Cos 47 Degrees: Value, Unit Circle And How To Find It

What Does Cos 47 Degrees Mean?

In a right triangle, the cosine of an angle is the adjacent side divided by the hypotenuse. So $\cos 47^\circ \approx 0.6820$ says that the side next to a $47^\circ$ angle is about $68.2$ as long as the hypotenuse.

On the unit circle, a circle of radius $1$ centred at the origin, cosine is the x-coordinate of the point where the angle's terminal side meets the circle. At $47^\circ$ that point is approximately $(0.6820, 0.7314)$.

A quadrant is one of the four regions the axes cut the plane into, numbered I to IV counterclockwise from the positive x-axis. The angle $47^\circ$ lies in Quadrant I, where both coordinates are positive, and it is its own reference angle (the acute angle the terminal side makes with the x-axis).

A radian measures an angle by the arc it cuts off on a unit circle. To convert degrees to radians, multiply by $\dfrac{\pi}{180}$.

$$47^\circ \times \frac{\pi}{180^\circ} = \frac{47\pi}{180} \approx 0.8203 \text{ rad}$$

Standard-Angle Cosine Reference Table

Cos 47 degrees sits just past $45^\circ$, so it is a little smaller than $\cos 45^\circ = \dfrac{\sqrt{2}}{2}$. The standard angles have exact forms; $47^\circ$ does not, which the last row makes plain.

The neighbours of $47^\circ$ show how quickly cosine falls in this range, about $0.012$ per degree.

Notice that $\cos 47^\circ$ and $\sin 43^\circ$ are the same number, $0.6820$, and that $\cos 43^\circ$ equals $\sin 47^\circ$.

Where Does Cos 47 Degrees Show Up, And Why Does It Matter?

Cos 47 degrees appears whenever something tilts at $47^\circ$ and you need its horizontal share. A $6$ m ladder leaning at $47^\circ$ to the ground reaches $6\cos 47^\circ \approx 4.09$ m out from the wall.

The same factor splits forces and velocities into components. A rope pulling at $47^\circ$ above the horizontal delivers $\cos 47^\circ \approx 68.2$ of its tension sideways, which is the kind of calculation heights and distances problems are built on.

• Surveying and slopes: the horizontal run of a $47^\circ$ incline is $0.682$ times its slope length.

• Navigation: a course $47^\circ$ off due east moves you east at $0.682$ of your speed.

• Checking methods: because $47^\circ$ has no tidy exact value, it is a fair check of how well a series or an approximation works.

What Is The Value Of Cos 47 Degrees?

The value of cos 47 degrees is $0.68199836$ to eight decimal places, which rounds to $0.6820$. To three decimals it is $0.682$.

This number is a decimal approximation of an irrational number. Its digits never end or repeat, so every written decimal is rounded, and the exact value is best written as $\cos 47^\circ$ itself.

Is There An Exact Value Of Cos 47 Degrees?

There is no exact value of cos 47 degrees built from whole numbers and real square roots, the way $\cos 45^\circ = \dfrac{\sqrt{2}}{2}$ is. A whole-number angle in degrees has a cosine expressible in real square roots only when the angle is a multiple of $3^\circ$, and $47$ is not.

This is a theorem about constructible angles, summarised in the list of trigonometric constants expressed in real radicals. For angles like $1^\circ$ or $47^\circ$, any radical formula needs cube roots of complex numbers, which is no help for computing a decimal.

There is a precise algebraic fact behind this. $\cos 47^\circ$ is a root of a polynomial with whole-number coefficients, but the smallest such polynomial has degree $48$, far beyond anything you would solve by hand.

What you can write exactly is a link to other angles. Splitting $47^\circ = 45^\circ + 2^\circ$ with the sum and difference formulas gives an exact identity, not a simpler number.

$$\cos 47^\circ = \cos 45^\circ \cos 2^\circ - \sin 45^\circ \sin 2^\circ$$

$$= \frac{\sqrt{2}}{2}\left(\cos 2^\circ - \sin 2^\circ\right)$$

With cos 2 degrees $\approx 0.99939083$ and sin 2 degrees $\approx 0.03489950$, this gives $0.70710678 \times 0.96449133 \approx 0.68199836$. The value of cos 47 matters as a calculator and approximation skill, not as a value to memorise.

Why Is Cos 47 Degrees Equal To Sin 43 Degrees?

