Answered

IDNStudy.com, ang iyong destinasyon para sa mabilis at kaugnay na mga sagot. Ang aming mga eksperto ay nagbibigay ng mabilis at eksaktong sagot upang tulungan kang maunawaan at malutas ang anumang problema.

Directions: Find the exact values of the following expressions:
2. [tex] \: \frac{5 { \sin}^{2} {30}^{ \circ} + { \cos}^{2}{45}^{ \circ} + 4 { \tan}^{2} {60}^{ \circ} }{2 \sin {30}^{ \circ} \cos {45}^{ \circ} + \tan {45}^{ \circ} } [/tex]​


Sagot :

[tex]\underline{\underline{\large{\red{\mathcal{✒GIVEN:}}}}}[/tex]

[tex]\bullet \: \: \rm{ \frac{5 { \sin}^{2} {30}^{ \circ} + { \cos}^{2}{45}^{ \circ} + 4 { \tan}^{2} {60}^{ \circ} }{2 \sin {30}^{ \circ} \cos {45}^{ \circ} + \tan {45}^{ \circ} }}[/tex]

[tex]\underline{\underline{\large{\red{\mathcal{REQUIRED:}}}}}[/tex]

Find the exact value.

[tex]\underline{\underline{\large{\red{\mathcal{SOLUTION:}}}}}[/tex]

Remember the six trigonometric ratios for [tex]\tt{\purple{special \: angles}}[/tex] [tex]\tt{{45}^{ \circ} , {30}^{ \circ} \: and \: {60}^{ \circ}}[/tex]:

[tex]\small{\boxed{ \bm{{ \red{ \sin {30}^{ \circ} = \dfrac{1}{2} }}}}}[/tex]

[tex]\small{\boxed{ \bm{{ \red{ \cos {45}^{ \circ} = \dfrac{ \sqrt{2} }{2} }}}}}[/tex]

[tex]\small{\boxed{ \bm{{ \red{ \tan {60}^{ \circ} = \sqrt{3} }}}}}[/tex]

[tex]\small{\boxed{ \bm{{ \red{ \tan {45}^{ \circ} = \sqrt{1} }}}}}[/tex]

Now, we substitute those values:

[tex]\small{\tt{ \frac{5( \frac{1}{2} {)}^{2} + ( \frac{ \sqrt{2} }{2} {)}^{2} + 4( \sqrt{3} {)}^{2} }{2 (\frac{1}{2}) ( \frac{ \sqrt{2} }{2} ) + 1} = \frac{ \frac{5}{4} + \frac{2}{4} + 12 }{ \frac{ \sqrt{2} }{2} + 1} }}[/tex]

[tex]\tt{ \frac{ \frac{7}{4} + 12}{ \frac{ \sqrt{2} + 2}{2} } = \frac{ \frac{7 + 48}{4} }{ \frac{ \sqrt{2} + 2}{2} } \div \frac{ \sqrt{2} + 2}{2} }[/tex]

[tex]\tt{ \dfrac{55(2)}{4( \sqrt{2} + 2)}}[/tex]

Simplify:

[tex]\large{\tt{\purple{ \dfrac{55}{2 \sqrt{2} + 4} }}}[/tex]

Final Answer:

[tex]\tt{\therefore}[/tex] The exact value is [tex]\large{\rm{\purple{ \dfrac{55}{2 \sqrt{2} + 4} }}}[/tex].