use a sum or difference formula to find the exact value of the trigonometric function.\n\\( \\sin 75 ^ {…
use a sum or difference formula to find the exact value of the trigonometric function.\n\\( \\sin 75 ^ { \\circ } \\)\nfind the exact value of the expression.\n\\( \\sin 75 ^ { \\circ } = \\square \\)\n(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. )
Answer
Explanation:
Step1: Express (75^{\circ}) as a sum of angles
We know that (75^{\circ}=45^{\circ}+30^{\circ}).
Step2: Use the sine sum formula
The sine sum formula is (\sin(A + B)=\sin A\cos B+\cos A\sin B). Here (A = 45^{\circ}) and (B=30^{\circ}). We know that (\sin45^{\circ}=\frac{\sqrt{2}}{2}), (\cos45^{\circ}=\frac{\sqrt{2}}{2}), (\sin30^{\circ}=\frac{1}{2}), (\cos30^{\circ}=\frac{\sqrt{3}}{2}). Substitute these values into the formula: [ \begin{align*} \sin(45^{\circ}+ 30^{\circ})&=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}\ &=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}\ &=\frac{\sqrt{6}}{4}+\frac{\sqrt{2}}{4}\ &=\frac{\sqrt{6}+\sqrt{2}}{4} \end{align*} ]
Answer:
(\frac{\sqrt{6}+\sqrt{2}}{4})