Sin 23 Degrees: Value, Unit Circle And How To Find It

What Does Sin 23 Degrees Mean?

In a right triangle, the sine of an angle is the opposite side divided by the hypotenuse. So $\sin 23^\circ \approx 0.3907$ says that the side facing a $23^\circ$ angle is about $39.07$ as long as the hypotenuse.

On the unit circle, a circle of radius $1$ centred at the origin, sine is the y-coordinate of the point where the angle's terminal side meets the circle. At $23^\circ$ that point is approximately $(0.9205, 0.3907)$.

A quadrant is one of the four regions the axes cut the plane into, numbered I to IV counterclockwise from the positive x-axis. The angle $23^\circ$ lies in Quadrant I, where both coordinates are positive, and it is its own reference angle (the acute angle the terminal side makes with the x-axis).

A radian measures an angle by the arc it cuts off on a unit circle, so multiply by $\dfrac{\pi}{180}$ to change degrees to radians.

$$23^\circ \times \frac{\pi}{180^\circ} = \frac{23\pi}{180} \approx 0.4014 \text{ rad}$$

Standard-Angle Sine Reference Table

Sin 23 degrees falls between $\sin 0^\circ = 0$ and $\sin 30^\circ = \dfrac{1}{2}$, a little under four-fifths of the way to $\dfrac{1}{2}$. The standard angles have exact forms; $23^\circ$ does not.

The neighbours of $23^\circ$ show sine climbing about $0.016$ per degree here.

Compare the radian column with the sine column. At $23^\circ$ the radian measure $0.4014$ is already about $0.011$ bigger than the sine, which matters for the small-angle shortcut later.

Where Does Sin 23 Degrees Show Up, And Why Does It Matter?

Sin 23 degrees gives the vertical share of anything tilted at $23^\circ$. A $4$ m playground slide set at $23^\circ$ to the ground rises $4\sin 23^\circ \approx 1.56$ m at its top.

The angle also sits close to Earth's axial tilt, about $23.44^\circ$. Seasonal sun-angle calculations use the sine of that tilt, $\sin 23.44^\circ \approx 0.3978$, so $\sin 23^\circ \approx 0.3907$ is a handy rough stand-in, about $2$ low.

• Ramps and slides: height reached equals length times $\sin 23^\circ$, or $0.39$ of the slope length.

• Angles of elevation: a kite string at $23^\circ$ above the ground puts the kite at $0.39$ of the string's length above the hand, a standard angle of elevation setup.

• Testing approximations: $23^\circ$ is just beyond where the small-angle rule $\sin\theta \approx \theta$ stays within $1$, so it shows that rule's limit clearly.

What Is The Value Of Sin 23 Degrees?

The value of sin 23 degrees is $0.39073113$ to eight decimal places, which rounds to $0.3907$. To three decimals it is $0.391$.

The digits never end or repeat, because $\sin 23^\circ$ is irrational. The OEIS entry A019832 lists the expansion as $0.3907311284892737\ldots$, so any decimal you write is a rounded value of the exact number $\sin 23^\circ$.

Is There An Exact Value Of Sin 23 Degrees?

There is no exact value of sin 23 degrees made from whole numbers and real square roots, the way $\sin 30^\circ = \dfrac{1}{2}$ or $\sin 45^\circ = \dfrac{\sqrt{2}}{2}$ are. A whole-number angle in degrees has sine and cosine expressible in real square roots only when it is a multiple of $3^\circ$, and $23$ is not.

That result comes from the theory of constructible angles, summarised in the list of trigonometric constants expressed in real radicals. The OEIS entry adds that $\sin 23^\circ$ is an algebraic number of degree $48$: it solves a polynomial equation with whole-number coefficients, but the smallest such polynomial has degree $48$.

This was the part of non-special angles that took me longest to accept, because $23^\circ$ looks no stranger than $24^\circ$. Yet $\sin 24^\circ$ has a (long) square-root form and $\sin 23^\circ$ does not, purely because $24$ is a multiple of $3$.

