use an identity to write the expression as a single trigonometric function value or as a single number.\n\\(…

Explanation:

Step1: Use the double - angle identity

Recall the double - angle identity (\cos(2\alpha)=1 - 2\sin^{2}\alpha), which can be rewritten as (\sin^{2}\alpha=\frac{1-\cos(2\alpha)}{2}). Here (\alpha = 77.3^{\circ}), so (\frac{1}{2}-\frac{1}{4}\sin^{2}(77.3^{\circ})=\frac{1}{2}-\frac{1}{4}\times\frac{1 - \cos(2\times77.3^{\circ})}{2}).

Step2: Simplify the expression

[ \begin{align*} \frac{1}{2}-\frac{1}{4}\sin^{2}(77.3^{\circ})&=\frac{1}{2}-\frac{1}{8}(1-\cos(154.6^{\circ}))\ &=\frac{1}{2}-\frac{1}{8}+\frac{1}{8}\cos(154.6^{\circ})\ &=\frac{4 - 1}{8}+\frac{1}{8}\cos(154.6^{\circ})\ &=\frac{3}{8}+\frac{1}{8}\cos(154.6^{\circ})\ \end{align*} ] Another way: We know that (\frac{1}{2}-\frac{1}{4}\sin^{2}(77.3^{\circ})=\frac{2 - \sin^{2}(77.3^{\circ})}{4}). Since (\sin^{2}(77.3^{\circ})=\frac{1-\cos(154.6^{\circ})}{2}), then (\frac{2-\frac{1 - \cos(154.6^{\circ})}{2}}{4}=\frac{4-(1 - \cos(154.6^{\circ}))}{8}=\frac{3+\cos(154.6^{\circ})}{8})

Also, using the identity (\frac{1}{2}-\frac{1}{4}\sin^{2}\theta=\frac{1}{4}(2-\sin^{2}\theta)) and (\cos(2\theta)=1 - 2\sin^{2}\theta\Rightarrow\sin^{2}\theta=\frac{1 - \cos(2\theta)}{2})

[ \begin{align*} \frac{1}{2}-\frac{1}{4}\sin^{2}(77.3^{\circ})&=\frac{1}{4}\left(2-\frac{1-\cos(154.6^{\circ})}{2}\right)\ &=\frac{1}{4}\times\frac{4-(1 - \cos(154.6^{\circ}))}{2}\ &=\frac{3+\cos(154.6^{\circ})}{8}\ \end{align*} ]

If we use the double - angle formula in another form:

We know that (\frac{1}{2}-\frac{1}{4}\sin^{2}(77.3^{\circ})=\frac{1}{4}(2-\sin^{2}(77.3^{\circ})))

Since (2-\sin^{2}\theta=1+(1 - \sin^{2}\theta)=1+\cos^{2}\theta)

[ \begin{align*} \frac{1}{2}-\frac{1}{4}\sin^{2}(77.3^{\circ})&=\frac{1}{4}(1+\cos^{2}(77.3^{\circ})) \end{align*} ]

But the most straightforward is using (\cos(2\alpha) = 1-2\sin^{2}\alpha\Rightarrow\sin^{2}\alpha=\frac{1 - \cos(2\alpha)}{2})

[ \begin{align*} \frac{1}{2}-\frac{1}{4}\sin^{2}(77.3^{\circ})&=\frac{1}{2}-\frac{1}{4}\times\frac{1-\cos(154.6^{\circ})}{2}\ &=\frac{4-(1 - \cos(154.6^{\circ}))}{8}\ &=\frac{3+\cos(154.6^{\circ})}{8}\ \end{align*} ]

If we use the co - function identity and other manipulations wrong.

Let's use a calculator:

(\sin(77.3^{\circ})\approx0.975), (\sin^{2}(77.3^{\circ})\approx0.9506)

(\frac{1}{2}-\frac{1}{4}\times0.9506=\frac{1}{2}-\frac{0.9506}{4}=0.5 - 0.23765 = 0.26235)

(\cos(154.6^{\circ})\approx - 0.902)

(\frac{3+\cos(154.6^{\circ})}{8}=\frac{3-0.902}{8}=\frac{2.098}{8}=0.26225\approx0.2623)

Answer:

(\frac{1}{4}\cos(154.6^{\circ})+\frac{3}{8})