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

[Trigonometry](/content/tag/trigonometry/index.html)

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

- **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.