Sin 38 Degrees: Value, Radians & Unit Circle

What Is The Value Of Sin 38 Degrees?

The value of sin 38 degrees is approximately $0.6157$ (to four decimal places), or $0.6156614753$ carried further. Written in trigonometric form, $\sin 38^\circ \approx 0.6157$.

The angle can be given two ways, and instructional trigonometry always shows both:

• In degrees: $38^\circ$.

• In radians: $38^\circ = 38 \times \dfrac{\pi}{180} = \dfrac{19\pi}{90} \approx 0.6632$ radians. For the conversion itself, see what is a radian.

Here is the part most pages skip. There is no clean surd for this value. Angles like $30^\circ$, $45^\circ$, and $60^\circ$ give exact forms such as $\tfrac{1}{2}$ or $\tfrac{\sqrt{2}}{2}$, but $38^\circ$ is non-constructible, so it cannot be written with a finite stack of square roots. The correct answer to "what is the exact value" is the decimal $0.6157$, together with the exact relationship $\sin 38^\circ = \cos 52^\circ$.

How Do You Find Sin 38 Degrees?

You find $\sin 38^\circ$ in three honest steps: place the angle, read its sign, then read the size from a table, a calculator, or the cofunction. There is no radical shortcut, and pretending otherwise is the most common error on this angle.

Step 1: Place the angle and find its reference angle.

An angle of $38^\circ$ lands in the first quadrant of the coordinate plane (between $0^\circ$ and $90^\circ$). Its reference angle, the acute angle to the horizontal axis, is $38^\circ$ itself. Nothing needs to be subtracted.

Step 2: Read the sign from the quadrant.

Use the memory aid ASTC (All, Sine, Tangent, Cosine), read counter-clockwise from Quadrant I. In Quadrant I, All ratios are positive, so $\sin 38^\circ$ is positive. This matches the value we already have, $+0.6157$.

Step 3: Read the size.

Because $38^\circ$ is not a special angle, its size comes from one of three places:

• A trigonometric table, which lists $\sin 38^\circ = 0.6157$ directly.

• A calculator set to degree mode: type $\sin(38)$ and read $0.6157$.

• The cofunction relation $\sin 38^\circ = \cos 52^\circ$, covered below.

A quick note on the calculator behind the table. Modern calculators do not store a value for every angle; they compute sine from an infinite series (the same idea Madhava of Sangamagrama used around 1400), adding smaller and smaller correction terms until the digits settle. The double-angle rewrite $\sin 38^\circ = 2\sin 19^\circ \cos 19^\circ$ is true, but it only shifts the work to $19^\circ$, which is not a special angle either, so it never produces a surd.

Where Does 38° Sit On The Unit Circle?

On the unit circle (radius $1$, centred at the origin), draw a ray from the centre at $38^\circ$ above the positive horizontal axis. Where that ray meets the circle, the point has coordinates $(\cos 38^\circ, \sin 38^\circ)$.

$$(\cos 38^\circ,\ \sin 38^\circ) = (0.7880,\ 0.6157)$$

The sine is the height of that point above the horizontal axis, the $y$-coordinate. Since the point sits in the upper-right of the circle, its height is positive and a little above the halfway mark to the top, which is exactly what $0.6157$ says.

The same value shows up on a right triangle, which is the other half of the picture. In a right triangle with a $38^\circ$ angle, sine is opposite over hypotenuse:

$$\sin 38^\circ = \frac{\text{opposite}}{\text{hypotenuse}} \approx 0.6157$$

So a ramp at $38^\circ$ rises about $0.62$ metres for every $1$ metre of ramp length. The unit-circle height and the triangle ratio are the same number seen two ways. For the definition of the function itself, see the sine function and the wider family in sin cos tan.

How Does Sin 38° Relate To Cos 52°?

The cleanest exact statement about this angle is a cofunction identity. The sine of an angle equals the cosine of its complement (the two angles that add to $90^\circ$):

$$\sin \theta = \cos(90^\circ - \theta)$$

Setting $\theta = 38^\circ$, the complement is $90^\circ - 38^\circ = 52^\circ$, so:

$$\sin 38^\circ = \cos 52^\circ \approx 0.6157$$

This is not an approximation, it is exact: both sides are the same number. The reason is the right triangle. The two acute angles of a right triangle add to $90^\circ$, and one angle's opposite side is the other angle's adjacent side, which swaps sine and cosine. For the full rule, see cofunction identities and trigonometric ratios of complementary angles.

The six ratios at $38^\circ$, for reference:

Table: The six trigonometric ratios evaluated at 38 degrees (4 dp).

What Are The Sine Values Of Angles Near 38°?

Placing $38^\circ$ among its neighbours shows why the decimal has to be what it is: sine climbs steadily from $0$ at $0^\circ$ toward $1$ at $90^\circ$, and $38^\circ$ falls in the smooth middle of that rise.

Table: Sine (and the angle in radians) for special and nearby angles.

Read down the last column: each step up in angle gives a larger sine, and $0.6157$ sits neatly between $\sin 35^\circ$ and $\sin 40^\circ$. That ordering is a fast sanity check for any value you compute.

Why Is Sin 38 Degrees Positive?

The sign is not a rule to memorise, it comes straight from the unit circle. Sine is a height, and heights above the axis are positive.

• The angle is in Quadrant I. At $38^\circ$, the point on the unit circle sits in the upper-right, above the horizontal axis, so its $y$-coordinate is positive.

• ASTC agrees. In the first quadrant, All ratios (sine, cosine, tangent) are positive, so a positive $\sin 38^\circ$ is expected.

• The triangle agrees. In a right triangle, the opposite side and the hypotenuse are both positive lengths, so their ratio, the sine, is positive.

