Use the angle addition formula for sine.

March 28, 2026Mathematics

ScanSolve AI Answer

March 28, 2026

Answer

$\frac{\sqrt{6} + \sqrt{2}}{4}$

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 $\boxed{\frac{\sqrt{6} + \sqrt{2}}{4}}$ .

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Use the angle addition formula for sine. We can express 75^ as the sum of two standard angles, 45^ + 30^.

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.

ScanSolve AI AnswerAnswered on March 28, 2026

Step 1: Use the angle addition formula for sine. We can express 75^ as the sum of two standard angles, 45^ + 30^. The addition formula for sine is (A+B) = A B + A B. Step 2: Substitute A=45^ and B=30^ into the formula. 75^ = (45^ + 30^) = 45^ 30^ + 45^ 30^ Step 3: Substitute the known exact values for 45^, 30^, 45^, and 30^. We know that: 45^ = sqrt(2)2 30^ = sqrt(3)2 45^ = sqrt(2)2 30^ = (1)/(2) 75^ = (sqrt(2)2)(sqrt(3)2) + (sqrt(2)2)((1)/(2)) Step 4: Multiply the terms and simplify. 75^ = sqrt(2) · sqrt(3)4 + sqrt(2) · 14 75^ = sqrt(6)4 + sqrt(2)4 75^ = sqrt(6) + sqrt(2)4 The value of 75^ in simple surd form is sqrt(6) + sqrt(2)4.