Sin 70 Degrees: Value, Unit Circle & How To Find It

What Is The Value Of Sin 70 Degrees?

The value of sin 70 degrees is approximately $0.9397$, correct to four decimal places. Written with the angle in both units, $\sin 70^\circ = \sin\frac{7\pi}{18} \approx 0.9397$, where $\frac{7\pi}{18} \approx 1.2217$ radians. The full decimal runs $0.93969262\ldots$, and $0.9397$ is the form used in nearly every calculation.

There is no neat closed form here. The special angles $30^\circ$, $45^\circ$, and $60^\circ$ give tidy surds like $\frac{1}{2}$, $\frac{\sqrt{2}}{2}$, and $\frac{\sqrt{3}}{2}$, but $70^\circ$ is not one of them. Its sine is a genuine irrational number that no simple square-root expression captures, which is why $0.9397$ is the correct answer to give.

Two facts pin the value down before we compute anything:

• It is positive. The angle $70^\circ$ lands in Quadrant I of the unit circle, where every sine value is positive.

• It is close to 1. Since $\sin 90^\circ = 1$ and $70^\circ$ is fairly near $90^\circ$, the sine should be large. A value of $0.9397$ fits, sitting just below $\sin 75^\circ \approx 0.9659$ and above $\sin 60^\circ \approx 0.8660$.

How Do You Find Sin 70 Degrees On A Right Triangle?

Sine begins with a right triangle. For an acute angle, $\sin \theta$ is the ratio of the side opposite the angle to the hypotenuse, the "opposite over hypotenuse" rule from sin cos tan.

$$\sin 70^\circ = \frac{\text{opposite}}{\text{hypotenuse}}$$

Picture a right triangle with one angle set to $70^\circ$ and a hypotenuse of length $1$. The side opposite the $70^\circ$ angle then has length $\sin 70^\circ \approx 0.9397$, and the side adjacent has length $\cos 70^\circ \approx 0.3420$. The opposite side is almost as long as the hypotenuse itself, which is what a steep $70^\circ$ angle looks like.

You cannot get $0.9397$ by hand from this triangle the way you can derive $\sin 30^\circ = \frac{1}{2}$ from a 30-60-90 triangle. A $70^\circ$ triangle has no such shortcut. What the triangle gives you is the meaning of the number: at $70^\circ$, the rise is about $94%$ of the slant length.

Where Does 70 Degrees Sit On The Unit Circle?

On the unit circle, a circle of radius $1$ centred at the origin, an angle is measured anticlockwise from the positive $x$-axis. The point where the angle's ray meets the circle has coordinates $(\cos\theta, \sin\theta)$. So the sine is simply the $y$-coordinate of that point.

For $70^\circ$, the point on the circle is:

$$(\cos 70^\circ, \sin 70^\circ) \approx (0.3420, ; 0.9397)$$

The $y$-coordinate, $0.9397$, is exactly sin 70 degrees. Because the point sits high and slightly right of centre in the upper-right quadrant, the height is large and positive, matching the triangle picture from the previous section.

Why Does Sin 70 Degrees Have No Simple Surd?

This is the part most pages skip. The reason $70^\circ$ has no clean square-root form, while $72^\circ$ or $75^\circ$ do, comes down to a cubic equation.

Start from the cofunction relation $\sin 70^\circ = \cos 20^\circ$. Now use the triple-angle identity $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$ with $\theta = 20^\circ$, since $3 \times 20^\circ = 60^\circ$ and $\cos 60^\circ = \tfrac{1}{2}$:

$$\cos 60^\circ = 4\cos^3 20^\circ - 3\cos 20^\circ$$

$$\tfrac{1}{2} = 4x^3 - 3x, \qquad \text{where } x = \cos 20^\circ = \sin 70^\circ$$

Clearing the fraction gives the cubic:

$$8x^3 - 6x - 1 = 0$$

So sin 70 degrees is a root of $8x^3 - 6x - 1 = 0$. Substituting $x = 0.9397$ satisfies it. This cubic has three real roots but is irreducible over the rationals, and its solutions cannot be written with real square roots alone, only with cube roots of complex numbers or with a cosine. That is the honest algebraic reason there is no tidy surd.

Here is the takeaway in plain terms:

• Constructible angles (like $30^\circ$, $45^\circ$, $60^\circ$, $75^\circ$) have sines built from square roots.

