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

What Is The Value Of Cos 70 Degrees?

Cos 70 Degrees is approximately $0.3420$ when rounded to four decimal places, written as $\cos 70^\circ \approx 0.3420$. The same angle in radians is $\frac{7\pi}{18} \approx 1.2217$, so you may also see it as $\cos\frac{7\pi}{18}$. Both notations point to the identical number.

The value is positive because 70° lands in the first quadrant, where every cosine is positive. It is smaller than $\cos 45^\circ$ and $\cos 60^\circ$ because cosine shrinks steadily as the angle grows from $0^\circ$ toward $90^\circ$. By 70° the cosine has already dropped below one-half.

The full-precision decimal is $0.3420201433\ldots$, an irrational number with no repeating pattern and no exact fraction. For almost all work, four decimal places ($0.3420$) is enough.

How Do You Find Cos 70 Degrees?

There is no special-angle shortcut for 70°, so you find $\cos 70^\circ$ one of three honest ways: read it off the unit circle, use the cofunction identity, or use a table or calculator. Each gives the same $0.3420$.

The quickest exact statement uses complementary angles. Since 70° and 20° add to 90°, the cofunction identity gives:

$$\cos 70^\circ = \sin(90^\circ - 70^\circ) = \sin 20^\circ$$

So $\cos 70^\circ$ and $\sin 20^\circ$ are the same number, $0.3420$. This is the relation every ranking page repeats, and it is genuinely useful: any cosine of an angle above 45° can be rewritten as the sine of a smaller angle.

To read the sign before you read the size, use the CAST rule (also taught as ASTC). In Quadrant I all three of sine, cosine, and tangent are positive, so $\cos 70^\circ$ is positive before you compute a single digit. Only the size is left to find.

Where Does 70° Sit On The Unit Circle?

On the unit circle, an angle of 70° is measured anticlockwise from the positive x-axis, and $\cos 70^\circ$ is the x-coordinate of the point where the angle's ray meets the circle. That point is approximately $(0.3420,\ 0.9397)$.

Read the coordinates directly:

• The x-coordinate is $\cos 70^\circ \approx 0.3420$.

• The y-coordinate is $\sin 70^\circ \approx 0.9397$.

• The point sits high and close to the top of the circle, because 70° is near the vertical, so its horizontal reach ($\cos$) is small and its vertical rise ($\sin$) is large.

The right-triangle view gives the same answer. Drop a vertical line from the point on the circle to the x-axis, and you get a right triangle with hypotenuse 1 (the radius). The angle at the origin is 70°, the side along the x-axis is the adjacent side, and:

$$\cos 70^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{0.3420}{1} = 0.3420$$

Whether you call it "the x-coordinate on the unit circle" or "adjacent over hypotenuse in a right triangle," the number is identical. That is the point of anchoring the cosine function in both pictures at once.

Is There An Exact Value For Cos 70 Degrees?

There is no simple surd for $\cos 70^\circ$. Unlike $\cos 30^\circ = \frac{\sqrt{3}}{2}$ or $\cos 45^\circ = \frac{\sqrt{2}}{2}$, the number $0.3420$ cannot be written with a finite stack of square roots and fractions. The honest exact statements are the cofunction form $\cos 70^\circ = \sin 20^\circ$ and the equation the value secretly satisfies.

Here is that equation. The triple-angle identity says $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$. Set $\theta = 70^\circ$, so $3\theta = 210^\circ$ and $\cos 210^\circ = -\frac{\sqrt{3}}{2}$. Writing $x = \cos 70^\circ$:

$$4x^3 - 3x = -\frac{\sqrt{3}}{2}$$

$$8x^3 - 6x + \sqrt{3} = 0$$

The value $x = \cos 70^\circ \approx 0.3420$ is a root of this cubic. The cubic is irreducible over the rationals and falls into the casus irreducibilis, the classic case where three real roots exist but none can be written using only real square roots. This is the same algebraic wall that makes trisecting a 60° angle impossible with compass and straightedge, and 70° is simply not a constructible angle. So the lack of a clean radical is not a gap in our knowledge, it is a proven fact about the number.