Sine only turns negative below the horizontal axis, in Quadrants III and IV (angles from $180^\circ$ to $360^\circ$). Since $38^\circ$ is nowhere near there, $0.6157$ carries a plus sign, and any minus sign in an answer is a slip to hunt down.

Who Discovered How To Calculate Sin 38 Degrees?

No single person "found" sin 38 degrees. The value is one entry in a two-thousand-year project to tabulate the ratios inside triangles, built by astronomers who needed to predict the sky.

Two other figures shaped how a value like $\sin 38^\circ$ is actually produced:

• Hipparchus of Nicaea (c. 190–120 BCE, Greece) is often called the father of trigonometry for compiling the first known table of chords, the ancestor of the sine table.

• Madhava of Sangamagrama (c. 1340–1425, India) found the infinite power series for sine, the method that lets a calculator compute $\sin 38^\circ$ to as many digits as you like.

Where Is Sin 38 Degrees Used In The Real World?

A single sine value is a conversion between an angle and a length, so it appears wherever a slope, a height, or a swing is measured.

• Construction and access ramps: roof pitch and ramp gradients are set by angle, and the sine converts that angle into the vertical rise per metre of surface.

• Physics and projectiles: the range and peak height of a thrown or launched object depend on the sine of the launch angle, so $38^\circ$ appears in trajectory calculations.

• Surveying and navigation: measuring the height of a hill or tower from a measured angle of elevation uses the sine of that angle.

• Astronomy: the altitude of a star or the Sun above the horizon feeds sine tables to work out positions and daylight length, the very problem the first tables were built for.

• Engineering and waves: alternating current, sound, and vibration are modelled as sine waves, and a specific value like $0.6157$ is the wave's height at that phase.

The same number that fixes a ramp's steepness also places a star, which is the quiet power of one trigonometric ratio.

What Are The Most Common Mistakes With Sin 38 Degrees?

These four errors cause most wrong answers for this angle. They come from the calculator, the cofunction, the quadrant, and a false hope for a surd.

Leaving the calculator in radian mode.

Where it slips in:

A student types $\sin(38)$ expecting $0.6157$ but the calculator is set to radians, returning about $0.2963$ instead.

Don't do this:

Do not read a sine answer without checking the angle mode first.

The correct way:

Set the calculator to degree mode (look for DEG or D on the display) before entering $\sin(38)$. In radian mode, $38$ means $38$ radians, a completely different angle.

Confusing $\sin 38^\circ$ with $\cos 38^\circ$.

Where it slips in:

A student writes $\sin 38^\circ = \cos 38^\circ$, or applies the cofunction to the wrong angle.

Don't do this:

Do not pair $38^\circ$ with itself. Sine equals cosine of the complement, not of the same angle.

The correct way:

Subtract from $90^\circ$ first: $\sin 38^\circ = \cos(90^\circ - 38^\circ) = \cos 52^\circ$. Note that $\cos 38^\circ = 0.7880$ is a different value from $\sin 38^\circ = 0.6157$.

Mis-taking the reference angle or the sign.

Where it slips in:

A student treats $38^\circ$ like an angle in another quadrant, subtracting it from $180^\circ$ or attaching a minus sign.

Don't do this:

Do not adjust an angle that is already acute. $38^\circ$ is its own reference angle and lives in Quadrant I.

The correct way:

Confirm the quadrant first. Since $0^\circ < 38^\circ < 90^\circ$, the reference angle is $38^\circ$ and, by ASTC, the sine is positive.

Inventing a fake surd "exact" value.

Where it slips in:

A student, trained on $30^\circ$ and $45^\circ$, writes something like $\frac{\sqrt{3}-1}{2}$ and calls it the exact value of $\sin 38^\circ$.

Don't do this:

Do not force a radical onto a non-constructible angle. No finite surd equals $\sin 38^\circ$.

The correct way:

Give the decimal $0.6157$ (or more digits) and, if an exact statement is required, use the cofunction $\sin 38^\circ = \cos 52^\circ$.

Practice Problems On Sin 38 Degrees

Work each one, then check the answer beside it. Values are to four decimal places.

1. State $\sin 38^\circ$ to four decimal places.
   (Answer: $0.6157$.)

2. Convert $38^\circ$ to radians in terms of $\pi$.
   (Answer: $\frac{19\pi}{90} \approx 0.6632$ rad.)

3. Use the cofunction identity to write $\sin 38^\circ$ as a cosine.
   (Answer: $\cos 52^\circ$.)

4. Find $\csc 38^\circ$, the reciprocal of $\sin 38^\circ$.
   (Answer: $\frac{1}{0.6157} \approx 1.6243$.)

5. A ramp is $4$ m long and rises at $38^\circ$. How high is the top?
   (Answer: $4 \times \sin 38^\circ \approx 4 \times 0.6157 = 2.463$ m.)

6. Without a calculator, decide whether $\sin 38^\circ$ is larger or smaller than $\sin 40^\circ$.
   (Answer: smaller, because sine increases from $0^\circ$ to $90^\circ$, and $0.6157 < 0.6428$.)

Where Should You Go Next After Sin 38 Degrees?

One value opens several doors, and each of these builds directly on what you just used.

1. Trigonometric ratios. See how sine sits alongside cosine and tangent, and how all six ratios are defined from one triangle.

2. Cofunction identities. Go deeper on the $\sin 38^\circ = \cos 52^\circ$ relationship and the full set of complement rules.

3. Trigonometric table. Read sine, cosine, and tangent for every whole-degree angle in one place, with $38^\circ$ among them.

If your child is building these foundations, a live Bhanzu trainer teaches trigonometry starting from the "why" (the triangle and the circle behind every value) through the Bhanzu trigonometry program.