Cos 24 Degrees: Exact Value, Decimal, Unit Circle And Examples

What Does Cos 24 Degrees Mean?

In a right triangle, the cosine of an acute angle is the side adjacent to that angle divided by the hypotenuse. So $\cos 24^\circ \approx 0.9135$ says the side next to a $24^\circ$ angle is about $91.35$ as long as the hypotenuse, whatever the triangle's size.

On the unit circle, a circle of radius $1$ centred at the origin, the cosine of an angle is the $x$-coordinate of the point the angle reaches. The point at $24^\circ$ is $(0.9135, 0.4067)$, so the two definitions agree: the triangle's hypotenuse is the radius $1$.

A quadrant is one of the four regions the axes cut the plane into, numbered I to IV counterclockwise from the top right. Angles between $0^\circ$ and $90^\circ$ lie in Quadrant I, where $x$ is positive, so $\cos 24^\circ$ is positive.

The reference angle of an angle is the acute angle its terminal side makes with the $x$-axis. For $24^\circ$ the reference angle is $24^\circ$ itself, and angles such as $156^\circ$, $204^\circ$ and $336^\circ$ share it, which is why their cosines are $\pm 0.9135$.

What is cos 24 degrees in radians? Multiply by $\frac{\pi}{180}$: $24 \times \frac{\pi}{180} = \frac{2\pi}{15} \approx 0.4189$ rad, so $\cos 24^\circ = \cos\left(\frac{2\pi}{15}\right)$ (see converting degrees to radians). The angle changes its unit; the cosine does not change at all.

Going backwards, the principal value of $\arccos$ is the one answer it returns in the range $0$ to $\pi$ (that is, $0^\circ$ to $180^\circ$). Since $24^\circ$ lies in that range, $\arccos(0.9135454576) = 24^\circ$ exactly.

Standard-Angle Cosine Reference Table

Twenty-four degrees sits between two famous angles, $15^\circ$ and $30^\circ$, and every row below has an exact radical form. The table runs through the first quadrant so you can see where $\cos 24^\circ$ falls.

Cosine falls as the angle opens, so $\cos 24^\circ$ must land between $\cos 15^\circ \approx 0.9659$ and cos 30 degrees, which is $0.8660$. The value $0.9135$ fits that window, and it is a quick check on any calculator answer.

Where Does Cos 24 Degrees Show Up, And Why Does It Matter?

Twenty-four degrees is the central angle of a regular 15-sided polygon (a pentadecagon), because $360^\circ \div 15 = 24^\circ$. Every side of a 15-gon inscribed in a circle of radius $r$ has length $r\sqrt{2 - 2\cos 24^\circ} \approx 0.4158r$.

That 15-gon is built from a triangle and a pentagon drawn in the same circle, which is exactly why the exact value of $\cos 24^\circ$ mixes $\sqrt{3}$ (from the triangle) with $\sqrt{5}$ (from the pentagon). Outside geometry, any force or velocity at $24^\circ$ to the horizontal has a horizontal part equal to the full amount times $0.9135$.

Can Cos 24 Degrees Be Written As A Fraction?

No. Cos 24 degrees is an irrational number, so no fraction of two whole numbers equals it, and $0.9135$ is only a rounded decimal.

Here is the proof sketch. Using $\cos 72^\circ = 4\cos^3 24^\circ - 3\cos 24^\circ$ and the golden-ratio value of $\cos 72^\circ$, the number $c = \cos 24^\circ$ turns out to satisfy

$$16c^4 - 8c^3 - 16c^2 + 8c + 1 = 0.$$

By the rational root theorem, a fraction root would have to be one of $\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm \frac{1}{16}$. Substituting each one leaves a non-zero remainder, so $\cos 24^\circ$ has no fraction form; the closest honest answer is the surd above.

How Do You Find The Exact Value Of Cos 24 Degrees?

The cleanest route writes $24^\circ = 60^\circ - 36^\circ$ and uses two angles whose cosines are already exact. Every other method below either repackages that idea or checks the number.

Step Zero: Why Is $\cos 36^\circ$ Equal To $\frac{1 + \sqrt{5}}{4}$?

Take an isosceles triangle $ABC$ with apex $\angle A = 36^\circ$, base angles $\angle B = \angle C = 72^\circ$, legs $AB = AC = 1$ and base $BC = x$. Bisect $\angle B$ and let the bisector meet $AC$ at $D$.

