Sin 75 Degrees - Exact Value (√6+√2)/4 Explained

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Sin 75 Degrees - Exact Value (√6+√2)/4 Explained

Trigonometry

TL;DR

The value of sin 75 degrees is exactly ( \frac{\sqrt{6}+\sqrt{2}}{4} ), about 0.9659. This article derives it by writing 75° as 45° + 30°, shows why ( \sin 75°=\cos 15° ), places the angle on the unit circle, and works through examples and common mistake.

The value of sin 75 degrees is ( \frac{\sqrt{6}+\sqrt{2}}{4} \approx 0.9659.

Quick Answer:

Result: ( \sin 75° = \frac{\sqrt{6}+\sqrt{2}}{4} )

Decimal: 0.9659 (to four places)

In radians: ( \sin\left(\frac{5\pi}{12}\right) = \frac{\sqrt{6}+\sqrt{2}}{4} )

Method shown: sum formula ( \sin(45°+30°) )

Same as: ( \cos 15° )

Quick Reference Table of Sine Values

Seventy-five degrees is built from 45° and 30°, and its sine sits high — the angle is close to 90°, where sine reaches 1. The table places it among the standard sine values in Quadrant I.

Angle (degrees) Angle (radians) ( \sin\theta ) (exact) ( \sin\theta ) (decimal)
0° 0 0 0.0000
15° ( \frac{\pi}{12} ) ( \frac{\sqrt{6}-\sqrt{2}}{4} ) 0.2588
30° ( \frac{\pi}{6} ) ( \frac{1}{2} ) 0.5000
45° ( \frac{\pi}{4} ) ( \frac{\sqrt{2}}{2} ) 0.7071
60° ( \frac{\pi}{3} ) ( \frac{\sqrt{3}}{2} ) 0.8660
75° ( \frac{5\pi}{12} ) ( \frac{\sqrt{6}+\sqrt{2}}{4} ) 0.9659
90° ( \frac{\pi}{2} ) 1 1.0000

The top and bottom non-trivial entries are linked: ( \sin 75° ) and ( \sin 15° ) carry the same surds with opposite middle signs, because 75° and 15° are complementary. That is also why ( \sin 75° ) matches ( \cos 15° ) exactly.

What Sin 75 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 75° counterclockwise from the positive x-axis lands a point high in Quadrant I, at ( \left(\cos 75°, \sin 75°\right) ), whose height above the x-axis is ( \sin 75° = \frac{\sqrt{6}+\sqrt{2}}{4} ).

The right-triangle definition — opposite over hypotenuse — applies because 75° is acute, but 75° is not an angle you can read off a 30-60-90 or 45-45-90 triangle. So the value is built from the standard angles 45° and 30°.

How to Find the Value of Sin 75 Degrees

The direct route writes 75° as a sum of two standard angles. Two questions come up most.

Why does sin 75 equal cos 15? Because 75° and 15° add to 90°, so they are complementary, and the cofunction identity says ( \sin\theta=\cos(90°−\theta) ). So ( \sin 75°=\cos(90°−75°)=\cos 15° ), which is why both equal ( \frac{\sqrt{6}+\sqrt{2}}{4} ).

Method 1: The 45° + 30° sum formula

The sine sum identity is
( \sin(A+B)=\sin A \cos B + \cos A \sin B. )
Set A=45°, B=30°, and substitute ( \sin 45°= \frac{\sqrt{2}}{2}, \cos 30°= \frac{\sqrt{3}}{2}, \cos 45°= \frac{\sqrt{2}}{2}, \sin 30°= \frac{1}{2} ):

( \sin 75°=\sin(45°+30°)=\sin 45° \cos 30° + \cos 45° \sin 30° )

( \sin 75°= \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2}\cdot\frac{1}{2} )

( \sin 75°= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6}+\sqrt{2}}{4}. )

Final answer: ( \sin 75°= \frac{\sqrt{6}+\sqrt{2}}{4} \approx 0.9659. )

Method 2: The complement shortcut

Since ( \sin 75°=\cos 15° ), and ( \cos 15°= \frac{\sqrt{6}+\sqrt{2}}{4} ) from the 45°−30° split, the value transfers directly:

( \sin 75°=\cos 15°=\frac{\sqrt{6}+\sqrt{2}}{4}. )

Examples of Sin 75 Degrees

Example 1

Evaluate 4( \sin 75°).

( 4\sin 75°=4\cdot\frac{\sqrt{6}+\sqrt{2}}{4} = \sqrt{6}+\sqrt{2} \approx 3.863. )

Example 2

Find ( \sin 75° ) by splitting the sine of a sum — the wrong way first.

A common first move is to add the sines: ( \sin 75°=\sin 45°+\sin 30°= \frac{\sqrt{2}}{2} + \frac{1}{2} \approx 1.207 ), which is impossible — sine never exceeds 1. Use the sum identity:

( \sin 75°=\sin 45° \cos 30° + \cos 45° \sin 30°= \frac{\sqrt{6}+\sqrt{2}}{4} \approx 0.9659. )

Example 3

Show that ( \sin 75°+\sin 15°= \frac{\sqrt{6}}{2} ), given ( \sin 15°= \frac{\sqrt{6}-\sqrt{2}}{4} ).

( \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 75° + \cos^2 75° = 1 ), using ( \cos 75°= \frac{\sqrt{6}-\sqrt{2}}{4} ).

( \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 75° in radians and state the value.

75°=75×( \frac{\pi}{180} = \frac{5\pi}{12} ), so ( \sin\left(\frac{5\pi}{12}\right) = \frac{\sqrt{6}+\sqrt{2}}{4} ).

Common Mistakes With Sin 75 Degrees

Mistake 1: Adding the sines of the parts

Where it slips in: The first instinct on ( \sin(45°+30°) ) is to add ( \sin 45° ) and ( \sin 30° ).

Don't do this: ( \sin 75°=\sin 45°+\sin 30° \approx 1.207.) The correct way: Sine is not additive.

Mistake 2: Assuming sin 75 equals cos 75

Where it slips in: Confusing the complement rule, which pairs ( \sin 75° ) with ( \cos 15° ), not ( \cos 75° ).

Mistake 3: Reporting only the decimal. The decimal is rounded; the surd is exact.

Key Takeaways