• $70^\circ$ is not constructible, because its value solves an irreducible cubic, not a chain of square roots.

• The practical value stays $\sin 70^\circ \approx 0.9397$, and that is not an approximation of some "real" surd answer. The decimal is the answer.

How Is Sin 70 Degrees Related To Cos 20 Degrees?

The cleanest exact statement about this angle is a cofunction identity, not a surd. Sine and cosine of complementary angles (two angles that add to $90^\circ$) are equal:

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

Setting $\theta = 70^\circ$, and since $90^\circ - 70^\circ = 20^\circ$:

$$\sin 70^\circ = \cos 20^\circ \approx 0.9397$$

This is why a value table for $\cos 20^\circ$ and a value table for $\sin 70^\circ$ agree to every decimal. If you know cos 20 degrees, you already know sin 70 degrees. The idea generalises through the cofunction identities and the wider rule for trigonometric ratios of complementary angles.

There is a second exact relationship worth knowing. Using the double-angle form $\sin 2\alpha = 2\sin\alpha\cos\alpha$ with $\alpha = 35^\circ$:

$$\sin 70^\circ = 2\sin 35^\circ \cos 35^\circ \approx 0.9397$$

Both statements are exact. Neither turns into a simple surd, which is the recurring theme of this angle.

How Does A Calculator Compute Sin 70 Degrees?

If there is no surd and no unit-circle shortcut, how does a calculator return $0.93969262$ instantly? It uses a power series, the same tool eighteenth-century table-makers used by hand.

First the angle is converted to radians, because the series only works in radians: $70^\circ = \frac{7\pi}{18} \approx 1.2217$. Then it evaluates the Maclaurin series for sine, where the denominators are the running products $3\times2\times1 = 6$, then $5\times4\times3\times2\times1 = 120$, then $7\times\cdots\times1 = 5040$:

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

Feeding in $x \approx 1.2217$ and adding a handful of terms already lands near $0.93969$. Each extra term shrinks quickly, so five or six terms give more precision than any school problem needs. A short list of DB-linked values shows where $70^\circ$ falls among its neighbours.

Table: Sine, cosine, and tangent for angles near 70°, with radian measure.

Notice the symmetry in the table: $\sin 20^\circ = \cos 70^\circ$ and $\sin 70^\circ = \cos 20^\circ$, the cofunction pattern at work. For the full set of standard values, see the trigonometric table.

Who Discovered How To Calculate Angles Like Sin 70 Degrees?

Nobody "discovered" $0.9397$. What people built, over roughly fifteen centuries, were the methods to compute the sine of any angle, including awkward ones like $70^\circ$ that hide from square roots.

Two threads meet in this value:

• Aryabhata turned scattered chord tables into a systematic sine table, the ancestor of every trig table since.

• Madhava of Sangamagrama (around 1340–1425) discovered the sine power series two centuries before it appeared in Europe, which is exactly how a calculator now evaluates a non-special angle like $70^\circ$.

Where Is Sin 70 Degrees Used In The Real World?

A $70^\circ$ angle is steep but common, and its sine (about $0.94$) shows up wherever a steep rise or a near-vertical component matters.

• Solar panels: in high-latitude winter, panels are tilted steeply, near $70^\circ$, so the sine sets how much of the low sun's rays strike the panel face-on.

• Ramps and roofs: a support strut at $70^\circ$ carries a vertical load of about $0.94$ of its length, which structural engineers read straight off the sine.

• Antennas and radar: a dish or beam raised to $70^\circ$ elevation points almost skyward, and the vertical reach of the signal scales with $\sin 70^\circ$.

• Projectile motion: a ball launched at $70^\circ$ sends about $94%$ of its speed upward, giving a high, short-range arc used in lobbed shots and mortar-style trajectories.

• Navigation and surveying: measuring the height of a tall tower from close up needs a steep sight line, and angles near $70^\circ$ feed directly into the sine ratio.

The same number, $0.9397$, links a rooftop strut, a winter solar panel, and a lobbed ball. One ratio quietly does the work across fields that never mention trigonometry by name.

What Are The Most Common Mistakes With Sin 70 Degrees?

These four errors cause most wrong answers for this angle. They come straight from the reader questions that surface around $70^\circ$: calculator confusion and the wish for a neat surd.

Leaving the calculator in radian mode.