Triangle $BCD$ has angles $36^\circ$, $72^\circ$, $72^\circ$, so it is similar to $ABC$ and $BD = BC = x$. Triangle $ABD$ has two $36^\circ$ angles, so $AD = BD = x$ and $DC = 1 - x$.

Similarity gives the proportion

$$\frac{x}{1} = \frac{1 - x}{x}$$

$$x^2 + x - 1 = 0$$

$$x = \frac{\sqrt{5} - 1}{2} \approx 0.6180$$

Now drop a perpendicular from $D$ to $AB$ in the isosceles triangle $ABD$. It lands at the midpoint of $AB$, so

$$\cos 36^\circ = \frac{\frac{1}{2}}{x} = \frac{1}{\sqrt{5} - 1} = \frac{\sqrt{5} + 1}{4} \approx 0.8090$$

From $\sin^2 36^\circ = 1 - \cos^2 36^\circ$ we also get $\sin 36^\circ = \frac{\sqrt{10 - 2\sqrt{5}}}{4} \approx 0.5878$.

Method 1: Subtract $36^\circ$ From $60^\circ$

The cos(A − B) formula says $\cos(A - B) = \cos A\cos B + \sin A\sin B$. With $A = 60^\circ$ and $B = 36^\circ$:

$$\cos 24^\circ = \cos 60^\circ\cos 36^\circ + \sin 60^\circ\sin 36^\circ$$

$$= \frac{1}{2} \times \frac{1 + \sqrt{5}}{4} + \frac{\sqrt{3}}{2} \times \frac{\sqrt{10 - 2\sqrt{5}}}{4}$$

$$= \frac{1 + \sqrt{5}}{8} + \frac{\sqrt{30 - 6\sqrt{5}}}{8}$$

$$= \frac{1 + \sqrt{5} + \sqrt{30 - 6\sqrt{5}}}{8} \approx 0.9135454576$$

The last step uses $\sqrt{3} \times \sqrt{10 - 2\sqrt{5}} = \sqrt{30 - 6\sqrt{5}}$. This is the form Wolfram MathWorld lists for $\cos\left(\frac{2\pi}{15}\right)$.

Method 2: Subtract $30^\circ$ From $54^\circ$

The same answer appears from $24^\circ = 54^\circ - 30^\circ$, because $\cos 54^\circ = \sin 36^\circ$ and $\sin 54^\circ = \cos 36^\circ$:

$$\cos 24^\circ = \cos 54^\circ\cos 30^\circ + \sin 54^\circ\sin 30^\circ = \frac{\sqrt{10 - 2\sqrt{5}}}{4} \times \frac{\sqrt{3}}{2} + \frac{1 + \sqrt{5}}{4} \times \frac{1}{2}$$

Each term matches a term from Method 1, so the two routes agree to every digit.

Method 3: Use The Co-Function Rule

The cofunction identities say $\cos\theta = \sin(90^\circ - \theta)$. So $\cos 24^\circ = \sin 66^\circ$, and any sine table entry for $66^\circ$ gives $0.9135$ as well.

Method 4: Read It Off The Unit Circle

Rotate the radius $24^\circ$ counterclockwise from the positive $x$-axis. The endpoint is $(\cos 24^\circ, \sin 24^\circ) \approx (0.9135, 0.4067)$, and the $x$-coordinate is the answer.

The same point is reached by $384^\circ$ and $-336^\circ$ (coterminal angles differ by $360^\circ$), and cosine is even, so $\cos(-24^\circ) = 0.9135$ too.

Method 5: Double-Angle And Half-Angle Checks

The double angle formula $\cos 2\theta = 2\cos^2\theta - 1$ with $\theta = 12^\circ$ returns $2\cos^2 12^\circ - 1 \approx 0.9135$. Going the other way, the half-angle form $\cos 24^\circ = \sqrt{\frac{1 + \cos 48^\circ}{2}}$ also returns $0.9135$.

The cos 3x identity gives a third check: $4\cos^3 24^\circ - 3\cos 24^\circ = \cos 72^\circ = \frac{\sqrt{5} - 1}{4} \approx 0.3090$.

Method 6: Calculator And Taylor Series

A calculator in degree mode returns $0.9135454576$. Inside, the angle is first converted to radians, $x = \frac{2\pi}{15} \approx 0.418879$, and then a series like $\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots$ (or an equivalent routine) is summed.

