Sin 42 Degrees: Value, Radians & Unit Circle

What Is The Value Of Sin 42 Degrees?

Sin 42 degrees is approximately $0.6691$, and to more places $\sin 42^\circ \approx 0.669131$. Written with the angle in radians, this is $\sin\frac{7\pi}{30} \approx 0.6691$, since $42^\circ = 42 \times \frac{\pi}{180} = \frac{7\pi}{30} \approx 0.7330$ radians.

There is no short, clean exact form. Angles such as $30^\circ$, $45^\circ$, and $60^\circ$ give tidy surds like $\frac{1}{2}$, $\frac{\sqrt{2}}{2}$, and $\frac{\sqrt{3}}{2}$. The angle $42^\circ$ does not. A closed expression exists in principle (any multiple of $3^\circ$ can be written with nested radicals), but for $42^\circ$ it is a long, nested-radical expression no one uses in practice. So the honest, working answer is the decimal:

$$\sin 42^\circ \approx 0.6691 \qquad \left(42^\circ = \tfrac{7\pi}{30} \text{ rad} \approx 0.7330\right)$$

For the wider family of values and where $42^\circ$ fits among them, the full trigonometric table is the fastest reference.

How Do You Find Sin 42 Degrees?

Finding sin 42 degrees is really two questions: what sign does it take, and what number does it equal? The sign comes from the quadrant; the number comes from a table, a calculator, or a series.

Step 1: Find the reference angle. The reference angle is the acute angle to the nearest horizontal axis. Since $42^\circ$ is already between $0^\circ$ and $90^\circ$, it is its own reference angle. No adjustment is needed.

Step 2: Fix the sign from the quadrant. Using the ASTC (also called CAST) rule, all six ratios are positive in Quadrant I, where $42^\circ$ lives. So $\sin 42^\circ$ is positive.

Step 3: Read off the value. Because $42^\circ$ has no simple surd, you take the number from the trigonometric ratios table, a calculator set to degrees, or an infinite series. Each route gives $\sin 42^\circ \approx 0.6691$.

The one relationship that does simplify things is the cofunction identity. Because $42^\circ$ and $48^\circ$ add to $90^\circ$, sine at one equals cosine at the other:

$$\sin 42^\circ = \cos(90^\circ - 42^\circ) = \cos 48^\circ \approx 0.6691$$

This is a genuine exact statement, and it is why the cofunction identities and the trigonometric ratios of complementary angles are worth knowing. It does not produce a surd, but it links two values you might otherwise treat as separate.

Where Does 42 Degrees Sit On The Unit Circle?

On the unit circle, an angle is measured counterclockwise from the positive $x$-axis, and the point where its ray meets the circle has coordinates $(\cos\theta, \sin\theta)$. The sine is always the $y$-coordinate.

For $42^\circ$, the ray lands in the first quadrant, up and to the right, at the point $(\cos 42^\circ, \sin 42^\circ) \approx (0.7431, 0.6691)$. The height of that point above the $x$-axis is sin 42 degrees.

The same picture is also a right triangle. The radius is the hypotenuse of length $1$, the vertical dashed line is the opposite side, and $\sin 42^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{0.6691}{1}$. This is the bridge between the sin, cos, tan ratios in a triangle and the coordinates on the circle: they are the same numbers seen two ways.

Why Is There No Simple Exact Value For Sin 42 Degrees?

The tidy angles get their tidy surds from clean geometry: $45^\circ$ comes from a square's diagonal, and $30^\circ$ and $60^\circ$ from an equilateral triangle cut in half. The angle $42^\circ$ has no such construction, so any attempt to write it exactly just moves the difficulty somewhere else.

• Angle-sum relabelling. Splitting $42^\circ = 45^\circ - 3^\circ$ and expanding gives $\sin 42^\circ = \sin 45^\circ \cos 3^\circ - \cos 45^\circ \sin 3^\circ$. That is correct, but $\sin 3^\circ$ is itself an ugly value, so nothing simplifies. The same happens with $42^\circ = 30^\circ + 12^\circ$: $\sin 12^\circ$ has no clean form either.

