cos³x - 6cos x.

Factor out 2:

8cos³x - 6cos x = 2(4cos³x - 3cos x) = 2cos 3x.

This is much shorter than turning every cosine into sine or introducing tangent. The coefficients tell you which pattern to recognise.

For comparison, 4sin³x - 3sin x is -sin 3x. Reversing the order changes the sign.

Example 10: use double angles to factor a mixed expression

Prove that:

sin 4x + 2sin 2x = 8sin x cos³x.

Starting on the left:

sin 4x + 2sin 2x

= 2sin 2x cos 2x + 2sin 2x

= 2sin 2x(1 + cos 2x)

= 2(2sin x cos x)(2cos²x)

= 8sin x cos³x.

The turning point was factoring out sin 2x. Expanding sin 4x immediately into a long expression in x would work, but it would hide that useful common factor.

Example 11: simplify without forgetting an excluded value

Simplify sin 2x/(1 + cos 2x).

Using double-angle forms:

sin 2x/(1 + cos 2x)

= (2sin x cos x)/(2cos²x)

= tan x, provided cos x is nonzero.

At x = 90°, the original denominator is zero, so that value was never allowed. Cancelling cos x does not bring it back into the domain.

Power reduction is a double-angle formula read backwards

From cos 2x = 1 - 2sin²x and cos 2x = 2cos²x - 1:

• sin²x = (1 - cos 2x)/2
• cos²x = (1 + cos 2x)/2

Use these when reducing powers makes the rest of an expression easier.

For instance:

sin²18° cos²18° = (sin 36°/2)²

= sin²36°/4

= (1 - cos 72°)/8.

There are two changes here: first combine the sine-cosine product, then reduce the resulting square. Keeping those steps separate makes the factor of 8 easy to check.

Do not lose solutions when an identity appears in an equation

An identity is a tool for solving an equation, but algebraic steps still need care.

Solve sin 2x = sin x for 0° ≤ x < 360°.

Replace sin 2x:

2sin x cos x = sin x.

Move everything to one side and factor:

sin x(2cos x - 1) = 0.

So either sin x = 0 or cos x = 1/2.

Within the given interval, the solutions are:

x = 0°, 60°, 180°, 300°.

If you had divided by sin x at the start, you would have lost 0° and 180°. Also, 360° is excluded by the question’s interval even though it satisfies the equation.

A short error checklist

Tempting mistake	A quick correction
Writing sin 2x = 2sin x	The missing factor is cos x
Reading sin²x as sin 2x	One squares a value; the other doubles an angle
Using 3cos x - 4cos³x for cos 3x	The cubic term comes first
Choosing the half-angle sign from x	Locate x/2 instead
Taking every square root as positive	A root gives magnitude; the ratio may be negative
Using a tangent formula with a zero denominator	Check whether the target or only the chosen form is undefined
Cancelling sin x while solving an equation	Treat sin x = 0 as a separate possibility
Reducing x by 360° before choosing half-angle signs	Preserve the original interval while halving

Practise choosing before calculating

Try writing the formula you intend to use before doing any arithmetic.

1. If sin x = 9/41 and x is acute, find cos 2x.
2. If tan x = 3/4, find sin 2x, cos 2x, and tan 2x.
3. If cos x = 1/4, find cos 3x.
4. If cos x = -15/17 and 180° < x < 270°, find sin(x/2), cos(x/2), and tan(x/2).
5. Simplify 12sin x - 16sin³x.
6. Find the exact value of cos 157.5°.
7. Solve cos 2x = cos x for 0° ≤ x < 360°.

Answers with the deciding step

1. Use the sine-only cosine formula.

cos 2x = 1 - 2(9/41)² = 1519/1681. No need to find cos x first.

2. Use tangent-based double-angle forms.

sin 2x = (3/2)/(25/16) = 24/25.

cos 2x = (7/16)/(25/16) = 7/25.

Their quotient gives tan 2x = 24/7.

3. Use the cosine cubic.

cos 3x = 4(1/4)³ - 3(1/4) = 1/16 - 12/16 = -11/16.

4. Halve the interval first.

The half angle lies in the second quadrant:

sin(x/2) = 4/sqrt(17),

cos(x/2) = -1/sqrt(17),

tan(x/2) = -4.

5. Factor out 4 and recognise the remaining pattern.

4(3sin x - 4sin³x) = 4sin 3x.

6. Regard 157.5° as half of 315°.

The required cosine is negative:

cos 157.5° = -sqrt[(1 + cos 315°)/2] = -sqrt(2 + sqrt(2))/2.

7. Turn the equation into a quadratic in cos x.

2cos²x - 1 = cos x gives (2cos x + 1)(cos x - 1) = 0.

Thus cos x = -1/2 or cos x = 1, giving x = 0°, 120°, 240° in the stated interval.

Frequently asked questions

Is sin 2x the same as 2sin x?

No. sin 2x means the sine of twice the angle. The correct identity is sin 2x = 2sin x cos x. For example, sin 60° is sqrt(3)/2, while 2sin 30° is 1.

Which cos 2x formula should I choose?

Use 1 - 2sin²x when sine is given, and 2cos²x - 1 when cosine is given. Use cos²x - sin²x when both ratios are already available or the expression naturally contains their squared difference.

Why do half-angle formulae have a plus-or-minus sign?

They come from squared identities. Taking a square root gives the magnitude of the ratio. Choose the sign using the quadrant of the half angle; for a fully specified angle, you do not normally keep both signs as the final answer.

Can I find sin(x/2) from cos x alone?

You can find its magnitude, but its sign may remain uncertain without an interval or other angle information. If the information guarantees 0° < x < 360°, then 0° < x/2 < 180° and the sine is positive.

Can I reduce an angle by 360° before halving it?

Be careful. Reducing x by 360° reduces x/2 by 180°, changing half-angle sine and cosine signs. Halve the actual angle or interval first, then use coterminal angles if helpful.

Are triple-angle formulae valid in radians?

Yes. They are identities about angle relationships. Keep one unit throughout. For instance, sin(3 × pi/6) and sin(3 × 30°) both equal 1.

Is sin³x the same as sin 3x?

No. sin³x means (sin x)³. sin 3x means sin(3x). The identity sin 3x = 3sin x - 4sin³x connects them, but they are different expressions.

What if a tangent formula has a zero denominator?

Do not divide. Check the original angle and target function. At x = 45°, tan 2x is undefined. At x = 90°, tan x is undefined but tan 2x equals zero, so the tangent-based double-angle formula cannot be used even though the target exists.

Must I rationalise every exact-value answer?

Follow the question’s instructions. Expressions such as 4/sqrt(17) and 4sqrt(17)/17 are equal exact values. Rationalising can make comparison easier, but a decimal approximation should not replace an exact answer when an exact value is requested.

What is the best way to revise these formulae?

Practise a small mixed set where you must choose the formula before calculating. Include one double-angle value, one cubic pattern, one half-angle sign question, and one equation with a possible zero factor. Explain your first step aloud, then check the final sign and domain.