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

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

Quick Answer:

Result: $\sin 75° = \dfrac{\sqrt{6}+\sqrt{2}}{4}$

Decimal: $0.9659$ (to four places)

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

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

Same as: $\cos 15°$ (complementary angle)

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.

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 $\frac{\sqrt{6}+\sqrt{2}}{4}$. See cos 15 degrees for the same value reached from the other angle.

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: $\sin 75° = \dfrac{\sqrt{6}+\sqrt{2}}{4} \approx 0.9659.$

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: $\sin 75° = \sin 45° + \sin 30° = \frac{\sqrt{2}}{2} + \frac{1}{2} \approx 1.207$. That is impossible — sine never exceeds $1$. Sine is not additive across angles. Use the sum identity:

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

A value above $1$ is the instant signal that the sine was split incorrectly.

Example 3

Show that $\sin 75° + \sin 15° = \dfrac{\sqrt{6}}{2}$, given $\sin 15° = \dfrac{\sqrt{6}-\sqrt{2}}{4}$.

$$\frac{\sqrt{6}+\sqrt{2}}{4} + \frac{\sqrt{6}-\sqrt{2}}{4} = \frac{2\sqrt{6}}{4} = \frac{\sqrt{6}}{2}.$$

The $\sqrt{2}$ terms cancel — the same cancellation seen on the sin 15 degrees page.

Example 4

Verify $\sin^2 75° + \cos^2 75° = 1$, using $\cos 75° = \dfrac{\sqrt{6}-\sqrt{2}}{4}$.

$$\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)^2 + \left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)^2 = \frac{(8+2\sqrt{12}) + (8-2\sqrt{12})}{16} = \frac{16}{16} = 1.$$

The Pythagorean identity holds; the cross terms cancel.

Example 5

Express $75°$ in radians and state the value.

$75° = 75 \times \frac{\pi}{180} = \frac{5\pi}{12}$ radians, so $\sin\left(\frac{5\pi}{12}\right) = \frac{\sqrt{6}+\sqrt{2}}{4}$. Converting between radians and degrees leaves the value the same.

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. Apply $\sin(A+B) = \sin A\cos B + \cos A\sin B$. The reliable habit is the range check — any sine result above $1$ or below $-1$ is wrong before you finish.

Mistake 2: Assuming sin 75 equals cos 75

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

Don't do this: Writing $\sin 75° = \cos 75°$.

The correct way: The cofunction identity pairs an angle with its complement: $\sin 75° = \cos(90° - 75°) = \cos 15°$. The learner who half-remembers "sin equals cos" forgets the $90° - \theta$ step — the fix is to always subtract the angle from $90°$.

Mistake 3: Reporting only the decimal

Where it slips in: A calculator returns $0.9659$, but the question asks for the exact value.

Don't do this: Writing $\sin 75° = 0.9659$ when "exact" is required.

The correct way: Keep $\frac{\sqrt{6}+\sqrt{2}}{4}$. The decimal is rounded; the surd is exact.

Key Takeaways

• Sin 75 degrees equals the exact surd $\frac{\sqrt{6}+\sqrt{2}}{4}$, about $0.9659$.

• It is built by writing $75°$ as $45° + 30°$ and applying the sine sum formula.

• $\sin 75° = \cos 15°$ because $75°$ and $15°$ are complementary.

• The most common error is adding the sines of the parts, which gives an impossible value above one.

Practice These Before Moving On

1. Derive $\cos 75°$ from the $45° + 30°$ split and confirm $\sin^2 75° + \cos^2 75° = 1$.

2. Use the complement rule to write $\sin 75°$ from $\cos 15°$ without re-deriving it.

3. Convert $75°$ to radians and write the value as $\sin\left(\frac{5\pi}{12}\right)$.

To go deeper into sum-and-difference values with a teacher, Bhanzu's trigonometry tutor and high school math tutor sessions cover the compound-angle and cofunction rules together, with math classes online for live practice.

Read More

• Sum and difference identities — the full compound-angle toolkit.

• Sin Cos Tan — the three core ratios and how they connect.

• Cos 30 degrees — one of the standard angles sin 75° is built from.

• Trigonometric ratios — how sine, cosine, and tangent are defined.

• Trigonometric table — every standard-angle value in one chart.