• A surd exists but is unusable. Since $42^\circ$ is a multiple of $3^\circ$, an exact expression can be built with nested square roots. It is so long that every textbook and calculator reports the decimal instead.

• Series and tables win. In practice the number comes from an infinite series or a stored table, both of which land on $0.6691$.

That is the honest picture. Sin 42 degrees is a perfectly definite number; it simply is not a pretty one, and pretending otherwise (for example, writing a fake short surd) is the mistake to avoid.

How Is Sin 42 Degrees Related To Other Sine Values?

Reading sin 42 degrees next to its neighbours shows how smoothly sine grows across the first quadrant. Every value below is positive because all of these angles sit in Quadrant I.

Table: Sine of nearby first-quadrant angles, in degrees and radians (four decimal places).

Two special anchors, $\sin 30^\circ = \tfrac{1}{2}$ and $\sin 60^\circ = \tfrac{\sqrt{3}}{2}$, bracket this range and confirm the trend: $\sin 42^\circ = 0.6691$ falls neatly between $0.5$ and $0.8660$.

The other five ratios at $42^\circ$ follow from this one value and its cosine, $\cos 42^\circ \approx 0.7431$:

Table: All six trigonometric ratios at $42^\circ$ (four decimal places).

Here $\tan 42^\circ = \frac{\sin 42^\circ}{\cos 42^\circ}$, and each reciprocal ratio is one divided by its partner. If you prefer to work in radians throughout, the trigonometric ratios in radians page uses the $\frac{7\pi}{30}$ form directly.

How Does A Calculator Actually Compute Sin 42 Degrees?

A calculator does not store $\sin 42^\circ$ in a table; it computes it from a series. The Taylor series for sine, with the angle in radians, is:

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

The denominators $6$, $120$, and $5040$ are the products $3 \times 2 \times 1$, $5 \times 4 \times 3 \times 2 \times 1$, and $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$. Feeding in $x = \frac{7\pi}{30} \approx 0.7330$:

$$\sin 42^\circ \approx 0.7330 - 0.0656 + 0.0018 - 0.0000 + \cdots \approx 0.6691$$

Just four terms already give $0.6691$. This is the same machinery, refined, that produced the very first trigonometric tables by hand centuries ago.

Who Discovered How To Calculate Sin 42 Degrees?

Long before calculators, astronomers needed the sine of every angle to predict the motion of the planets, so they built the tables by hand.

Two names anchor that history:

• Hipparchus of Nicaea (c. 190–120 BCE, Greece) built the first systematic chord table, treating angles as a computational tool for astronomy.

• Aryabhata (476–550 CE, India) produced an early sine table and named the function jya, the root of the modern word "sine."

Where Is Sin 42 Degrees Used In The Real World?

A specific value like sin 42 degrees appears wherever a real angle happens to be around $42^\circ$, and the sine converts that angle into a usable ratio.

• Ramps and inclines: for a slope tilted at $42^\circ$, the sine gives the vertical rise per unit of slope length, which sets how steep a ramp, roof, or hillside actually is.

• Projectile motion: the vertical part of a launch speed is (speed) $\times \sin(\text{angle})$, so a ball or jet of water launched at $42^\circ$ uses $\sin 42^\circ \approx 0.6691$ of its speed going upward.

• Alternating current: voltage in an AC circuit follows a sine wave, and the instantaneous value at a phase of $42^\circ$ is the peak times $\sin 42^\circ$.

• Structural engineering: a support beam or cable set at $42^\circ$ carries a vertical load component proportional to $\sin 42^\circ$.

• Navigation and surveying: resolving a distance or bearing measured at $42^\circ$ into north-south and east-west parts uses sine and cosine of that angle.

The pattern is the same each time: an angle in the world, a sine to turn it into a ratio, and a number near $0.6691$ doing the work.

What Are The Most Common Mistakes With Sin 42 Degrees?

These are the errors that most often turn a correct method into a wrong answer for sin 42 degrees. Each one is quick to avoid once you have seen it.

