Use the angle addition formula for sine.

Question

Use the angle addition formula for sine.

Asked on March 28, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-03-28T21:04:15.860Z

Step 1: Use the angle addition formula for sine. We can express $75^\circ$ as the sum of two standard angles, $45^\circ + 30^\circ$. The addition formula for sine is $\sin(A+B) = \sin A \cos B + \cos A \sin B$. Step 2: Substitute $A=45^\circ$ and $B=30^\circ$ into the formula. $$ \sin 75^\circ = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ $$ Step 3: Substitute the known exact values for $\sin 45^\circ$, $\cos 30^\circ$, $\cos 45^\circ$, and $\sin 30^\circ$. We know that: $\sin 45^\circ = \frac{\sqrt{2}}{2}$ $\cos 30^\circ = \frac{\sqrt{3}}{2}$ $\cos 45^\circ = \frac{\sqrt{2}}{2}$ $\sin 30^\circ = \frac{1}{2}$ $$ \sin 75^\circ = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) $$ Step 4: Multiply the terms and simplify. $$ \sin 75^\circ = \frac{\sqrt{2} \cdot \sqrt{3}}{4} + \frac{\sqrt{2} \cdot 1}{4} $$ $$ \sin 75^\circ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} $$ $$ \sin 75^\circ = \frac{\sqrt{6} + \sqrt{2}}{4} $$ The value of $\sin 75^\circ$ in simple surd form is $\boxed{\frac{\sqrt{6} + \sqrt{2}}{4}}$.

Accepted Answer · 2026-03-28T21:04:15.860Z

Step 1: Use the angle addition formula for sine. We can express $75^\circ$ as the sum of two standard angles, $45^\circ + 30^\circ$. The addition formula for sine is $\sin(A+B) = \sin A \cos B + \cos A \sin B$. Step 2: Substitute $A=45^\circ$ and $B=30^\circ$ into the formula. $$ \sin 75^\circ = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ $$ Step 3: Substitute the known exact values for $\sin 45^\circ$, $\cos 30^\circ$, $\cos 45^\circ$, and $\sin 30^\circ$. We know that: $\sin 45^\circ = \frac{\sqrt{2}}{2}$ $\cos 30^\circ = \frac{\sqrt{3}}{2}$ $\cos 45^\circ = \frac{\sqrt{2}}{2}$ $\sin 30^\circ = \frac{1}{2}$ $$ \sin 75^\circ = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) $$ Step 4: Multiply the terms and simplify. $$ \sin 75^\circ = \frac{\sqrt{2} \cdot \sqrt{3}}{4} + \frac{\sqrt{2} \cdot 1}{4} $$ $$ \sin 75^\circ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} $$ $$ \sin 75^\circ = \frac{\sqrt{6} + \sqrt{2}}{4} $$ The value of $\sin 75^\circ$ in simple surd form is $\boxed{\frac{\sqrt{6} + \sqrt{2}}{4}}$.