Tan 37 Degrees: Value, 3-4-5 Approximation And Examples

What Does Tan 37 Degrees Mean?

Tan 37 degrees is the tangent function evaluated at an angle of $37^\circ$. In a right triangle with a $37^\circ$ angle, it is the opposite side divided by the adjacent side, $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$.

On the unit circle, the circle of radius $1$ centred at the origin, the point at angle $\theta$ is $(\cos\theta, \sin\theta)$, and $\tan\theta = \frac{\sin\theta}{\cos\theta}$. At $37^\circ$ that point is about $(0.7986, 0.6018)$.

Two terms used later:

• Quadrant. The axes cut the plane into four quadrants, numbered counterclockwise from the upper right. An angle between $0^\circ$ and $90^\circ$, like $37^\circ$, lies in Quadrant I, where every trig function is positive.

• Reference angle. The acute angle between an angle's terminal side and the x-axis. For a Quadrant I angle it is the angle itself, so $\tan 217^\circ$, $\tan 143^\circ$ and $\tan 323^\circ$ all borrow the size $0.7536$ from $37^\circ$, with signs set by their quadrants.

What Is Tan 37 Degrees In Radians?

Multiply by $\frac{\pi}{180}$, the step in converting degrees to radians.

$$37^\circ \times \frac{\pi}{180^\circ} = \frac{37\pi}{180} \approx 0.6458 \text{ rad}$$

So $\tan 37^\circ = \tan(0.6458\ldots) \approx 0.7536$. The angle is the same; only its unit changes.

Standard-Angle Tangent Reference Table

Thirty-seven degrees sits between the standard angles $30^\circ$ and $45^\circ$. The table places it among its neighbours; the two rows without an exact form are marked plainly.

Notice how fast tangent climbs. The step from $36^\circ$ to $37^\circ$ adds about $0.027$, while the step from $53^\circ$ to $60^\circ$ adds about $0.405$.

Where Tan 37 Degrees Shows Up: Which Problems Use It?

Tan 37 degrees shows up wherever a slope rises about 3 units for every 4 units across. Three common homes:

• Physics problem sets. Inclines and projectile questions often use $37^\circ$ and $53^\circ$ with $\sin 37^\circ \approx \frac{3}{5}$, $\cos 37^\circ \approx \frac{4}{5}$ and $\tan 37^\circ \approx \frac{3}{4}$, so the arithmetic stays clean.

• Roof pitch and ramps. A 9-in-12 roof pitch has slope $0.75$, an angle of $36.87^\circ$, which builders round to $37^\circ$. The same idea sits behind any slope written as rise over run.

• Grades on roads and trails. A $37^\circ$ incline is a $75.36$ grade, because percent grade is $100 \times \tan\theta$.

Is There An Exact Value Of Tan 37 Degrees?

No simple one. Tan 37 degrees cannot be written using whole numbers, fractions and square roots, because an angle of a whole number of degrees has values of that kind only when the number is a multiple of $3$ (Exact trigonometric values, Wikipedia).

The number is still algebraic: it is a root of a polynomial with integer coefficients, of degree $24$, according to the OEIS entry A019935. A degree-24 root is not something anyone writes out by hand, so in practice the honest answer is the decimal.

Compare $36^\circ$, a multiple of $3$. Its tangent has the exact form $\sqrt{5 - 2\sqrt{5}} \approx 0.7265$, which comes from the regular pentagon.

What Is Tan 37 Degrees In Fraction Form?

Tan 37 degrees has no exact fraction form, because it is irrational. The fractions people use are approximations, and some are far better than $\frac{3}{4}$.

The best small-denominator fractions come from the continued fraction of $0.7535540501\ldots$:

For quick mental work $\frac{3}{4}$ is fine. For anything that has to be accurate to four decimal places, $\frac{3}{4}$ is not enough and the decimal $0.7536$ is the safe answer.

Why Is Tan 37 Taken As 3/4 In The 3-4-5 Triangle?

Tan 37 is taken as $\frac{3}{4}$ because the smaller acute angle of a 3-4-5 right triangle is $36.87^\circ$, close enough to $37^\circ$ to round. The 3-4-5 triangle is the smallest of the Pythagorean triples, with $3^2 + 4^2 = 5^2$.

In that triangle, the angle opposite the side $3$ has:

$$\tan\theta = \frac{3}{4} = 0.75, \quad \theta = \tan^{-1}(0.75) \approx 36.8699^\circ$$

The rounding hides a gap of $0.13^\circ$. Here is what that gap does to each ratio:

The $\frac{3}{4}$ shortcut is a modelling choice, not a fact about $37^\circ$. Use it when a problem says so, and say so when you use it.

How Do You Find Tan 37 Degrees?

You find tan 37 degrees with a calculator in degree mode, which returns $0.7535540\ldots$. The other methods below either check that number or explain where it comes from.

