Tan 23 Degrees: Value, Unit Circle, Methods And Examples

What Does Tan 23 Degrees Mean?

In a right triangle, the tangent function of an angle is the opposite side divided by the adjacent side. So $\tan 23^\circ$ asks how many units a line rises for every 1 unit it runs when it leans $23^\circ$ above the horizontal.

On the unit circle (radius 1, centred at the origin), the same angle lands on the point $(\cos 23^\circ, \sin 23^\circ) \approx (0.9205, 0.3907)$. The tangent is the $y$-coordinate divided by the $x$-coordinate:

$$\tan 23^\circ = \frac{\sin 23^\circ}{\cos 23^\circ} \approx \frac{0.3907}{0.9205} \approx 0.4245$$

A quadrant is one of the four regions the axes cut the plane into, numbered I to IV counterclockwise from the upper right. A quadrant I angle has positive sine, cosine and tangent, and $23^\circ$ is its own reference angle (the acute angle to the $x$-axis), so no sign adjustment is needed.

In radians, $23^\circ = \frac{23\pi}{180} \approx 0.4014$, because a full turn of $360^\circ$ equals $2\pi$. The guide to what a radian is shows why that conversion factor is $\frac{\pi}{180}$.

Standard-Angle Tangent Reference Table

The angle $23^\circ$ sits just past $22.5^\circ$, one of the few angles near it with a tidy exact tangent. Here is where it falls among the standard first-quadrant angles.

Tangent grows faster and faster as the angle climbs toward $90^\circ$. Between $0^\circ$ and $30^\circ$, though, it rises almost in a straight line, which is why $\tan 23^\circ$ lands only a little above $\tan 22.5^\circ$.

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

Road and ramp builders describe slopes as a percent grade, which is $100 \times \tan\theta$ (Grade (slope)). A $23^\circ$ incline is therefore a $42.4$ grade, far steeper than any public road.

In physics, a block that just starts to slide on a ramp tilted at angle $\theta$ has a static friction coefficient $\mu_s = \tan\theta$ (Friction). A crate that slips at exactly $23^\circ$ tells you $\mu_s \approx 0.42$, a typical value for wood on wood.

Why Is There No Simple Exact Value For Tan 23 Degrees?

An angle with a whole number of degrees has trig values made only of real square roots exactly when the number of degrees is a multiple of 3 (Exact trigonometric values). Since 23 is not a multiple of 3, $\tan 23^\circ$ cannot be written with square roots alone.

The number is still algebraic: it is a root of a polynomial with integer coefficients of degree 24 (OEIS A019921). Any closed form would need cube roots of complex numbers, which is no help for a calculation by hand.

This one took me a while to accept as a student, because every table I had seen was full of neat surds. The neat ones are the exceptions, and $23^\circ$ is the ordinary case.

Can Tan 23 Degrees Be Written As A Fraction?

No exact fraction exists, because $\tan 23^\circ$ is irrational. What you can write is a fraction that approximates it.

• Two decimals: $\frac{42}{100} = \frac{21}{50} = 0.42$, off by about $0.0045$.

• Four decimals: $\frac{4245}{10000} = \frac{849}{2000} = 0.4245$, off by about $0.00003$.

• Neighbour surd: $\sqrt{2} - 1 \approx 0.4142$ is the exact $\tan 22.5^\circ$, close but not equal.

When a question says "give the value in fraction form" for $23^\circ$, the honest answer is the decimal, or the symbol $\tan 23^\circ$ itself.

How Do You Find The Value Of Tan 23 Degrees?

Six routes give the same $0.4245$. The first two are what you would use on homework; the rest show where the number comes from.

Method 1: Calculator In Degree Mode

Set the calculator to degrees (DEG on the screen), then press tan and enter 23.

$$\tan 23^\circ = 0.4244748162\ldots$$

Method 2: Sine Over Cosine On The Unit Circle

Read the unit-circle coordinates for $23^\circ$, then divide.

$$\sin 23^\circ \approx 0.3907311, \quad \cos 23^\circ \approx 0.9205049$$

$$\tan 23^\circ = \frac{0.3907311}{0.9205049} \approx 0.4244748$$

Method 3: Co-Function With Cot 67 Degrees

The cofunction identities say $\tan\theta = \cot(90^\circ - \theta)$. With $90^\circ - 23^\circ = 67^\circ$:

$$\tan 23^\circ = \cot 67^\circ = \frac{1}{\tan 67^\circ} = \frac{1}{2.3558524} \approx 0.4244748$$

So a table that lists $\tan 67^\circ$ gives you $\tan 23^\circ$ for free, and that is why "cot 67 degrees" returns the same $0.4245$.