Leaving the calculator in radian mode.

Where it slips in:

A student types "sin 42" while the calculator is set to radians and reads off $-0.9165$, the sine of $42$ radians, not $42$ degrees.

Don't do this:

Do not trust the display until you have checked the angle mode.

The correct way:

Set the calculator to degrees (look for DEG or D on screen) before entering $42$. The correct result is $\sin 42^\circ \approx 0.6691$. If you must stay in radians, enter $\frac{7\pi}{30}$ instead.

Inventing a fake exact value.

Where it slips in:

Expecting every angle to behave like $30^\circ$ or $45^\circ$, a student writes something like $\frac{\sqrt{2}}{2}$ or a made-up surd for $\sin 42^\circ$.

Don't do this:

Do not force a short surd onto a non-special angle.

The correct way:

Report the decimal $\sin 42^\circ \approx 0.6691$. The only exact simplification available is the cofunction statement $\sin 42^\circ = \cos 48^\circ$.

Confusing the cofunction partner.

Where it slips in:

A student remembers "sine equals cosine of the complement" but subtracts from the wrong number, writing $\sin 42^\circ = \cos 42^\circ$ or $\cos 58^\circ$.

Don't do this:

Do not skip the complement step. The partner angle must add to $90^\circ$.

The correct way:

Compute the complement first: $90^\circ - 42^\circ = 48^\circ$, so $\sin 42^\circ = \cos 48^\circ$, and both equal $0.6691$.

Assigning the wrong sign.

Where it slips in:

Working with a related angle such as $138^\circ$ or $222^\circ$, a student forgets to check the quadrant and drops or adds a minus sign.

Don't do this:

Do not carry the Quadrant I sign into other quadrants.

The correct way:

For $42^\circ$ itself, ASTC gives a positive value. For a related angle, find the reference angle first, then set the sign from the quadrant, so $\sin 138^\circ = +\sin 42^\circ$ but $\sin 222^\circ = -\sin 42^\circ$.

Practice Problems On Sin 42 Degrees

Try each one, then check the answer beside it.

1. Write $42^\circ$ in radians in the form $\frac{a\pi}{b}$.
   (Answer: $\frac{7\pi}{30} \approx 0.7330$.)

2. Use the cofunction identity to rewrite $\sin 42^\circ$ as a cosine.
   (Answer: $\sin 42^\circ = \cos 48^\circ \approx 0.6691$.)

3. Given $\cos 42^\circ \approx 0.7431$, find $\tan 42^\circ$ to four decimal places.
   (Answer: $\tan 42^\circ = \frac{0.6691}{0.7431} \approx 0.9004$.)

4. A ramp $10$ m long rises at $42^\circ$. How high is its top end?
   (Answer: height $= 10 \sin 42^\circ \approx 10 \times 0.6691 = 6.691$ m.)

5. Is $\sin 138^\circ$ equal to $\sin 42^\circ$ or to $-\sin 42^\circ$?
   (Answer: $\sin 138^\circ = +\sin 42^\circ \approx 0.6691$, since $138^\circ$ is in Quadrant II where sine is positive and its reference angle is $42^\circ$.)

6. Find $\csc 42^\circ$ to four decimal places.
   (Answer: $\csc 42^\circ = \frac{1}{\sin 42^\circ} = \frac{1}{0.6691} \approx 1.4945$.)

Where Should You Go Next After Sin 42 Degrees?

Sin 42 degrees connects to several natural next steps, and each door below leads to a real Bhanzu page.

1. Trigonometric table. See sin 42 degrees alongside every standard angle in one place, with cosine and tangent too.

2. Cofunction identities. Understand fully why $\sin 42^\circ = \cos 48^\circ$, and use the same idea for any complementary pair.

3. What is a radian. Get comfortable with the $\frac{7\pi}{30}$ form so calculator and formula work stops tripping you up.

If your child is building these foundations, a live Bhanzu trainer teaches trigonometric values from the reasoning up, starting with the unit circle and the "why," in the Bhanzu trigonometry program.