Triple angle Identities

Triple angle identities express trigonometric functions of $3x$ in terms of functions of $x$.

They are useful when a problem contains angles such as $3x$, $60^\circ-3x$, or when equations involve cubic expressions in sine or cosine.

Sine Triple angle Formula

Start with

Using the angle addition formula,

Now substitute

and

This gives

Since $\cos^2 x=1-\sin^2 x$,

Simplifying,

Cosine Triple angle Formula

Similarly,

Using the angle addition formula,

Substitute

and

Then

Using $\sin^2 x=1-\cos^2 x$,

Simplifying,

Tangent Triple angle Formula

Using the tangent addition formula, you get

provided the expressions are defined.

This formula is less commonly memorised, but it can be useful in certain equation solving problems.

Sums of Trigonometric Series

Some problems ask us to evaluate sums such as

or

There are several ways to approach these sums. A useful method is to multiply by a carefully chosen trigonometric factor so that the sum telescopes.

Sum of Sines

Let

Multiply both sides by $2\sin\left(\frac{x}{2}\right)$.

Using the identity

we get

Now apply product to sum:

So the sum becomes

Most terms cancel. We are left with

Using sum to product,

Therefore,

This formula is valid when $\sin\left(\frac{x}{2}\right)\neq 0$. Special cases where the denominator is zero should be handled separately.

Sum of Cosines

A similar telescoping argument gives

again assuming $\sin\left(\frac{x}{2}\right)\neq 0$.

These two formulas are especially useful when evaluating long trigonometric sums.

Worked Examples

Example 1

Simplify $\sin 5x\cos 2x$.

Use the product to sum identity

With $A=5x$ and $B=2x$,

Example 2

Express $\sin^2 x$ in terms of $\cos 2x$.

From the double angle identity,

Rearranging,

This identity is useful when reducing powers of trigonometric functions.

Example 3

Solve $4\cos^3 x - 3\cos x = \frac{1}{2}$ for $0^\circ \le x < 360^\circ$.

Recognise the left hand side as the cosine triple angle formula:

So the equation becomes

We know that $\cos \theta = \frac{1}{2}$ when $\theta = 60^\circ + 360^\circ k$ or $\theta = 300^\circ + 360^\circ k$.

So

From the first family: $x = 20^\circ, 140^\circ, 260^\circ$.

From the second family: $x = 100^\circ, 220^\circ, 340^\circ$.

The six solutions in $[0^\circ, 360^\circ)$ are

Example 4

Evaluate

$$\sin 10^\circ+\sin 20^\circ+\cdots+\sin 80^\circ.$$

This is a sine series with first angle $10^\circ$, common step $10^\circ$, and last angle $80^\circ$.

Using the sum formula with $x=10^\circ$ and $n=8$,

This is already exact form.

Example 5

Show that

$$\sin x+\sin 3x+\sin 5x+\cdots+\sin(2n-1)x = \frac{\sin^2(nx)}{\sin x}.$$

Let

Multiply by $2\sin x$:

Using

we get

Therefore the sum telescopes:

All middle terms cancel, leaving

Since

we obtain

Key Contest Insight

Most trigonometric problems become easier after changing the form of the expression.

Use:

• Product to sum when products appear
• Sum to product when sums need to be factored
• Double angle identities when squares appear
• Auxiliary angle when an expression has the form $a\sin x+b\cos x$
• Triple angle identities when cubes or $3x$ appear
• Telescoping when a long trigonometric series appears

The main challenge is recognising which identity makes the expression simpler.

Strategy Tips

• Do not expand blindly. Look for structure first.
• Products such as $\sin A\cos B$ often suggest product to sum.
• Squares such as $\sin^2 x$ and $\cos^2 x$ often suggest double angle identities.
• For expressions like $a\sin x+b\cos x$, use the auxiliary angle method to find maximum and minimum values quickly.
• Cubes such as $\sin^3 x$ and $\cos^3 x$ often suggest triple angle identities.
• Long sums with evenly spaced angles often telescope after multiplying by a suitable sine factor.
• Always check whether special cases make denominators zero.
• In degree notation, keep track of whether angles are in degrees or radians.

Common Pitfalls

• Mixing degrees and radians.
• Forgetting the negative sign in

• Using a formula without checking whether division by zero occurs.
• Applying a product to sum identity with the wrong signs.
• Forgetting that triple angle identities contain cubic terms.
• Expanding trigonometric expressions when a direct identity would simplify them.

Practice Problems