What you can write exactly is a link to other angles. With the sum and difference formulas, $23^\circ = 25^\circ - 2^\circ$ gives an identity, not a simpler number.

$$\sin 23^\circ = \sin 25^\circ \cos 2^\circ - \cos 25^\circ \sin 2^\circ$$

Using sin 25 degrees $\approx 0.42261826$, $\cos 25^\circ \approx 0.90630779$, cos 2 degrees $\approx 0.99939083$ and $\sin 2^\circ \approx 0.03489950$, the right side is $0.39073113$. Sin 23 degrees is therefore a calculator and approximation skill, not a value to memorise.

Why Is Sin 23 Degrees Equal To Cos 67 Degrees?

Sin 23 degrees equals $\cos 67^\circ$ because $23^\circ$ and $67^\circ$ add to $90^\circ$. In a right triangle with those two acute angles, the side opposite $23^\circ$ is the side adjacent to $67^\circ$.

$$\sin 23^\circ = \cos(90^\circ - 23^\circ) = \cos 67^\circ \approx 0.3907$$

This is the co-function rule from the cofunction identities. It also gives the partner pair: $\cos 23^\circ = \sin 67^\circ \approx 0.9205$.

How Do You Find Sin 23?

Find sin 23 degrees with a calculator in degree mode, then confirm it with a series or an approximation you can check by hand. The methods below go from quickest to most revealing.

Method 1: Calculator In Degree Mode

With DEG showing on the screen, enter $\sin(23)$. The display reads $0.3907311285\ldots$, which rounds to $0.3907$.

In radian mode the same keys give $\sin(23 \text{ rad}) \approx -0.8462$. A negative result is the warning sign, since a Quadrant I angle always has a positive sine.

Method 2: Read It From The Unit Circle

Turn a radius $23^\circ$ counterclockwise from the positive x-axis. Its endpoint's height, the y-coordinate, is $\sin 23^\circ \approx 0.3907$.

Method 3: Use The Co-Function Value

If a table gives $\cos 67^\circ \approx 0.3907$, that is already $\sin 23^\circ$.

Method 4: Sum The Sine Series

The Maclaurin series for sine, listed in the NIST Digital Library of Mathematical Functions, needs the angle in radians, $x = \dfrac{23\pi}{180} \approx 0.40142573$.

$$\sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \dots$$

Two terms give $0.3906$, three give $0.3907$ correct to six places. The terms shrink fast because $x$ is well under $1$.

Method 5: Step Back From $30^\circ$ With A Tangent Line

Near a known angle, sine behaves almost like a straight line. From $\sin 30^\circ = \dfrac{1}{2}$, step back $h = 7^\circ \approx 0.122173$ rad, using the slope $\cos 30^\circ = \dfrac{\sqrt{3}}{2}$.

$$\sin(30^\circ - h) \approx \frac{1}{2} - \frac{\sqrt{3}}{2} \times 0.122173 \approx 0.39420$$

That is high by $0.0035$, because $7^\circ$ is a fairly long step. Adding the curvature correction $-\dfrac{1}{2} \cdot \dfrac{h^2}{2}$ brings it to $0.39046$, within $0.0003$.

Method 6: Interpolate In A Table

From sin 20 degrees $\approx 0.3420$ and $\sin 25^\circ \approx 0.4226$, go three-fifths of the way up.

$$0.3420 + \frac{3}{5}(0.4226 - 0.3420) \approx 0.3904$$

That is $0.0004$ low, because the sine curve bends downward between the two table entries.

Sin 23 In Terms Of Other Functions

Every other ratio of $23^\circ$ can rebuild the sine. In Quadrant I all the signs are positive, so the $\pm$ that some pages print resolves to $+$.

Which Angles Have The Same Sine As 23 Degrees?

Sine is the y-coordinate, so any angle whose point sits at height $0.3907$ shares the value. Sine repeats every $360^\circ$ and is unchanged by $\theta \to 180^\circ - \theta$.