Method 4: Taylor Series In Radians

Calculators use fast routines related to the series $\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \cdots$, valid only when $x$ is in radians. With $x = 0.4014257$:

Four terms already give $0.4245$ to four decimal places. The table also shows that the small-angle shortcut $\tan x \approx x$ is off by more than 5% at $23^\circ$, so it does not work here (it is reliable only for angles under about $10^\circ$).

Method 5: Linear Step From Tan 22.5 Degrees

Start from the exact $\tan 22.5^\circ = \sqrt{2} - 1 \approx 0.4142136$. The slope of the tangent curve there is $\sec^2 22.5^\circ = 1 + (\sqrt{2} - 1)^2 \approx 1.1715729$, and the extra half degree is $\frac{\pi}{360} \approx 0.0087266$ radians.

$$\tan 23^\circ \approx 0.4142136 + 1.1715729 \times 0.0087266 \approx 0.4244375$$

The error is about $0.000037$, so this hand method is good to four decimal places after rounding. (Actually, it slightly undershoots, because the tangent curve bends upward.)

Method 6: Interpolating A Trig Table

A printed table lists $\tan 22^\circ \approx 0.4040$ and $\tan 24^\circ \approx 0.4452$. Their average is $0.4246$, within $0.0002$ of the true value, which is how people found values before calculators.

How Can Tan 23 Degrees Be Written Using Other Trig Functions?

Every one of these equals $0.4244748$, and the positive square root applies throughout because $23^\circ$ is in quadrant I.

The secant row is the Pythagorean identity $1 + \tan^2\theta = \sec^2\theta$ rearranged.

Why Does My Calculator Say Tan 23 Is 1.588?

Because it is in radian mode, so it computed the tangent of 23 radians, not 23 degrees. Type "tan 23" into a search bar and you see the same $1.5882$ for the same reason.

Twenty-three radians is about $1317.8^\circ$, which is $3.66$ full turns. The fix is to switch to DEG, or to convert first: $\tan\left(\frac{23\pi}{180}\right) = 0.4245$.

How Does Tan 23 Degrees Compare With Nearby Angles?

Tangent changes by roughly $0.02$ per degree in this stretch, so neighbours give a quick plausibility check. The tan 20 degrees page covers the start of this run in more depth.

The gaps grow slowly, which is the upward bend of the curve showing up in the numbers. By $30^\circ$ the value reaches $0.5774$, the exact $\frac{\sqrt{3}}{3}$ on the tan 30 degrees page.

Examples Of Tan 23 Degrees

Example 1

A ramp rises at $23^\circ$ over a horizontal run of $5$ m. How high does it climb?

$$\text{height} = 5 \times \tan 23^\circ$$

$$\text{height} = 5 \times 0.4244748 \approx 2.12 \text{ m}$$

Example 2

Find $\tan 23^\circ$ from $\tan 20^\circ \approx 0.3639702$ and $\tan 3^\circ \approx 0.0524078$.

Wrong attempt. Add the two tangents: $0.3639702 + 0.0524078 = 0.4163780$.

That cannot be right, because the true value is $0.4244748$ and the sum falls short by about $0.0081$. Tangent is not additive.

Correct. Use the sum and difference identities:

$$\tan(20^\circ + 3^\circ) = \frac{\tan 20^\circ + \tan 3^\circ}{1 - \tan 20^\circ \times \tan 3^\circ}$$

$$\tan 23^\circ = \frac{0.4163780}{1 - 0.0190747} = \frac{0.4163780}{0.9809253} \approx 0.4244748$$

Example 3

Given that $\tan 23^\circ = k$, write $\tan 67^\circ$, $\tan 157^\circ$, $\tan 203^\circ$ and $\tan(-23^\circ)$ in terms of $k$, without a calculator.

$$\tan 67^\circ = \cot 23^\circ = \frac{1}{k}$$

$$\tan 157^\circ = \tan(180^\circ - 23^\circ) = -k$$

$$\tan 203^\circ = \tan(180^\circ + 23^\circ) = k$$

$$\tan(-23^\circ) = -k$$

Tangent repeats every $180^\circ$ and is negative in quadrants II and IV, which decides each sign.

Example 4

Evaluate $\sec^2 23^\circ - \tan^2 23^\circ$.

From $1 + \tan^2\theta = \sec^2\theta$:

$$\sec^2 23^\circ - \tan^2 23^\circ = 1$$

Numerical check: $1.1801789 - 0.1801789 = 1$.