Cos 47 degrees equals $\sin 43^\circ$ because $47^\circ$ and $43^\circ$ are complementary: they add to $90^\circ$. In a right triangle, the side adjacent to one acute angle is the side opposite the other, so the two ratios are the same fraction.

$$\cos 47^\circ = \sin(90^\circ - 47^\circ) = \sin 43^\circ \approx 0.6820$$

This is the co-function identity, covered in full under trigonometric ratios of complementary angles. The partner pair works the other way too: $\cos 43^\circ = \sin 47^\circ \approx 0.7314$, which is the value on the sin 47 degrees page.

How Do You Find Cos 47?

Find cos 47 degrees with a calculator in degree mode, then check it with an identity or a series. The methods below run from quickest to most revealing.

Method 1: Calculator In Degree Mode

Set the calculator to degrees (DEG on the screen) and enter $\cos(47)$. It returns $0.6819983600\ldots$, which rounds to $0.6820$.

In radian mode the same keystrokes compute the cosine of $47$ radians, about $-0.9923$. A negative answer is the giveaway, because Quadrant I cosines are always positive.

Method 2: Read It From The Unit Circle

Rotate a radius $47^\circ$ counterclockwise from the positive x-axis. The x-coordinate of its endpoint is $\cos 47^\circ \approx 0.6820$, slightly left of the $45^\circ$ point at $0.7071$.

Method 3: Use A Co-Function Value You Already Have

If a table lists $\sin 43^\circ \approx 0.6820$, you already have $\cos 47^\circ$.

$$\cos 47^\circ = \sin 43^\circ \approx 0.6820$$

Method 4: Sum The Maclaurin Series

Calculators do not look values up in a giant table. Many pocket calculators use an algorithm called CORDIC, which builds the angle out of tiny rotations, while computers usually reduce the angle and evaluate a polynomial.

The series for cosine, listed in the NIST Digital Library of Mathematical Functions, shows how a polynomial can do the job. It needs the angle in radians, $x = \dfrac{47\pi}{180} \approx 0.82030475$.

$$\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \frac{x^8}{40320} - \dots$$

Four terms already give $0.6820$ correct to four places. Each new term shrinks the error by roughly a factor of $100$ here, because $x$ is less than $1$.

Method 5: Approximate From $45^\circ$ With A Tangent Line

Near a known angle, cosine changes almost in a straight line. Starting at $45^\circ$ and stepping $h = 2^\circ = 0.0349066$ rad, the slope of cosine is $-\sin 45^\circ$.

$$\cos(45^\circ + h) \approx \frac{\sqrt{2}}{2}(1 - h) = 0.70710678 \times 0.96509340 \approx 0.68242$$

That is high by $0.00043$, so it is good to three decimal places. Adding the next correction, $-\dfrac{\sqrt{2}}{2} \cdot \dfrac{h^2}{2}$, gives $0.681993$, within $0.000005$ of the true value.

Method 6: Interpolate Between Table Values

From cos 45 degrees $\approx 0.7071$ and cos 50 degrees $\approx 0.6428$, go two-fifths of the way down.

$$0.7071 - \frac{2}{5}(0.7071 - 0.6428) \approx 0.6814$$

The answer is about $0.0006$ too low, because the cosine curve bends between the two table points. This is how people used printed trigonometric tables before calculators.

Sec 47 And The Other Functions

The other five ratios follow from $\cos 47^\circ$ and $\sin 47^\circ$. Because $47^\circ$ is in Quadrant I, every sign is positive.

Some pages write a $\pm$ in front of the square-root forms. For $47^\circ$ the plus sign is the only correct choice, since cosine is positive in Quadrant I.

Which Angles Have The Same Cosine As 47 Degrees?

Cosine depends only on the x-coordinate on the unit circle, so several angles share the value $0.6820$. Cosine is an even function, $\cos(-\theta) = \cos\theta$, and it repeats every $360^\circ$.

• $\cos(-47^\circ) = 0.6820$, a reflection in the x-axis.

• $\cos 313^\circ = 0.6820$, the same point reached the long way round, since $313^\circ = 360^\circ - 47^\circ$.

• $\cos 407^\circ = 0.6820$, one full turn more; such angles are coterminal angles.

• $\cos 133^\circ = -0.6820$ and $\cos 227^\circ = -0.6820$, the same size with the sign flipped in Quadrants II and III.