• $\sin 157^\circ = 0.3907$, because $157^\circ = 180^\circ - 23^\circ$ mirrors the point into Quadrant II.

• $\sin 383^\circ = 0.3907$, one full turn later; such angles are coterminal angles.

• $\sin(-23^\circ) = -0.3907$, $\sin 203^\circ = -0.3907$ and $\sin 337^\circ = -0.3907$, the same height below the x-axis.

Examples Of Sin 23 Degrees

Example 1

A right triangle has a hypotenuse of $12$ cm and an angle of $23^\circ$. Find the side opposite that angle.

$$\sin 23^\circ = \frac{\text{opposite}}{12}$$

$$\text{opposite} = 12 \times \sin 23^\circ$$

$$= 12 \times 0.390731 \approx 4.69 \text{ cm}$$

Final answer: about $4.69$ cm

Example 2

If $\sin 23^\circ = m$, find $\cos 113^\circ$ in terms of $m$.

The tempting route spots that $113^\circ = 90^\circ + 23^\circ$ and switches cosine to sine, writing $\cos 113^\circ = \sin 23^\circ = m$.

That answer is positive, since $m \approx 0.39$. But $113^\circ$ lies in Quadrant II, where the x-coordinate, and so the cosine, is negative.

The fix is to expand with the angle-sum formula, which carries the sign automatically.

$$\cos(90^\circ + 23^\circ) = \cos 90^\circ \cos 23^\circ - \sin 90^\circ \sin 23^\circ$$

$$= 0 - 1 \times m = -m$$

Final answer: $\cos 113^\circ = -m \approx -0.3907$

Example 3

Show that $\sin 23^\circ + \sin 37^\circ = \cos 7^\circ$.

Use the sin A + sin B formula, $\sin A + \sin B = 2\sin\dfrac{A + B}{2}\cos\dfrac{A - B}{2}$.

$$\sin 23^\circ + \sin 37^\circ = 2\sin 30^\circ \cos(-7^\circ)$$

$$= 2 \times \frac{1}{2} \times \cos 7^\circ = \cos 7^\circ$$

Check: $0.3907 + 0.6018 = 0.9925$ and $\cos 7^\circ \approx 0.9925$.

Example 4

Evaluate $\dfrac{\sin 23^\circ}{\cos 67^\circ} + \dfrac{\sin 157^\circ}{\sin 23^\circ}$.

$$\frac{\sin 23^\circ}{\cos 67^\circ} = 1 \quad \text{because } \cos 67^\circ = \sin 23^\circ$$

$$\frac{\sin 157^\circ}{\sin 23^\circ} = 1 \quad \text{because } \sin 157^\circ = \sin(180^\circ - 23^\circ) = \sin 23^\circ$$

$$1 + 1 = 2$$

Final answer: $2$, exactly, with no decimals needed.

Example 5

A $4$ m slide makes a $23^\circ$ angle with the ground. How high is the top of the slide?

$$\text{height} = 4 \times \sin 23^\circ$$

$$= 4 \times 0.390731 \approx 1.56 \text{ m}$$

Final answer: about $1.56$ m

Example 6

Estimate $\sin 23^\circ$ without a calculator in two ways, and state the error of each.

Convert first: $x = \dfrac{23\pi}{180} \approx 0.401426$.

$$\text{small-angle rule: } \sin x \approx x = 0.4014$$

$$\text{two series terms: } x - \frac{x^3}{6} \approx 0.401426 - 0.010781 = 0.3906$$

The true value is $0.3907$. The small-angle rule is $0.0107$ (about $2.7$) too high; two terms are only $0.00009$ too low.

Final answer: about $0.3906$ from two terms, and the one-term rule is not accurate enough at $23^\circ$.

What Are The Most Common Mistakes With Sin 23 Degrees?

The value itself is easy to get; the slips happen in the calculator, in the shortcut, or in the co-function swap. Each one has a quick check.

