Sin 40 Degrees : Value 0.6428 Explained

What Does Sin 40 Degrees Mean?

Sine is one of the three trigonometric ratios, in a right triangle, the sine of an angle is the side opposite it divided by the hypotenuse. So $\sin 40^\circ$ asks: in a right triangle with a $40^\circ$ angle, what fraction of the hypotenuse is the opposite side?

On the unit circle, sine is the $y$-coordinate of the point at $40^\circ$, which is about $0.6428$. A "special angle" is one like $30^\circ$, $45^\circ$, or $60^\circ$ whose sine comes out as a tidy fraction or radical; $40^\circ$ is not one of them.

That is the key honesty here: $\sin 40^\circ$ has no clean closed form. Its value is an irrational decimal that a calculator reports, and treating it as a memorisation target the way you would $\sin 30^\circ$ is a mistake.

Where Does Sin 40 Degrees Show Up?

A ramp, roof, or hillside pitched at $40^\circ$ has a vertical rise proportional to $\sin 40^\circ$, so a $10$ m slope climbs $10 \times 0.6428 \approx 6.43$ m. Surveyors and builders reach for $\sin 40^\circ$ constantly, and none of them need an exact fraction, a decimal to a few places is what the job uses.

The angle also appears in angle of elevation problems, where a line of sight tilted $40^\circ$ above the horizontal fixes a height-to-distance relationship. This is the honest home of a non-special angle: applied work, where a calculator reading is the answer.

Standard-Angle Reference Table

$40^\circ$ sits between two standard angles, $30^\circ$ and $45^\circ$, and its sine sits between theirs. Here are sine values for the nearby angles, exact where one exists and decimal otherwise.

Read down the decimal column and sine climbs steadily from $0.5$ toward $0.87$. The value $0.6428$ for $40^\circ$ is exactly where you would expect it, just above the $30^\circ$ mark, just below $45^\circ$.

How Do You Find The Value Of Sin 40 Degrees?

Because $40^\circ$ is not a special angle, the practical answer comes from a calculator, with two supporting checks.

Method 1: Calculator in degree mode.

Set the calculator to degree mode and enter $\sin(40)$. It returns

$$\sin 40^\circ = 0.642787\ldots \approx 0.6428$$

Degree mode matters: in radian mode the same keystrokes give $\sin(40\ \text{rad}) \approx 0.745$, a completely different number.

Method 2: The cofunction check.

Sine and cosine are cofunctions: $\sin\theta = \cos(90^\circ - \theta)$. So

$$\sin 40^\circ = \cos(90^\circ - 40^\circ) = \cos 50^\circ$$

Entering $\cos 50^\circ$ returns the same $0.6428$, which confirms the reading. This is also how trigonometric tables pair angles that add to $90^\circ$.

Method 3: Bracket it against known angles.

Since $30^\circ < 40^\circ < 45^\circ$, the value must sit between $\sin 30^\circ = 0.5$ and $\sin 45^\circ \approx 0.7071$. A reading of $0.6428$ passes that sanity range; a reading like $0.34$ would not, and would flag a mode error.

A note on the "exact value": there is a nested-radical expression for $\sin 40^\circ$ tied to solving a cubic, but it is not exam material and offers no practical advantage over the decimal. For every real purpose, $0.6428$ is the value.

Examples Of Sin 40 Degrees

Example 1

Evaluate $10\sin 40^\circ$, rounded to two decimals.

$$10\sin 40^\circ = 10 \times 0.6428 = 6.428 \approx 6.43$$

Example 2

A student wants $\sin 40^\circ$ and reasons "$40^\circ$ is close to $45^\circ$, so I will just use $\dfrac{\sqrt{2}}{2}$."

Wrong attempt. Rounding the angle up to the nearest special angle and using its exact value.

That breaks on precision: $\dfrac{\sqrt{2}}{2} \approx 0.7071$, but $\sin 40^\circ \approx 0.6428$. The gap of about $0.064$ is far too large to ignore, a $6.4$% error on a real measurement.