Four terms already give eight correct decimal places, because $0.4189$ rad is a small number to raise to high powers.

How Accurate Are Quick Approximations Of Cos 24 Degrees?

Shortcut estimates of $\cos 24^\circ$ land close but not on the value, and the size of the miss matters. One search-engine summary line during our research quoted $0.9184$, which is exactly the linear estimate from $30^\circ$, not the true value.

The small-angle rule $\cos x \approx 1 - \frac{x^2}{2}$ is reliable below about $10^\circ$ to $15^\circ$; at $24^\circ$ it already misses in the third decimal place. For an answer to 4 dp, use the exact form or the calculator, not a shortcut.

What Is Cos 24 Degrees In Terms Of Other Trig Functions?

Every trig function of $24^\circ$ can be rewritten as $\cos 24^\circ$, which helps when a problem hands you a different ratio. The table uses $\sin 24^\circ \approx 0.4067$ and $\tan 24^\circ \approx 0.4452$.

The autosuggest pairing "cos 24 sin 66" is the first row: the two are equal because $24^\circ + 66^\circ = 90^\circ$. The square-root rows take the positive root because $24^\circ$ is in Quadrant I.

Examples Of Cos 24 Degrees

Example 1

Evaluate $1 - \sin^2 24^\circ$.

By the Pythagorean identity, $1 - \sin^2 24^\circ = \cos^2 24^\circ$.

$$\cos^2 24^\circ = (0.9135454576)^2 \approx 0.8346$$

Example 2

Find $\cos 24^\circ$ by writing $24^\circ = 60^\circ - 36^\circ$.

Wrong attempt: split the cosine across the subtraction.

$$\cos 60^\circ - \cos 36^\circ = 0.5 - 0.8090 = -0.3090$$

That breaks at once. An angle of $24^\circ$ lies in Quadrant I, so its cosine must be positive, and a negative answer is impossible.

Correct: cosine does not distribute over subtraction. Use the difference formula instead.

$$\cos 24^\circ = \cos 60^\circ\cos 36^\circ + \sin 60^\circ\sin 36^\circ$$

$$= 0.5 \times 0.8090 + 0.8660 \times 0.5878 \approx 0.9135$$

Example 3

Simplify $\frac{3\cos 24^\circ}{\sin 114^\circ}$.

Since $114^\circ = 90^\circ + 24^\circ$, $\sin 114^\circ = \cos 24^\circ$.

$$\frac{3\cos 24^\circ}{\sin 114^\circ} = \frac{3\cos 24^\circ}{\cos 24^\circ} = 3$$

Example 4

A 6 m ladder leans against a wall and makes $24^\circ$ with the ground. How far is its foot from the wall?

The ladder is the hypotenuse and the ground distance is adjacent to the $24^\circ$ angle.

$$\text{distance} = 6 \times \cos 24^\circ$$

$$= 6 \times 0.9135454576 \approx 5.48 \text{ m}$$

Example 5

Prove that $\cos 24^\circ + \cos 55^\circ + \cos 125^\circ + \cos 204^\circ + \cos 300^\circ = \frac{1}{2}$.

Pair each angle with its reference angle.

$$\cos 125^\circ = \cos(180^\circ - 55^\circ) = -\cos 55^\circ$$

$$\cos 204^\circ = \cos(180^\circ + 24^\circ) = -\cos 24^\circ$$

$$\cos 300^\circ = \cos(360^\circ - 60^\circ) = \cos 60^\circ = \frac{1}{2}$$

The first four terms cancel in pairs, which leaves $\frac{1}{2}$.

Example 6

Show that $\cos 12^\circ \cos 24^\circ \cos 48^\circ \cos 84^\circ = \frac{1}{16}$.

Use the identity $\cos\theta \cos(60^\circ - \theta)\cos(60^\circ + \theta) = \frac{\cos 3\theta}{4}$ twice.

With $\theta = 12^\circ$: $\cos 12^\circ \cos 48^\circ \cos 72^\circ = \frac{\cos 36^\circ}{4}$.

With $\theta = 24^\circ$: $\cos 24^\circ \cos 36^\circ \cos 84^\circ = \frac{\cos 72^\circ}{4}$.

Multiply the two lines and divide both sides by $\cos 36^\circ \cos 72^\circ$:

$$\cos 12^\circ \cos 24^\circ \cos 48^\circ \cos 84^\circ = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$$

A calculator confirms $0.0625$. The rusher who multiplies four rounded decimals usually lands on $0.0624$ or $0.0626$ and then doubts the identity; keeping the cosines symbolic until the last line is what makes the clean $\frac{1}{16}$ visible.