Mistake 1: Calculator Left In Radian Mode

Where it slips in: The rusher who types $\sin(23)$ without looking at the mode.

Don't do this: Accept $\sin 23 \approx -0.8462$. That is the sine of $23$ radians, more than three and a half full turns.

The correct way: Use degree mode, or type $\sin\left(\dfrac{23\pi}{180}\right)$ in radian mode. Both give $0.3907$, and the positive sign is your Quadrant I check.

Students who estimate first ("a bit under $\dfrac{1}{2}$, since $23^\circ$ is a bit under $30^\circ$") catch a $-0.85$ immediately. That one-second estimate is the habit worth building.

Mistake 2: Using $\sin\theta \approx \theta$ At $23^\circ$

Where it slips in: Physics and engineering shortcuts that work beautifully for small angles.

Don't do this: Write $\sin 23^\circ \approx 0.4014$ and treat it as accurate.

The correct way: Keep the rule for angles up to about $14^\circ$, where it stays within $1$. At $23^\circ$ it is $2.7$ high, so add the next term, $-\dfrac{x^3}{6}$, or use a calculator.

The memorizer who learned "$\sin\theta \approx \theta$" without its range is the one this catches. The rule is an approximation with a limit, not an identity.

Mistake 3: Dropping The $90^\circ$ In The Co-Function Rule

Where it slips in: Rewriting sine as cosine under time pressure.

Don't do this: Write $\sin 23^\circ = \cos 23^\circ$, which would make the two values $0.3907$ and $0.9205$ equal.

The correct way: Switch the function and subtract from $90^\circ$ together: $\sin 23^\circ = \cos 67^\circ$.

The first instinct is to swap the function and keep the angle, because the swap feels like the whole rule. Saying the full sentence, "sine of an angle is cosine of its complement", while writing it fixes this for most students within a few problems.

Where Can You Get Help With Trigonometry Values Like Sin 23 Degrees?

Values like sin 23 degrees test judgement more than memory: when a shortcut is good enough, and how to spot a calculator answer that cannot be right. A trigonometry tutor or a high school math tutor can build those checks with you on real problems.

For steady practice across the syllabus, math classes online and math tutoring cover identities, the unit circle and approximation in order.

Key Takeaways

• Sin 23 degrees is approximately $0.3907$ ($0.39073113$), positive because $23^\circ$ is in Quadrant I.

• In radians the angle is $\dfrac{23\pi}{180} \approx 0.4014$.

• It has no simple surd form, because $23$ is not a multiple of $3$, but it equals $\cos 67^\circ$ exactly.

• Two or three terms of the sine series, or a tangent line from $30^\circ$ with a correction, reach the value to three or more decimals.

• The common errors are radian mode ($-0.8462$), trusting $\sin\theta \approx \theta$ past about $14^\circ$, and writing $\cos 23^\circ$ instead of $\cos 67^\circ$.

Practice These Before Moving On

1. Evaluate $\dfrac{4\sin 23^\circ}{\cos 67^\circ}$.

2. Find $\sin 157^\circ$ to four decimal places.

3. Find $\sin(-23^\circ)$ and $\sin 203^\circ$.

4. Estimate $\sin 23^\circ$ with $\sin\theta \approx \theta$, then give the percentage error.

5. If $\sin 23^\circ = m$, write $\sin 67^\circ$ in terms of $m$.

Answers:

1. $4$, since $\cos 67^\circ = \sin 23^\circ$

2. $0.3907$, because $157^\circ = 180^\circ - 23^\circ$

3. Both are $-0.3907$

4. $0.4014$, which is about $2.7$ above $0.3907$

5. $\sin 67^\circ = \cos 23^\circ = \sqrt{1 - m^2}$, taking the positive root because $23^\circ$ is in Quadrant I

If question 5 tempted you to write $m$, return to Mistake 3: the complement of $67^\circ$ is $23^\circ$, so the function must switch to cosine.

Practice these to solidify your understanding, and if you would like a teacher to check your working live, book a free demo class.

Read More

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