That leaves the practical question: how does a table or a calculator produce $0.3420$? By adding up a power series. In radians, with $x = \frac{7\pi}{18} \approx 1.2217$, cosine is the infinite sum:

$$\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots$$

Feed in $x \approx 1.2217$ and the running total settles near $0.3420$ after a handful of terms. Every "$\cos$" button and every printed trigonometric table rests on some version of this idea.

Table: Cosine of 70° next to the nearby special angles, in degrees and radians.

The pattern reads top to bottom: as the angle climbs toward 90°, the cosine falls toward 0. The special angles cos 30°, cos 45°, and cos 60° have tidy surds; 70° and 20° do not, which is exactly why $\cos 70^\circ = \sin 20^\circ$ is the neatest form you will get. For the matching sine, see sin 20 degrees.

Why Is Cos 70 Degrees Positive?

Cos 70 Degrees is positive because 70° lands in the first quadrant of the unit circle, where the horizontal coordinate is on the right-hand, positive side of the origin. Cosine is that horizontal coordinate, so its sign follows the quadrant.

The quadrant sign rule is worth carrying for every angle, not just this one:

• Quadrant I (0° to 90°): cosine is positive. 70° lives here, so $\cos 70^\circ > 0$.

• Quadrant II (90° to 180°): cosine turns negative, because the x-coordinate crosses to the left of the origin.

• Quadrants III and IV: cosine is negative in III and positive again in IV.

A second reason is size, not sign. Cosine starts at $1$ when the angle is $0^\circ$ and slides down to $0$ at $90^\circ$. Seventy degrees is most of the way along that slide, so the value is positive but small, just above one-third. Both facts, the positive sign and the small size, come straight from where the point sits on the circle.

Who Discovered The Cosine Function?

Nobody woke up one morning and invented cosine. It grew out of centuries of astronomers trying to predict where the stars and planets would be, and the earliest versions were tables of chord lengths, not the ratios we use today.

Two earlier figures built the ground Madhava stood on:

• Hipparchus of Nicaea (c. 190–120 BCE, Greece) compiled the first known trigonometric table, a table of chords, to model the motion of the Sun and Moon. He is often called the founder of trigonometry.

• Aryabhata (476–550 CE, India) tabulated the sine function, which he called jya, in his astronomical work the Aryabhatiya. The word "sine" itself is a mistranslation of jya that travelled through Arabic into Latin.

Where Is Cos 70 Degrees Used In The Real World?

A single cosine value like $0.3420$ rarely stars on its own, but the cosine of a steep angle shows up wherever a slanted quantity has to be split into a flat, horizontal part.

• Waves and alternating current: the voltage in a mains socket rises and falls as a cosine wave, and engineers read the value at specific phase angles, 70° among them, to find the instantaneous voltage.

• Navigation and GPS: turning a heading and a distance into east-west and north-south movement uses cosine, so a bearing near 70° contributes only a small horizontal step.

• Architecture and roofing: a rafter or ramp set at 70° covers little horizontal ground for its length, and $\cos 70^\circ$ is the factor that tells a builder exactly how little.

• Computer graphics: rotating a point or a game character by an angle multiplies its coordinates by sines and cosines, so $\cos 70^\circ$ appears inside the rotation of any object turned 70°.

• Astronomy: the brightness of sunlight on a tilted surface, or on the ground near sunrise, scales with the cosine of the angle from vertical.

The thread through all five is the same: cosine measures the horizontal, "flat" share of something tilted, and at 70° that share is small.

What Are The Most Common Mistakes With Cos 70 Degrees?

These four errors account for most wrong answers on $\cos 70^\circ$, verified against the radian-mode value that appears on calculator-reference pages and the cofunction relation that dominates competitor FAQs.