Example 5

The Sun stands $23^\circ$ above the horizon. How long is the shadow of a $4$ m pole?

The pole is the opposite side and the shadow is the adjacent side, so divide by the tangent:

$$\text{shadow} = \frac{4}{\tan 23^\circ} = \frac{4}{0.4244748} \approx 9.42 \text{ m}$$

The first instinct on shadow problems is to multiply by $\tan 23^\circ$ and get $1.70$ m, a shadow shorter than the pole, which is impossible for a low Sun. The angle of elevation guide walks through more of these set-ups.

Example 6

Use $\tan 23^\circ \approx 0.4244748$ to find $\tan 46^\circ$.

The tan 2x formula gives $\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$:

$$\tan 46^\circ = \frac{2 \times 0.4244748}{1 - 0.1801789}$$

$$\tan 46^\circ = \frac{0.8489496}{0.8198211} \approx 1.0355$$

Final answer: $\tan 46^\circ \approx 1.0355$, just above $\tan 45^\circ = 1$, as expected.

Where Do Students Trip Up On Tan 23 Degrees?

Mistake 1: Forcing A Special-Angle Surd

Where it slips in: Students who learned the special-angle table first try to squeeze $23^\circ$ into it.

Don't do this: Writing $\tan 23^\circ = \sqrt{2} - 1$ because $23^\circ$ is "about" $22.5^\circ$.

The correct way: Treat $\sqrt{2} - 1 \approx 0.4142$ as a neighbour, not the answer, and report $\tan 23^\circ \approx 0.4245$.

The memorizer is most at risk here: the table feels like the whole subject, so a gap in it looks like a mistake in the question.

Mistake 2: Reading Tan Inverse As One Over Tan

Where it slips in: The $\tan^{-1}$ key sits right above tan on most calculators, and the superscript looks like a power.

Don't do this: Treating $\tan^{-1} 23$ as $\frac{1}{\tan 23^\circ} = 2.3559$.

The correct way: $\frac{1}{\tan 23^\circ}$ is $\cot 23^\circ \approx 2.3559$, while $\tan^{-1} 23 \approx 87.51^\circ$ is an angle.

That second answer is a principal value, meaning the single output between $-90^\circ$ and $90^\circ$ the calculator returns; the inverse tan page explains the range. Students who can say "reciprocal gives a number, inverse gives an angle" out loud almost never make this swap.

Mistake 3: Rounding Before The Last Step

Where it slips in: Multi-step problems where the rusher writes $\tan 23^\circ = 0.42$ on line one.

Don't do this: Carrying $0.42$ through Example 5, which gives a shadow of $9.52$ m instead of $9.42$ m.

The correct way: Keep $\tan 23^\circ$ (or at least $0.4245$) until the final line, then round once.

Where Can You Get Help With Trigonometry Values?

Values like $\tan 23^\circ$ are a calculator and estimation skill rather than a memory item, and the reasoning behind them (co-functions, series, signs) is where a teacher helps most. A trigonometry tutor can work through unit-circle and identity problems at your pace.

For broader support, a high school math tutor covers trigonometry alongside algebra and geometry, and live math classes online build the same skills in a small group.

Key Takeaways

• Tan 23 degrees is approximately $0.4245$ ($0.4244748162$ to ten places) and is positive because $23^\circ$ lies in quadrant I.

• No simple surd exists, because 23 is not a multiple of 3; the decimal is the working value.

• It equals $\frac{\sin 23^\circ}{\cos 23^\circ}$ and also $\cot 67^\circ$ by the co-function identity.

• The small-angle guess $\tan x \approx x$ is off by over 5% at $23^\circ$, while four series terms reach four decimal places.

• A reading of $1.588$ means the calculator is in radian mode.

Practice These Before Moving On

1. A ramp has a horizontal run of $8$ m and rises at $23^\circ$. Find its height to two decimal places.

2. Given $\tan 23^\circ = k$, write $\tan 337^\circ$ in terms of $k$.

3. Use $\tan(45^\circ - 23^\circ)$ to find $\tan 22^\circ$ to four decimal places.

4. A slope rises at $23^\circ$. What is its percent grade to one decimal place?

Answers: 1. $8 \times 0.4244748 \approx 3.40$ m. 2. $\tan 337^\circ = \tan(-23^\circ) = -k$. 3. $\frac{1 - 0.4244748}{1 + 0.4244748} \approx 0.4040$. 4. $100 \times 0.4244748 \approx 42.4$.

If Question 3 goes wrong, check the sign pattern in the subtraction formula from Example 2 before redoing the arithmetic.

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