Where it slips in:

A student types sin(70) expecting $0.9397$, but the calculator is set to radians and returns $0.7739$, the sine of $70$ radians, not $70$ degrees.

Don't do this:

Do not trust the display before checking the angle unit. The two answers are completely different numbers.

The correct way:

Set the calculator to degree mode (look for DEG), or convert first: $70^\circ = \frac{7\pi}{18}$ and evaluate $\sin\frac{7\pi}{18} \approx 0.9397$.

Inventing a surd form for $70^\circ$.

Where it slips in:

Because $30^\circ$, $45^\circ$, and $60^\circ$ have square-root answers, a student writes something like $\frac{\sqrt{3}}{2}$ or $\frac{\sqrt{7}}{3}$ for $\sin 70^\circ$ to look exact.

Don't do this:

Do not fabricate a radical. $70^\circ$ is not a constructible angle, and no simple surd equals its sine.

The correct way:

Give the decimal $\sin 70^\circ \approx 0.9397$, or the exact identity $\sin 70^\circ = \cos 20^\circ$. Both are correct; a made-up surd is not.

Confusing the cofunction partner.

Where it slips in:

A student recalls "sine equals cosine of the complement" but pairs $70^\circ$ with the wrong angle, writing $\sin 70^\circ = \cos 70^\circ$ or $\sin 70^\circ = \sin 20^\circ$.

Don't do this:

Do not use the same angle or the wrong function. $\cos 70^\circ \approx 0.3420$, which is nowhere near $0.9397$.

The correct way:

Subtract from $90^\circ$ and switch the function: $\sin 70^\circ = \cos(90^\circ - 70^\circ) = \cos 20^\circ \approx 0.9397$.

Assuming sin 70 degrees could be negative.

Where it slips in:

A student applies a quadrant rule carelessly and attaches a minus sign, or confuses $70^\circ$ with a second- or third-quadrant angle.

Don't do this:

Do not add a negative sign. $70^\circ$ is a first-quadrant angle.

The correct way:

Use the ASTC rule: in Quadrant I, all ratios are positive, so $\sin 70^\circ = +0.9397$.

Practice Problems On Sin 70 Degrees

Use $\sin 70^\circ \approx 0.9397$, $\cos 70^\circ \approx 0.3420$, and the identities above. Answers follow each problem.

1. A ladder $5$ m long leans against a wall at $70^\circ$ to the ground. How high up the wall does it reach?
   (Answer: height $= 5\sin 70^\circ \approx 5 \times 0.9397 = 4.70$ m.)

2. Evaluate $\cos 20^\circ$ without a calculator by using a cofunction identity.
   (Answer: $\cos 20^\circ = \sin 70^\circ \approx 0.9397$.)

3. Find $\tan 70^\circ$ from the sine and cosine values.
   (Answer: $\tan 70^\circ = \frac{\sin 70^\circ}{\cos 70^\circ} \approx \frac{0.9397}{0.3420} = 2.7475$.)

4. Convert $70^\circ$ to radians, leaving the answer as a multiple of $\pi$.
   (Answer: $70^\circ = 70 \times \frac{\pi}{180} = \frac{7\pi}{18} \approx 1.2217$ rad.)

5. Verify that $x = 0.9397$ approximately satisfies $8x^3 - 6x - 1 = 0$.
   (Answer: $8(0.9397)^3 - 6(0.9397) - 1 \approx 6.638 - 5.638 - 1 = 0$, confirming sin 70 degrees is a root.)

6. A projectile is launched at $70^\circ$ with speed $20$ m/s. Find the vertical component of its initial velocity.
   (Answer: $20\sin 70^\circ \approx 20 \times 0.9397 = 18.79$ m/s.)

Where Should You Go Next After Sin 70 Degrees?

This one value opens onto the wider structure of trigonometry, and a few natural doors lead outward.

1. Cos 20 degrees. The cofunction partner of sin 70 degrees, sharing every decimal, and the cleanest way to state this value exactly.

2. Trigonometric ratios of specific angles. See which angles give clean surds and which, like $70^\circ$, do not, and why.

3. Sine function. Step back from one angle to the whole sine wave, its period, and how what is a radian reframes every angle.

If your child is building these foundations, a live Bhanzu trainer teaches trigonometric values starting from the "why", the unit circle and the identities behind each number, in the Bhanzu trigonometry program.