Examples Of Cos 47 Degrees

Example 1

A right triangle has a hypotenuse of $10$ cm and an angle of $47^\circ$. Find the side adjacent to that angle.

$$\cos 47^\circ = \frac{\text{adjacent}}{10}$$

$$\text{adjacent} = 10 \times \cos 47^\circ$$

$$= 10 \times 0.68200 \approx 6.82 \text{ cm}$$

Final answer: about $6.82$ cm

Example 2

If $\cos 47^\circ + \sin 47^\circ = k$, express $\cos^2 47^\circ - \sin^2 47^\circ$ in terms of $k$.

Factor the difference of squares and square the sum.

$$\cos^2 47^\circ - \sin^2 47^\circ = (\cos 47^\circ + \sin 47^\circ)(\cos 47^\circ - \sin 47^\circ)$$

$$(\cos 47^\circ - \sin 47^\circ)^2 = 1 - 2\sin 47^\circ \cos 47^\circ = 2 - k^2$$

The tempting next line takes the positive root, $\cos 47^\circ - \sin 47^\circ = \sqrt{2 - k^2}$, which makes the answer $k\sqrt{2 - k^2}$, a positive number.

That cannot be right. The expression equals $\cos 94^\circ$ by the double-angle formula, and $94^\circ$ is in Quadrant II, where cosine is negative.

The fix is to check the sign of the bracket. Past $45^\circ$ cosine is smaller than sine, so $\cos 47^\circ - \sin 47^\circ$ is negative and takes the negative root.

$$\cos 47^\circ - \sin 47^\circ = -\sqrt{2 - k^2}$$

$$\cos^2 47^\circ - \sin^2 47^\circ = -k\sqrt{2 - k^2}$$

Final answer: $-k\sqrt{2 - k^2}$, which is about $-0.0698$ for $k \approx 1.4134$, matching $\cos 94^\circ \approx -0.0698$.

Example 3

Evaluate $\dfrac{22\cos 47^\circ}{\sin(-43^\circ)}$.

Sine is odd, so $\sin(-43^\circ) = -\sin 43^\circ$, and $\sin 43^\circ = \cos 47^\circ$.

$$\frac{22\cos 47^\circ}{\sin(-43^\circ)} = \frac{22\cos 47^\circ}{-\sin 43^\circ} = \frac{22\cos 47^\circ}{-\cos 47^\circ}$$

$$= -22$$

Final answer: $-22$, exactly, with no decimals needed.

Example 4

Find the exact value of $\cos^2 73^\circ + \cos^2 47^\circ + \cos 73^\circ \cos 47^\circ$.

The angles add to $120^\circ$, which is the key. Use two identities with $A = 73^\circ$ and $B = 47^\circ$.

$$\cos^2 A + \cos^2 B = 1 + \cos(A + B)\cos(A - B)$$

$$\cos A \cos B = \frac{\cos(A + B) + \cos(A - B)}{2}$$

Substitute $\cos(A + B) = \cos 120^\circ = -\dfrac{1}{2}$ and write $c = \cos 26^\circ$.

$$1 - \frac{c}{2} + \frac{-\frac{1}{2} + c}{2} = 1 - \frac{c}{2} - \frac{1}{4} + \frac{c}{2}$$

$$= \frac{3}{4}$$

Final answer: $\dfrac{3}{4}$, even though neither $\cos 73^\circ$ nor $\cos 47^\circ$ has a simple exact form.

Example 5

A ramp's horizontal run is $0.682$ times its length along the slope. At what angle does it rise?

Inverse cosine, written $\cos^{-1}$, gives the angle in $[0^\circ, 180^\circ]$ whose cosine is a given number. Here the run over the length is the cosine of the angle.

$$\theta = \cos^{-1}(0.682) \approx 47.0^\circ$$

Final answer: about $47^\circ$, which is the reverse of finding cos 47 degrees.

Example 6

Estimate $\cos 47^\circ$ without a calculator, using only $\cos 45^\circ = \dfrac{\sqrt{2}}{2}$, and state the error.

Convert the step to radians: $h = 2^\circ = \dfrac{2\pi}{180} \approx 0.034907$.

$$\cos 47^\circ \approx \frac{\sqrt{2}}{2}(1 - h)$$

$$\approx 0.707107 \times 0.965093 \approx 0.68242$$

The true value is $0.68200$, so the estimate is $0.00043$ too high.

