Trigonometric Identity Rewrites for Limits

Why Rewrite a Trigonometric Quotient?

When direct substitution into a trigonometric quotient gives \(0/0\), the expression does not by itself determine the limit. A useful next step is to rewrite it with a trigonometric identity. The goal is to expose a standard limit, especially \(\lim_{u\to 0}\frac{\sin u}{u}=1\), established in the earlier material on the sine quotient limit.

This tutorial focuses on identities that turn expressions involving \(1-\cos u\), \(\sin(2u)\), and squared trigonometric functions into forms that can be evaluated using that standard limit and the continuity of cosine. Angles are measured in radians throughout; the standard sine quotient limit in this form requires radians.

Identities Used in Limit Rewrites

The half-angle identity is particularly useful when a numerator contains one minus cosine. It follows from the double-angle identity \(\cos(2t)=1-2\sin^2 t\): setting \(t=u/2\) gives the equivalent formula below.

Other useful identities are the double-angle formula for sine and the Pythagorean identity. Each is valid for every real \(u\), so it may be substituted into an expression wherever that expression is defined.

To see why the half-angle identity is well suited to limits near zero, put it into the quotient with \(u^2\) in the denominator. For \(u\ne 0\), the sine quotient can be isolated:

As \(u\to 0\), \(u/2\to 0\), so the standard sine quotient limit applies. Squaring and multiplying by \(1/2\) then gives the quadratic cosine limit.

Worked Example: A One-Minus-Cosine Quotient

Scaled Sine Quotients After Rewriting

Many problems reduce to a ratio of sine functions whose arguments are constant multiples of the variable. If \(a\) and \(b\) are nonzero constants, write each sine in terms of its own argument divided by that argument. For \(x\ne0\) sufficiently near zero,

As \(x\to0\), both \(ax\) and \(bx\) tend to zero. The first sine quotient tends to \(1\), and the reciprocal of the second tends to \(1\), since \(\sin(bx)/(bx)\to1\). Thus the ratio tends to \(a/b\). This result is useful after applying an identity, but its conditions matter: both scale factors must be nonzero, and the quotient is considered near zero where its denominator is defined.

Worked Example: A Squared-Sine Quotient

Worked Example: Combining a Double-Angle Identity with a Cosine Factor

Cancellation and a Common Mistake

Rewriting identities does not make cancellation automatically valid at the point of the limit. For example, in the first worked example the factor \(x\) in \(\sin(2x)/(2x)\) is nonzero when \(x\ne0\). A limit as \(x\to0\) concerns nearby inputs, not necessarily the value at \(x=0\). It is therefore legitimate to use an equality that holds for all sufficiently close nonzero \(x\), even if the original expression is undefined at zero.

A frequent error is to confuse the half-angle identity with \(1-\cos x=2\sin^2 x\). That formula is incorrect: the correct sine argument is \(x/2\). For a concrete check, let \(x=\pi/3\). Then \(1-\cos(\pi/3)=1-\frac12=\frac12\), whereas \(2\sin^2(\pi/3)=2(\frac{\sqrt3}{2})^2=\frac32\). The incorrect identity would also give a wrong limit: applying it to \((1-\cos x)/x^2\) would produce \(2(\sin x/x)^2\), whose limit is \(2\), rather than the correct value \(\frac12\).

A Practical Rewrite Routine

When a trigonometric quotient gives \(0/0\) at a point, use a deliberate sequence rather than trying identities at random.

These rewrites matter because a quotient that initially appears indeterminate can conceal a straightforward product of known limiting factors. The identity supplies the algebraic bridge; the standard limits supply the values. A reliable solution shows both, so that its coefficients, arguments, and conditions can all be checked.