Evaluate sin(75°) without calculator

Question

Evaluate $\sin 75°$ without using a calculator, showing all steps.

Solution — Step by Step

We know exact values for 30°, 45°, 60°, and 90°. Write 75° as a sum of two of these:

75° = 45° + 30°

Now we can apply the sum formula for sine:

\sin(A + B) = \sin A \cos B + \cos A \sin B

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

Angle	sin	cos
30°	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$
45°	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$

Substituting:

\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}

\boxed{\sin 75° = \frac{\sqrt{6} + \sqrt{2}}{4}}

Numerically: $\sqrt{6} \approx 2.449$ , $\sqrt{2} \approx 1.414$ , sum $\approx 3.863$ , divided by 4 $\approx 0.966$ . Check: $\sin 75° \approx 0.966$ ✓

Why This Works

The addition formula $\sin(A+B) = \sin A \cos B + \cos A \sin B$ is a fundamental identity derived from the unit circle or using the rotation matrix. It holds for any angles $A$ and $B$ .

By expressing 75° as 45° + 30°, we reduce the problem to values we already know. This technique works for any angle that can be written as a sum or difference of the standard angles (30°, 45°, 60°, 90°, 180°).

Memorising $\sin 75° = (\sqrt{6}+\sqrt{2})/4$ is useful; understanding why is more useful.

Alternative Method

You could also write $75° = 90° - 15°$ and use $\sin(90° - \theta) = \cos\theta$ :

\sin 75° = \cos 15°

Then find $\cos 15° = \cos(45° - 30°)$ using the subtraction formula:

\cos(45° - 30°) = \cos 45° \cos 30° + \sin 45° \sin 30° = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}+\sqrt{2}}{4}

Same answer, as expected.

Common Mistake

Students often write $\sin 75° = \sin 45° + \sin 30°$ — treating sine as a linear function. Sine is not linear: $\sin(A+B) \neq \sin A + \sin B$ . Always use the full addition formula. If you use the shortcut, you get $\frac{\sqrt{2}}{2} + \frac{1}{2} \approx 1.21$ — which is impossible since sine values lie between $-1$ and $+1$ . That alone should tell you the shortcut is wrong.

The result $\sin 75° = \cos 15° = (\sqrt{6}+\sqrt{2})/4$ appears surprisingly often in JEE Main and CBSE Class 11 trigonometry problems. Similarly useful: $\cos 75° = \sin 15° = (\sqrt{6}-\sqrt{2})/4$ , $\tan 75° = 2+\sqrt{3}$ .

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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