Use a sum or difference identity to find the exact value.
-Use the exact value $\cos 36 ^ { \circ } = \frac { \sqrt { 5 } + 1 } { 4 }$ and identities to find the exact value of $\sin 36 ^ { \circ }$.
A) $\frac { ( \sqrt { 10 + 2 \sqrt { 5 } } ) ( 10 - 2 \sqrt { 5 } ) } { 5 }$
B) $\sqrt { 5 } + 1$
C) $\frac { \sqrt { 10 - 2 \sqrt { 5 } } } { 4 }$
D) $\frac { \sqrt { 10 - 2 \sqrt { 5 } } } { 8 }$

Question

Quizplus · Accepted Answer

Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, we can find $\sin 36^\circ$ given $\cos 36^\circ = \frac{\sqrt{5} + 1}{4}$. Solving for $\sin 36^\circ$, we get $\sin 36^\circ = \sqrt{1 - \cos^2 36^\circ} = \sqrt{1 - \left(\frac{\sqrt{5} + 1}{4}\right)^2} = \frac{\sqrt{10 - 2\sqrt{5}}}{4}$.

Use a Sum or Difference Identity to Find the Exact cos⁡36∘=5+14\cos 36 ^ { \circ } = \frac { \sqrt { 5 } + 1 } { 4 }cos36∘=45​+1​

Use a Sum or Difference Identity to Find the Exact $\cos 36 ^ { \circ } = \frac { \sqrt { 5 } + 1 } { 4 }$