Method 1: Calculator In Degree Mode

Set degree mode, then enter $\tan(37)$.

$$\tan 37^\circ = 0.7535540501\ldots \approx 0.7536$$

Method 2: Sine Over Cosine

Use the unit-circle coordinates at $37^\circ$.

$$\tan 37^\circ = \frac{\sin 37^\circ}{\cos 37^\circ} = \frac{0.6018150}{0.7986355} \approx 0.7536$$

Method 3: The Taylor Series In Radians

The series $\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \frac{62x^9}{2835} + \cdots$ works only with $x$ in radians. With $x = 0.6457718$:

Five terms reach $0.7535$, and the total keeps creeping up toward $0.75355$. Calculators use faster algorithms, but the series shows the value is computed, not looked up.

Method 4: The Co-Function Route

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

$$\tan 37^\circ = \cot 53^\circ = \frac{1}{\tan 53^\circ} = \frac{1}{1.3270448} \approx 0.7536$$

Method 5: Interpolating A Tangent Table

Average the table values on either side, from a trigonometric table.

$$\frac{\tan 36^\circ + \tan 38^\circ}{2} = \frac{0.7265 + 0.7813}{2} \approx 0.7539$$

The estimate is $0.0004$ high, because the tangent curve bends upward and a straight-line average overshoots it.

Method 6: The 3-4-5 Approximation

$$\tan 37^\circ \approx \frac{3}{4} = 0.75$$

This is $0.0036$ low (about $0.47$), fine for estimates and wrong for four-decimal answers.

Why The Small-Angle Shortcut Fails Here

The shortcut $\tan\theta \approx \theta$ (in radians) works well only below about $10^\circ$ to $15^\circ$. At $37^\circ$ it gives $0.6458$, off by $0.1078$, roughly $14$, so it is not a usable method for this angle.

How Is Tan 37 Degrees Written In Terms Of Other Trig Functions?

Tan 37 degrees can be written through each of the other five functions. Each expression below evaluates to about $0.7536$; the square-root forms take the positive root because $37^\circ$ is in Quadrant I.

Is Tan 37 Negative?

No, tan 37 degrees is positive, about $0.7536$. A result of $-0.8408$ means the calculator read $37$ as radians.

An angle of $37$ radians is more than five full turns and lands in a different quadrant, where tangent is negative.

$$\tan(37 \text{ rad}) \approx -0.8408, \qquad \tan(37^\circ) \approx 0.7536$$

When this page was researched, search suggestions for "what is tan 37" displayed "= -0.840771255", the radian answer. Always add the degree sign, or set degree mode.

Examples Of Tan 37 Degrees

Example 1

From a point $20$ m from the foot of a tree, the angle of elevation to the top is $37^\circ$. Find the height of the tree.

An angle of elevation is measured upward from the horizontal.

$$\tan 37^\circ = \frac{h}{20}$$

$$h = 20 \times \tan 37^\circ = 20 \times 0.7536$$

$$h \approx 15.07 \text{ m}$$

With the shortcut $\frac{3}{4}$ the answer would be $15$ m, about $7$ cm short.

Example 2

Is $\tan 37^\circ$ exactly equal to $\frac{3}{4}$?

Wrong attempt. A student draws a 3-4-5 triangle, reads the angle opposite $3$ as $37^\circ$, and concludes:

$$\tan 37^\circ = \frac{3}{4} \quad \text{(claimed exact)}$$

Why it breaks. Run the claim backwards. If $\tan\theta = \frac{3}{4}$ exactly, then $\theta = \tan^{-1}(0.75) = 36.8699^\circ$, not $37^\circ$.

Correct. The triangle's angle was rounded. The true value is:

$$\tan 37^\circ = 0.75355\ldots \neq 0.75$$

Example 3

Prove that $\dfrac{\cos 8^\circ - \sin 8^\circ}{\cos 8^\circ + \sin 8^\circ} = \tan 37^\circ$.

Divide the numerator and the denominator by $\cos 8^\circ$.

$$\frac{1 - \tan 8^\circ}{1 + \tan 8^\circ}$$

Since tan 45 degrees is $1$, write the $1$s as $\tan 45^\circ$.

$$\frac{\tan 45^\circ - \tan 8^\circ}{1 + \tan 45^\circ \tan 8^\circ}$$

This is the tan(A minus B) formula with $A = 45^\circ$, $B = 8^\circ$.

$$= \tan(45^\circ - 8^\circ) = \tan 37^\circ$$

A numeric check gives $0.7536$ on both sides.

Example 4

Given $\tan 18.5^\circ \approx 0.33460$, find $\tan 37^\circ$.

