Sin 15 Degrees - Exact Value (√6−√2)/4 Explained

Sin 15 Degrees - Exact Value (√6−√2)/4 Explained

TL;DR

The value of sin 15 degrees is exactly ( \frac{\sqrt{6}-\sqrt{2}}{4} ), about 0.2588. This article derives it by writing 15° as 45° − 30°, places the angle on the unit circle, gives a standard-angle table, and works through examples and the mistakes that cost marks.

The value of sin 15 degrees is ( \frac{\sqrt{6}-\sqrt{2}}{4} \approx 0.2588 ).

Quick Answer:
Result: ( \sin 15°= \frac{\sqrt{6}-\sqrt{2}}{4} )
Decimal: 0.2588 (to four places)
In radians: ( \sin(\frac{\pi}{12})= \frac{\sqrt{6}-\sqrt{2}}{4} )
Method shown: difference formula ( \sin(45°-30°) )
Exact form: ( \frac{\sqrt{6}-\sqrt{2}}{4} )

What Sin 15 Degrees Means

On the unit circle — a circle of radius 1 centred at the origin — the sine of an angle is the y-coordinate of the point where the angle's radius meets the circle. Rotating 15° counterclockwise from the positive x-axis lands just above it in Quadrant I, at the point ( (\cos 15°,\sin 15°) ), whose height above the x-axis is ( \sin 15°=\frac{\sqrt{6}-\sqrt{2}}{4} ).

The right-triangle definition — opposite over hypotenuse — agrees here because 15° is acute, but it is not one of the angles whose ratio you can read straight off a 30-60-90 or 45-45-90 triangle. That is why the value has to be built from angles you do know, rather than looked up.

How to Find the Value of Sin 15 Degrees

The cleanest route writes 15° as the difference of two standard angles.

Should I use the difference formula or the half-angle formula?

Both give the same answer. The difference formula treats 15° as 45°−30° and is the more direct of the two; the half-angle formula treats 15° as half of 30° and is handy when the target is naturally half of a known angle. This article uses the difference formula.

Method 1: The 45° − 30° difference formula

The sine difference identity is [\sin(A−B)=\sin A\cos B−\cos A\sin B.] Set ( A=45° ) and ( B=30° ) and substitute the known standard values:
[ \sin 15°=\sin(45°−30°)=\sin 45°\cos 30°−\cos 45°\sin 30° ]
[ = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2}\cdot\frac{1}{2} ]
[ = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6}-\sqrt{2}}{4}. ]

Final answer: ( \sin 15°=\frac{\sqrt{6}-\sqrt{2}}{4} \approx 0.2588. )

Method 2: From the unit circle (the check)

Mark 15° on the unit circle and drop a vertical from the point to the x-axis. The height of that segment is the sine. Measured, it is about 0.2588, which matches the surd above once you substitute ( \sqrt{6} \approx 2.449 ) and ( \sqrt{2} \approx 1.414 ):
[ \frac{2.449 - 1.414}{4} = \frac{1.035}{4} \approx 0.2588. ]

Examples of Sin 15 Degrees

Example 1

Evaluate 4 sin 15°.
[ 4\sin 15°=4\cdot\frac{\sqrt{6}-\sqrt{2}}{4} = \sqrt{6}-\sqrt{2} \approx 1.035. ]

Example 2

Find sin 15° by writing 15° as 60°−45° instead.
[ \sin 15°=\sin(60°−45°)=\sin 60°\cos 45°−\cos 60°\sin 45° ]
[ = \frac{\sqrt{3}}{2}\cdot\frac{\sqrt{2}}{2} - \frac{1}{2}\cdot\frac{\sqrt{2}}{2} = \frac{\sqrt{6}-\sqrt{2}}{4}. ]

Example 3

Show that ( \sin 15°+\cos 15°=\frac{\sqrt{6}}{2} ).
[ \frac{\sqrt{6}-\sqrt{2}}{4} + \frac{\sqrt{6}+\sqrt{2}}{4} = \frac{2\sqrt{6}}{4} = \frac{\sqrt{6}}{2}. ]

Example 4

Verify ( \sin^2 15°+\cos^2 15°=1 ).
[ \left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)^2 + \left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)^2 = \frac{16}{16} = 1. ]

Example 5

Express 15° in radians and state the value.
[ 15°=15×\frac{\pi}{180}=\frac{\pi}{12}. ]so ( \sin(\frac{\pi}{12})=\frac{\sqrt{6}-\sqrt{2}}{4} ).

Common Mistakes With Sin 15 Degrees

Mistake 1: Splitting the sine of a difference

Where it slips in: The first instinct on ( \sin(45°−30°) ) is to subtract the sines.
Don't do this: ( \sin 15°=\sin 45°−\sin 30°=\frac{\sqrt{2}}{2}−\frac{1}{2} ).
The correct way: Sine does not distribute over subtraction.

Mistake 2: Subtracting the angles in the wrong order

Where it slips in: Writing 15° as 30°−45° and applying the formula literally.
Don't do this: ( \sin(30°−45°)=-\sin 15° ).

Mistake 3: Reporting only the decimal

Where it slips in: Exam questions ask for the exact value, but a calculator hands back 0.2588.

Key Takeaways