Where Students Trip Up On Cos 24 Degrees?

Mistake 1: Calculator Left In Radian Mode

Where it slips in: Typing cos(24) on a phone or a calculator that defaults to radians.

Don't do this: Accept $\cos(24) \approx 0.4242$. That is the cosine of 24 radians (about $1375^\circ$), which is the number Google's search suggestions return for "is cos 24".

The correct way: Switch to degree mode, or enter $\cos\left(\frac{2\pi}{15}\right)$ in radian mode, and expect $0.9135$. The reference table gives an instant check: the answer must sit between $0.8660$ and $0.9659$.

Mistake 2: Writing A Rounded Decimal As The Exact Value

Where it slips in: Questions that say "find the exact value of cos 24 degrees".

Don't do this: Write $\cos 24^\circ = 0.9135$, or worse, $\cos 24^\circ = \frac{9135}{10000}$.

The correct way: Give $\frac{1 + \sqrt{5} + \sqrt{30 - 6\sqrt{5}}}{8}$ and add $\approx 0.9135$ only as an approximation. The memorizer who has only met $30^\circ$, $45^\circ$ and $60^\circ$ tends to say "24 has no exact value"; it does, it is just longer.

Mistake 3: Losing The Sign On Related Angles

Where it slips in: Angles that share the $24^\circ$ reference angle: $156^\circ$, $204^\circ$, $336^\circ$.

Don't do this: Copy $0.9135$ for all of them.

The correct way: Find the quadrant first, then attach the sign: $\cos 156^\circ = \cos 204^\circ = -0.9135$ (Quadrants II and III), while $\cos 336^\circ = +0.9135$ (Quadrant IV). Students who skip the quadrant step get the size right and the sign wrong, and writing the quadrant number beside the angle before touching the calculator fixes it.

Where Can You Get Help With Trigonometry Values?

You get help fastest from someone who can watch you work a value like $\cos 24^\circ$ and spot exactly which step breaks, whether it is the golden triangle, the difference formula or the sign. A trigonometry tutor can take you from special angles to derived ones like this, and a high school math tutor can tie it to the rest of your course.

If you prefer a structured group setting, math classes online cover trigonometry in order, and one-to-one math tutoring suits students who need a specific gap closed.

Key Takeaways

• Cos 24 degrees equals $\frac{1 + \sqrt{5} + \sqrt{30 - 6\sqrt{5}}}{8} \approx 0.9135$, and in radians it is $\cos\left(\frac{2\pi}{15}\right)$.

• The exact value comes from $\cos(60^\circ - 36^\circ)$, with $\cos 36^\circ = \frac{1 + \sqrt{5}}{4}$ from the golden triangle.

• It is irrational, so any fraction such as $\frac{9135}{10000}$ is only an approximation.

• $\cos 24^\circ = \sin 66^\circ$, and $\cos 204^\circ = \cos 156^\circ = -\cos 24^\circ$.

• A calculator answer of $0.4242$ means radian mode; the degree value sits between $\cos 30^\circ$ and $\cos 15^\circ$.

Practice These Before Moving On

1. Evaluate $4\cos 24^\circ$ to four decimal places.

2. Write $\cos 24^\circ$ as the sine of another acute angle.

3. Find $\cos 156^\circ$ and $\cos 336^\circ$ using the reference angle $24^\circ$.

4. A 12 m slide makes $24^\circ$ with the ground. Find its horizontal length to two decimal places.

5. Use $\cos 2\theta = 2\cos^2\theta - 1$ with $\theta = 24^\circ$ to find $\cos 48^\circ$ to four decimal places.

Answers: (1) $3.6542$. (2) $\sin 66^\circ$. (3) $\cos 156^\circ \approx -0.9135$ and $\cos 336^\circ \approx 0.9135$. (4) $12 \times 0.9135454576 \approx 10.96$ m. (5) $2 \times 0.8345653 - 1 \approx 0.6691$.

If question 3 tripped you up, go back to Mistake 3 and name the quadrant before you write the number.

Book a free demo class to work through derived trig values like this one with a live teacher.

Read More

• Cos 15 Degrees

• Cos 20 Degrees

• Cos 25 Degrees

• Trigonometric Table

• Pythagorean Identities