What Cos 15 Degrees Means
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 where the angle's radius meets the circle. Rotating $15°$ counterclockwise from the positive $x$-axis lands a point in Quadrant I, at $\left(\cos 15°, \sin 15°\right)$, whose horizontal distance from the origin is $\cos 15° = \frac{\sqrt{6}+\sqrt{2}}{4}$.
The right-triangle definition — adjacent over hypotenuse — also applies because $15°$ is acute. But $15°$ is not an angle whose ratio you can read off a $30$-$60$-$90$ or $45$-$45$-$90$ triangle, so the value must be built from angles you already know rather than looked up.
How to Find the Value of Cos 15 Degrees
The direct route writes $15°$ as the difference of two standard angles. Readers usually arrive with one question.
Difference formula or half-angle formula? Both reach the same value. The difference formula reads $15°$ as $45° - 30°$ and is the more direct; the half-angle formula reads $15°$ as half of $30°$ and suits angles that are naturally half of a known one. This article uses the difference formula.
Method 1: The 45° − 30° difference formula
The cosine difference identity is
$$\cos(A - B) = \cos A\cos B + \sin A\sin B.$$
Note the sign: cosine of a difference uses a plus, where sine of a difference used a minus. Set $A = 45°$, $B = 30°$, and substitute $\cos 45° = \frac{\sqrt{2}}{2}$, $\cos 30° = \frac{\sqrt{3}}{2}$, $\sin 45° = \frac{\sqrt{2}}{2}$, $\sin 30° = \frac{1}{2}$:
$$\cos 15° = \cos(45° - 30°) = \cos 45°\cos 30° + \sin 45°\sin 30°$$
$$\cos 15° = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2}\cdot\frac{1}{2}$$
$$\cos 15° = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6}+\sqrt{2}}{4}.$$
Final answer: $\cos 15° = \dfrac{\sqrt{6}+\sqrt{2}}{4} \approx 0.9659.$
Method 2: From the unit circle (the check)
Mark $15°$ and measure the horizontal distance from the origin to the foot of the point — that is the cosine. It reads about $0.9659$, which matches the surd once you substitute $\sqrt{6} \approx 2.449$ and $\sqrt{2} \approx 1.414$:
$$\frac{2.449 + 1.414}{4} = \frac{3.863}{4} \approx 0.9659.$$
Examples of Cos 15 Degrees
Example 1
Evaluate $4\cos 15°$.
$$4\cos 15° = 4\cdot\frac{\sqrt{6}+\sqrt{2}}{4} = \sqrt{6}+\sqrt{2} \approx 3.863.$$
Example 2
Find $\cos 15°$ using the difference form $\cos(45° - 30°)$, but watch the operation sign.
The tempting move is to copy the sine rule and subtract the products. Try it: $\cos 45°\cos 30° - \sin 45°\sin 30° = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6}-\sqrt{2}}{4} \approx 0.2588$. That cannot be cos 15° — it is far too small for an angle so close to $0°$, where cosine should be near $1$. The cosine difference formula uses a plus:
$$\cos 15° = \cos 45°\cos 30° + \sin 45°\sin 30° = \frac{\sqrt{6}+\sqrt{2}}{4} \approx 0.9659.$$
The $0.2588$ that the wrong sign produced is actually $\sin 15°$, a useful tell that the sign was flipped.
Example 3
Show that $\cos 15° - \sin 15° = \dfrac{\sqrt{2}}{2}$, given $\sin 15° = \dfrac{\sqrt{6}-\sqrt{2}}{4}$.
$$\frac{\sqrt{6}+\sqrt{2}}{4} - \frac{\sqrt{6}-\sqrt{2}}{4} = \frac{2\sqrt{2}}{4} = \frac{\sqrt{2}}{2}.$$
The $\sqrt{6}$ terms cancel — the mirror of the sin 15 degrees sum, where the $\sqrt{2}$ terms cancelled instead.
Example 4
Verify $\cos^2 15° + \sin^2 15° = 1$.
$$\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)^2 + \left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)^2 = \frac{(8+2\sqrt{12}) + (8-2\sqrt{12})}{16} = \frac{16}{16} = 1.$$
The Pythagorean identity holds: the cross terms cancel.
Example 5
Express $15°$ in radians and state the value.
$15° = 15 \times \frac{\pi}{180} = \frac{\pi}{12}$ radians, so $\cos\left(\frac{\pi}{12}\right) = \frac{\sqrt{6}+\sqrt{2}}{4}$. Converting between radians and degrees does not change the value.
Common Mistakes With Cos 15 Degrees
Mistake 1: Using the minus sign from the sine formula
Where it slips in: The cosine difference formula sits one line below the sine one, and the signs are opposite.
Don't do this: Writing $\cos(45° - 30°) = \cos 45°\cos 30° - \sin 45°\sin 30°$.
The correct way: Cosine of a difference uses a plus: $\cos(A-B) = \cos A\cos B + \sin A\sin B$. The reliable habit is a magnitude check — cos 15° is for an angle near $0°$, so the answer must be close to $1$; a result near $0.26$ means the sign was wrong.
Mistake 2: Splitting the cosine of a difference
Where it slips in: Reaching for $\cos 45° - \cos 30°$ as if cosine were linear.
Don't do this: $\cos 15° = \cos 45° - \cos 30° = \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \approx -0.159$.
The correct way: Cosine does not distribute over subtraction, and a negative answer for a Quadrant I angle is impossible. Apply the full identity. The learner meeting compound angles for the first time often subtracts the function values directly — the negative result is the giveaway.
Mistake 3: Leaving the answer as a decimal
Where it slips in: A calculator returns $0.9659$, but the question asks for the exact value.
Don't do this: Writing $\cos 15° = 0.9659$ when "exact" is required.
The correct way: Keep $\frac{\sqrt{6}+\sqrt{2}}{4}$. The decimal is rounded; the surd is exact and is what the question is checking.
Key Takeaways
Cos 15 degrees equals the exact surd $\frac{\sqrt{6}+\sqrt{2}}{4}$, about $0.9659$.
It is built by writing $15°$ as $45° - 30°$ and applying the cosine difference formula — note the plus sign.
The angle sits in Quadrant I on the unit circle, so the value is positive and close to one.
The most common error is borrowing the minus sign from the sine difference formula.
Practice These Before Moving On
Derive $\sin 15°$ from the $45° - 30°$ split and confirm $\cos^2 15° + \sin^2 15° = 1$.
Use $\cos 2\theta = 2\cos^2\theta - 1$ with $\theta = 15°$ to recover $\cos 30° = \frac{\sqrt{3}}{2}$.
Convert $15°$ to radians and write the value as $\cos\left(\frac{\pi}{12}\right)$.
To work through compound-angle cosines with a teacher, Bhanzu's trigonometry tutor and high school math tutor sessions go through the difference and double-angle formulas step by step, with math classes online for live practice.
Read More
Cos 2x formula — the double-angle identity that links cos 15° to cos 30°.
Cosine formula — the cosine ratio and its rules.
Cos 30 degrees — one of the standard angles cos 15° is built from.
Sum and difference identities — the full compound-angle toolkit.
Trigonometric table — every standard-angle value in one chart.
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