Leaving the calculator in radian mode.

Where it slips in:

A student types cos(70) while the calculator is set to radians and reads off $0.6333$, then writes that as the answer.

Don't do this:

Do not trust the display until you have checked the angle mode. The number $0.6333$ is $\cos(70\text{ radians})$, a completely different angle.

The correct way:

Set the calculator to degree mode (look for DEG) before entering 70, and confirm you get $0.3420$. If you want radians, enter $\frac{7\pi}{18}$, not 70.

Expecting a clean surd like the special angles.

Where it slips in:

A student assumes $\cos 70^\circ$ must equal something tidy such as $\frac{\sqrt{3}}{2}$, because 30°, 45°, and 60° all do.

Don't do this:

Do not force 70° into a radical. It is not a constructible angle, and no finite surd equals $\cos 70^\circ$.

The correct way:

Give the decimal $0.3420$, or the exact cofunction form $\cos 70^\circ = \sin 20^\circ$. Those are the honest exact answers.

Getting the sign wrong when 70° hides inside a bigger angle.

Where it slips in:

A student meets $\cos 110^\circ$ or $\cos 250^\circ$, correctly finds the reference angle 70°, but then keeps the value positive out of habit.

Don't do this:

Do not apply the reference angle without checking the quadrant. $\cos 110^\circ = -\cos 70^\circ = -0.3420$, because 110° sits in Quadrant II where cosine is negative.

The correct way:

Find the reference angle first, then set the sign from the CAST rule for the quadrant the full angle lives in.

Swapping the cofunction to the wrong ratio.

Where it slips in:

A student remembers a "90° minus" trick but writes $\cos 70^\circ = \cos 20^\circ$, keeping cosine on both sides.

Don't do this:

Do not keep the same function. The cofunction identity trades cosine for sine, not cosine for cosine.

The correct way:

Use $\cos 70^\circ = \sin(90^\circ - 70^\circ) = \sin 20^\circ$. Cosine of an angle equals sine of its complement.

Practice Problems On Cos 70 Degrees

Give answers to four decimal places unless told otherwise.

1. Evaluate $\cos 70^\circ$.
   (Answer: $0.3420$.)

2. Rewrite $\cos 70^\circ$ as a sine using the cofunction identity.
   (Answer: $\sin 20^\circ$.)

3. Convert 70° to radians.
   (Answer: $\frac{7\pi}{18} \approx 1.2217$.)

4. Given $\cos 70^\circ \approx 0.3420$, find $\sin 70^\circ$ using $\sin^2\theta + \cos^2\theta = 1$.
   (Answer: $\sqrt{1 - 0.3420^2} = \sqrt{0.8830} \approx 0.9397$.)

5. Find $\tan 70^\circ$ from $\dfrac{\sin 70^\circ}{\cos 70^\circ}$.
   (Answer: $\dfrac{0.9397}{0.3420} \approx 2.7475$.)

6. Which is larger, $\cos 70^\circ$ or $\cos 20^\circ$, and why?
   (Answer: $\cos 20^\circ \approx 0.9397$ is larger, because cosine decreases as the angle grows from $0^\circ$ to $90^\circ$.)

Where Should You Go Next After Cos 70 Degrees?

Cos 70 Degrees opens onto the wider toolkit of angle values and identities, and a few natural doors lead outward from here.

1. Cofunction identities. The rule behind $\cos 70^\circ = \sin 20^\circ$, generalised to every complementary pair, with its sibling page on trigonometric ratios of complementary angles.

2. The unit circle. Where every angle's cosine and sine come from, and the picture that makes signs and sizes obvious, extended in the unit circle with tangent.

3. What is a radian. Why $70^\circ = \frac{7\pi}{18}$, and how trigonometric ratios in radians keep calculators and code honest.

If your child is learning to move between degrees, radians, and the unit circle with confidence, a live Bhanzu trainer teaches these connections from the ground up in the Bhanzu trigonometry program.