Correct. Keep the angle at $40^\circ$ and read the calculator: $\sin 40^\circ \approx 0.6428$. Special-angle values substitute only for the special angles themselves.

Example 3

A ladder leans against a wall at $40^\circ$ to the ground. If the ladder is $5$ m long, how high up the wall does it reach?

$$\text{height} = 5 \times \sin 40^\circ = 5 \times 0.6428 \approx 3.21 \text{ m}$$

Example 4

Verify $\sin 40^\circ = \cos 50^\circ$ numerically.

$$\sin 40^\circ \approx 0.6428, \qquad \cos 50^\circ \approx 0.6428$$

They match, as the cofunction identity $\sin\theta = \cos(90^\circ - \theta)$ requires.

Example 5

Estimate $\sin 40^\circ$ using the small-angle rule $\sin\theta \approx \theta$ (in radians), then say why the estimate is poor.

In radians $40^\circ = \dfrac{2\pi}{9} \approx 0.698$, so the rule predicts $\sin 40^\circ \approx 0.698$. The true value is $0.6428$, an error of about $8$%.

The small-angle rule is only trustworthy below roughly $10^\circ$ to $15^\circ$; at $40^\circ$ it has drifted well off, which is exactly why $40^\circ$ needs a real evaluation, not an approximation.

Where Students Trip Up On Sin 40 Degrees

Mistake 1: Hunting for an exact fraction that does not exist

Where it slips in: Assuming every angle has a tidy value like the special ones.

Don't do this: Writing $\sin 40^\circ = \dfrac{2}{3}$ or some invented fraction because "there must be one".

The correct way: $40^\circ$ is not a special angle. Its sine is the irrational decimal $0.6428\ldots$; report the decimal to the precision the problem asks for.

Mistake 2: Leaving the calculator in radian mode

Where it slips in: A calculator left in radians returns $\sin(40) \approx 0.745$ for what should be a degree entry.

Don't do this: Trusting the screen without checking the angle unit. The memorizer who never checks the mode is the one most surprised by a wrong answer here.

The correct way: Confirm degree mode before entering $\sin(40)$, and bracket the result against $\sin 30^\circ$ and $\sin 45^\circ$ to catch a mode slip.

Mistake 3: Rounding the angle to a special value

Where it slips in: Swapping $40^\circ$ for $45^\circ$ to reuse a memorised radical.

Don't do this: Using $\dfrac{\sqrt{2}}{2}$ for $\sin 40^\circ$.

The correct way: The angles are $5^\circ$ apart, which shifts the sine by about $0.064$, a real error. Evaluate the actual angle.

Key Takeaways

• Sin 40 degrees is approximately $0.6428$, an irrational decimal with no clean fraction or radical.

• Because $40^\circ$ is not a special angle, this is a calculator or approximation skill, not a memorisation target.

• Set the calculator to degree mode, and check against $\sin 40^\circ = \cos 50^\circ$ and the range $0.5$ to $0.7071$.

• The small-angle rule fails here: at $40^\circ$ it is off by about $8$%.

To get comfortable with when to compute versus when to recall, work alongside Bhanzu's trigonometry tutor or a high school math tutor.

Practice These Before Moving On

1. A slope of length $12$ m is pitched at $40^\circ$. Use $\sin 40^\circ$ to find its vertical rise.

2. Confirm $\sin 40^\circ = \cos 50^\circ$ on your calculator, then find $\sin 50^\circ$.

3. Without a calculator, state two special-angle sines that $\sin 40^\circ$ must lie between, and say which it is closer to.

Want a live Bhanzu trainer to show when an angle needs a calculator and when it does not? Book a free demo class, online with an expert, anywhere.

For a broader treatment of how sine is tabulated, see the reference on trigonometric functions.

Read More

• Sin 47 degrees, another non-special angle read the same way.

• Sin 15 degrees, an angle that does have an exact form.

• Sin 75 degrees, the cofunction partner of sin 15°.

• Sin cos tan, the three ratios and how they relate.

• Radians to degrees, the conversion behind 40° = 2π/9.