Question

Find the value of each of the following: 5 sin square 30°+ cos square 45°-4 tan square 30° / 2 sin square 30° cos square 30°+ tan 45°

Answer 1

Question :-

$\frac{5 {sin}^{2}30 + {cos}^{2}45 - 4 {tan}^{2}30 }{2 {sin}^{2}30. {cos}^{2} 30 + tan45 }$

Solution :-

we know that,

• sin30 = 1/2
• cos45 = 1/√2
• tan30 = 1/√3
• tan45 = 1

_______________________

substituting values in the question,

= 5(1/2)^2 + (1/√2)^2 - 4(1/√3)^2/2(1/2)^2 x (√3/2)^2 + 1

= 5(1/4) + (1/2) - 4(1/3) /2(1/4) x (3/4) + 1

= 5/4 + 1/2 - 4/3 / 1/2 x 3/4 + 1

= 7/4 - 4/3 / 3/8 + 1

= 5/12 / 3/8 + 1

= 5/12 / 11/8

= 5/12 x 8/11

= 40/132

= 10/33

Answer 2

${\huge{\boxed{\mathtt{\purple{Question}}}}}$

➾ Find the value of :-

$\frac{5 \sin( {30})^{2} + { \cos(45) }^{2} - 4 { \tan(30) }^{2} }{2 { \sin(30) }^{2} . { \cos(30) }^{2} + \tan(45) }$

${\huge{\boxed{\mathtt{\purple{Solution}}}}}$

$\mathtt{\: Let's \: write \: the \: value \: of \: these \: trigonometric\: functions}$

$\: \sin(30) = \frac{1}{2}$

$\: \cos(45) = \frac{1}{ \sqrt{2} }$

$\tan(45) = 1$

$\: \tan(30) = \frac{1}{ \sqrt{3} }$

$\: \cos(30) = \frac{ \sqrt{3} }{2}$

$\mathtt{\:Putting \:these \:values\: in \:the\: given \:Question }$

$\: \frac{5 { (\frac{1}{2}) }^{2} + { (\frac{1}{ \sqrt{2} }) }^{2} - 4 ({ \frac{1}{ \sqrt{3} } })^{2} }{2 { (\frac{1}{2} })^{2} . { \frac{ \sqrt{3} }{2} }^{2} + 1}$

$\: \frac{5 (\frac{1}{4}) + \frac{1}{2} - 4 (\frac{1}{3} ) }{2( \frac{1}{4} ).( \frac{3}{4}) + 1 }$

$\: \frac{ \frac{5}{4} + \frac{1}{2} - \frac{4}{3} }{ \frac{3}{8} + 1}$

$\: \frac{ \frac{ \frac{15 + 6 - 16}{12} }{3 + 8} }{8}$

$\: \frac{ \frac{5}{12} }{ \frac{11}{8} }$

$\: \frac{5}{12} \times \frac{8}{11}$

$\: \frac{5 \times 2}{3 \times 11}$

${ \boxed {\mathtt{ \red{ \: \frac{10}{33} }}}}$