The double angle formula for tangent is $\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$.

$$\tan 37^\circ = \frac{2(0.33460)}{1 - (0.33460)^2}$$

$$= \frac{0.66920}{0.88804} \approx 0.7536$$

Example 5

Evaluate $\sec^2 37^\circ - 1$.

The identity $1 + \tan^2\theta = \sec^2\theta$ rearranges to $\sec^2\theta - 1 = \tan^2\theta$.

$$\sec^2 37^\circ - 1 = \tan^2 37^\circ = (0.75355)^2$$

$$\approx 0.5678$$

Example 6

Find the exact value of $\tan 37^\circ \times \tan 53^\circ$.

The two angles add to $90^\circ$, so $\tan 53^\circ = \cot 37^\circ = \frac{1}{\tan 37^\circ}$.

$$\tan 37^\circ \times \tan 53^\circ = \tan 37^\circ \times \frac{1}{\tan 37^\circ} = 1$$

The product is exactly $1$, even though neither factor has a simple exact form.

The question most students get wrong first in this set is Example 6: the instinct is to multiply two decimals and round, which hides the exact answer that the co-function identity gives for free.

Where Do Students Trip Up On Tan 37 Degrees?

Mistake 1: Writing 3/4 As The Exact Value

Where it slips in: Questions that ask for "the value" of $\tan 37^\circ$ after a run of physics problems that used $\frac{3}{4}$.

Don't do this: Writing $\tan 37^\circ = \frac{3}{4}$ as if it were exact.

The correct way: Write $\tan 37^\circ \approx 0.7536$, or $\tan 37^\circ \approx \frac{3}{4}$ with the approximation sign when a problem allows it. The memorizer who learned "37 is 3/4" as a fact never asks where it came from, and that is exactly why the 3-4-5 angle of $36.87^\circ$ is worth seeing once.

Mistake 2: Leaving The Calculator In Radian Mode

Where it slips in: Any calculator check where the last problem used radians.

Don't do this: Accepting $-0.8408$, the tangent of $37$ radians.

The correct way: Set degree mode or type the degree sign. The rusher who skips that glance gets a negative answer for a Quadrant I angle, and a negative tangent at $37^\circ$ is the signal to stop and check.

Mistake 3: Swapping The Co-Function Pair

Where it slips in: Problems that pair $37^\circ$ with $53^\circ$.

Don't do this: Writing $\tan 37^\circ = \tan 53^\circ$ or $\tan 53^\circ = 0.7536$.

The correct way: The partner of $\tan 37^\circ$ is $\cot 53^\circ$, not $\tan 53^\circ$; $\tan 53^\circ \approx 1.3270$ is the reciprocal. The two close ideas get read as one, and the fix is to say the full identity aloud: tangent of an angle equals cotangent of its complement.

A rounding gap that looks tiny on paper grows with distance. Over a $500$ m horizontal run, using $\frac{3}{4}$ instead of $0.7536$ for a $37^\circ$ slope misstates the rise by about $1.8$ m.

Where Can You Get Help With Trigonometry Values Like Tan 37 Degrees?

Non-special angles like tan 37 degrees are where calculator habits and approximation sense get built. A trigonometry tutor can work through exact-versus-approximate thinking, and a high school math tutor can link it to heights-and-distances problems.

Structured math classes online cover tangent values in sequence. Short sessions of math tutoring suit students who only need the calculator and co-function habits fixed.

Key Takeaways

• Tan 37 degrees is approximately $0.7536$ ($0.75355405\ldots$), and it has no exact form in fractions and square roots.

• The $\frac{3}{4}$ value comes from the 3-4-5 triangle, whose angle is $36.87^\circ$, so it runs about $0.47$ low.

• A calculator in degree mode is the standard method; the Taylor series, the co-function $\cot 53^\circ$ and table interpolation all confirm it.

• A result of $-0.8408$ means radian mode, and $\tan 37^\circ \times \tan 53^\circ$ is exactly $1$.

Practice These Before Moving On

1. A ladder leans so that it makes $37^\circ$ with the ground and its foot is $3$ m from the wall. How high up the wall does it reach, to two decimal places?

2. Evaluate $\tan 37^\circ + \cot 53^\circ$ to four decimal places.

3. Without a calculator, find $\tan 37^\circ \times \cot 37^\circ$.

4. By how much does $\frac{3}{4}$ differ from $\tan 37^\circ$, to four decimal places?

5. What is $\tan 217^\circ$, to four decimal places?

Answers:

1. $3 \times 0.75355 \approx 2.26$ m.

2. $0.75355 + 0.75355 \approx 1.5071$, because $\cot 53^\circ = \tan 37^\circ$.

3. $1$.

4. $0.0036$.

5. $0.7536$, since $217^\circ = 180^\circ + 37^\circ$ and tangent repeats every $180^\circ$.

If Question 1 came out below $2.25$, check whether $\frac{3}{4}$ crept in, then revisit Mistake 1.

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