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Small angle approximations

Use small angle approximations for sine, cosine and tangent in radians. Check scaled angles, percentage error, approximate equations and cancellation.

Before you startRadians, trigonometric values, algebraic fractions and binomial expansion

01 / Small angles make simpler formulas useful

An approximation needs a unit, a scale and an accuracy check.

sin θ ≈ θ
tan θ ≈ θ
cos θ ≈ 1 − θ²/2

θ must be small in magnitude and measured in radians.

These are approximations near zero, not identities for arbitrary angles. They are useful because a trigonometric expression becomes a polynomial. The same formulas apply for small negative angles.

Use the model to compare the exact function and its approximation. Increase the multiplier to see why “θ is small” is not enough if the function actually contains 3θ.

Compare the approximation with the functionExplore

Blue: exact function. Dashed gold: approximation. Green: the gap at the selected angle.

Actual angle 0.200000 rad. Exact ≈0.198669; approximation =0.200000.

Absolute error ≈0.001331; percentage error ≈0.669791%.

Judge accuracy using the actual angle and the error your calculation can tolerate.

FunctionAngle multiplierθ in radians 0.20

Watch the sine curve approach its linear approximation

Pause, replay or seek freely. The notes explain the same idea and stay in view.

02 / Why radians are essential

A small arc, its chord and its tangent length are close.

On a unit circle and for 0<θ<π/2, geometry gives sin θ<θ<tan θ. The middle quantity is an arc length because the radius is 1 and the angle is in radians. Near zero these lengths become close, which motivates sin θ≈θ and tan θ≈θ.

Cosine is near 1, but keeping its first correction gives cos θ≈1−θ²/2. This correction is essential in expressions such as 1−cos θ.

Approximate sin 3°Worked example

Convert 3° to π/60 radians

π/60≈0.052360.

sin 3°≈π/60

The exact value is about 0.052336.

Do not use the number 3 as the approximation

Sine cannot equal 3; the radian conversion supplies the correct scale.

03 / Substitute a small radian angle

Preserve the squared term in cosine.

Use θ=0.2 radiansWorked example

sin(0.2)≈0.2

Exact value ≈0.198669.

tan(0.2)≈0.2

Exact value ≈0.202710.

cos(0.2)≈1−0.2²/2=0.98

Exact value ≈0.980067.

For θ=−0.2, the sine and tangent approximations become −0.2; cosine’s approximation stays 0.98 because the square is unchanged. This agrees with sine/tangent being odd and cosine being even.

04 / The whole argument must be small

Replace θ by the entire angle, including its coefficient.

sin(4x) ≈ 4x
tan(3x) ≈ 3x
cos(3x) ≈ 1 − 9x²/2

The squared coefficient matters: (3x)²=9x². The condition for the last formula is that |3x| is small. An x value of 0.1 makes 10x=1 radian, which is not especially close to zero; using sin(10x)≈10x there gives a substantial error.

A shifted argument x+c is small only when x+c is near zero. A small x by itself does not justify approximating sin(x+π/3) by x+π/3.

05 / Simplify a quotient after substitution

Retain enough terms for the numerator and denominator.

Approximate sin(3x)/tan(2x) near x=0Worked example

sin(3x)≈3x; tan(2x)≈2x

Both scaled angles must be small.

The ratio is approximately 3x/(2x)=3/2

This cancellation is for x≠0.

The original ratio is undefined at x=0

Its limit as x approaches zero is 3/2; a limit is not the original function value.

Approximate (cos(4x)−1)/(x sin(2x))Worked example

cos(4x)−1≈−8x²

Keep the cosine correction and its minus sign.

x sin(2x)≈2x²

For nonzero x near zero.

The ratio approaches −4

Both numerator and denominator vanish, but their leading nonzero terms give a negative ratio.

06 / Do not let cancellation erase the information

A zero result can mean the approximation was too coarse.

Replacing cos x by 1 in 1−cos x would give zero and lose its main variation. The quadratic approximation instead gives 1−cos x≈x²/2.

Replacing sin x by x in x−sin x also gives zero. That does not show that x−sin x is exactly zero, or that a small circular segment has no area. To estimate this difference meaningfully, you need a more accurate sine approximation than the basic formulas in this lesson.

Simplify symbolically first; keep the first nonzero terms needed by the expression.

07 / Combine trig and binomial approximations

Keep terms consistently to the requested power.

Approximate √(1+sin x) through x²Worked example

sin x=x plus terms starting at order x³

So replacing sin x by x preserves the constant, linear and quadratic terms here.

Use (1+t)½≈1+t/2−t²/8

The binomial expansion is valid near t=0.

√(1+sin x)≈1+x/2−x²/8

Omitted terms start at cubic order.

Approximate 1/cos x through x²Worked example

cos x≈1−x²/2

Write the reciprocal as (1−x²/2)⁻¹.

Use (1−t)⁻¹≈1+t for small t

Here t=x²/2.

1/cos x≈1+x²/2

Terms beyond quadratic are omitted.

08 / Solve an approximate equation, then check its roots

An approximation can create a root far outside its useful range.

Find a small positive solution of sin x=0.1cos(2x). This is an approximate-equation task: the simplified roots are estimates for the original equation.

Keep the plausible small rootWorked example

x≈0.1(1−2x²)

Use sin x≈x and cos(2x)≈1−2x².

2x²+10x−1≈0

Multiply by 10 after rearranging.

x≈(−5±3√3)/2

The roots are about 0.098076 and −5.098076.

Keep x≈0.098076

The other value is far from zero and invalidates both small-angle substitutions.

Check the actual argument 2x≈0.196152

The numerical root of the original equation near zero is about 0.098234, so the estimate differs by about 0.161%.

At the approximate root, sin x−0.1cos(2x) is about −0.000163, not exactly zero. State an approximation honestly and assess whether its error is acceptable.

09 / Measure the error against the exact value

There is no single angle threshold that guarantees every task’s accuracy.

Percentage error is undefined when the exact value is zero.

At θ=0.2, sin θ≈θ has about 0.670% error, tan θ≈θ about 1.337%, and cos θ≈1−θ²/2 about 0.00679%. The same angle produces different accuracy for different functions.

The model intentionally includes angles large enough to show deterioration. Its display is evidence to assess, not a claim that every setting is a good approximation. For sine/tangent at zero, both values equal zero and the absolute error is zero, while the relative-error formula divides by zero.

10 / Small-angle geometry needs the same care

Arc and chord lengths become close, but they remain different.

A chord subtending angle θ in a radius r circle has length 2r sin(θ/2). For a small radian θ, this is approximately 2r(θ/2)=rθ, the arc length.

Radius 20 cm and angle 0.1 radianWorked example

Arc length =20(0.1)=2 cm

This arc formula is exact.

Chord =40sin(0.05)≈1.999167 cm

Its small-angle estimate is 2 cm.

A student uses sin x≈x to conclude x−sin x=0 exactly. Explain the error.

Hint

An approximation does not become an identity after subtraction.

Worked solution

The leading linear terms cancel; the remaining smaller terms were discarded. A finer approximation is needed to estimate the difference, and it is not identically zero.

12 / Recap

Use an approximation with its conditions and its error in view.

Use radians and the complete trigonometric argument.
Keep the quadratic correction in cosine when needed.
Do not confuse a limit with an undefined function value at zero.
Cancellation may require more accurate terms.
Check approximate roots in the original equation and reject distant artefacts.
Compare with exact values when judging accuracy.

Section 1 of 12 · Small angles make simpler formulas useful