Final answer: about $0.682$, correct to three decimal places.

What Are The Most Common Mistakes With Cos 47 Degrees?

Most errors on cos 47 degrees come from the calculator or from mixing up the co-function partner. The value itself is never the hard part.

Mistake 1: Pairing Cos 47 With The Wrong Sine

Where it slips in: Using the co-function identity in a hurry.

Don't do this: Write $\cos 47^\circ = \sin 47^\circ$ because both angles are "near $45^\circ$", or write $\cos 47^\circ = \cos 43^\circ$.

The correct way: Subtract from $90^\circ$ and switch the function: $\cos 47^\circ = \sin 43^\circ \approx 0.6820$, while $\sin 47^\circ \approx 0.7314$.

The first instinct with any angle close to $45^\circ$ is to assume sine and cosine are equal, because at exactly $45^\circ$ they are. Checking which is larger (past $45^\circ$, sine wins) catches the swap before it spreads into later steps.

Mistake 2: Leaving The Calculator In Radian Mode

Where it slips in: The rusher who types $\cos(47)$ without looking at the mode indicator.

Don't do this: Accept $\cos 47 \approx -0.9923$. That is the cosine of $47$ radians, an angle of more than seven full turns.

The correct way: Switch to degree mode, or enter $\cos\left(\dfrac{47\pi}{180}\right)$ in radian mode. Both give $0.6820$, and a positive value is the quick check for a Quadrant I angle.

Mistake 3: Using The Small-Angle Formula Too Far Out

Where it slips in: Approximating without a calculator, using $\cos\theta \approx 1 - \dfrac{\theta^2}{2}$.

Don't do this: Apply it at $47^\circ$, which gives $1 - \dfrac{0.8203^2}{2} \approx 0.6636$, off by about $0.018$.

The correct way: Keep that formula for small angles (it is accurate to about $0.001$ only up to roughly $10^\circ$). For $47^\circ$, expand around $45^\circ$ instead, as in Example 6, or keep more series terms.

Students who check every approximation against one known value before trusting it stop shipping answers like $0.66$ for this angle. The memorizer who learned "$1 - \dfrac{\theta^2}{2}$" as a universal shortcut is the one this mistake catches.

Where Can You Get Help With Trigonometry Values Like Cos 47 Degrees?

Non-special values like cos 47 degrees reward understanding over memory: knowing which identity to reach for, and how to check a calculator answer against the unit circle. A trigonometry tutor or a high school math tutor can practise exactly that habit with you, one problem at a time.

For a broader routine across topics, math classes online and math tutoring cover approximations, identities and the unit circle in a structured order.

Key Takeaways

• Cos 47 degrees is approximately $0.6820$ ($0.68199836$), positive because $47^\circ$ is in Quadrant I.

• In radians the angle is $\dfrac{47\pi}{180} \approx 0.8203$.

• It has no simple surd form, since $47$ is not a multiple of $3$, but it equals $\sin 43^\circ$ and $\dfrac{\sqrt{2}}{2}(\cos 2^\circ - \sin 2^\circ)$ exactly.

• A calculator in degree mode, four Maclaurin terms, or a tangent line from $45^\circ$ all reach the value or come within $0.0005$ of it.

• The common errors are pairing it with $\sin 47^\circ$, radian mode ($-0.9923$), and stretching the small-angle formula past about $10^\circ$.

Practice These Before Moving On

1. Evaluate $\dfrac{3\cos 47^\circ}{\sin 43^\circ}$.

2. Find $\cos 313^\circ$ to four decimal places.

3. Estimate $\cos 46^\circ$ with a tangent line from $45^\circ$, then compare with a calculator.

4. Given $\sec 47^\circ \approx 1.4663$, find $\cos 47^\circ$.

5. Find $\cos(-47^\circ)$ and $\cos 133^\circ$.

Answers:

1. $3$, since $\sin 43^\circ = \cos 47^\circ$

2. $0.6820$, because $313^\circ = 360^\circ - 47^\circ$

3. $0.7071 \times (1 - 0.017453) \approx 0.6948$; the calculator gives $0.6947$

4. $\dfrac{1}{1.4663} \approx 0.6820$

5. $0.6820$ and $-0.6820$

If question 1 did not come out as a whole number, go back to the co-function section: the two values cancel exactly.

Practice these to solidify your understanding, and if you would like a teacher to check your working live, book a free demo class.

Read More

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