Trigonometry - Study Mode

[#101] If $$x{sin ^2}{60^ circ }$$xa0 - $$frac{3}{2}{ ext{sec}}{60^ circ }$$ . $${ ext{ta}}{{ ext{n}}^2}{30^ circ }$$xa0 + $$frac{4}{5}{sin ^2}{45^ circ }$$ . $${ ext{ta}}{{ ext{n}}^2}{60^ circ }$$xa0 = 0, then x is?
Correct Answer

(C) $$ - frac{4}{{15}}$$

Explanation

Solution: $${ ext{ }}x{sin ^2}{60^ circ } - frac{3}{2}{ ext{sec}}{60^ circ }{ ext{.ta}}{{ ext{n}}^2}{30^ circ } + $$ xa0 xa0 xa0 $$frac{4}{5}{sin ^2}{45^ circ }.$$ xa0 $${ ext{ta}}{{ ext{n}}^2}{60^ circ }$$ xa0 $$ = 0$$ $$ Rightarrow { ext{ }}x{left( {frac{{sqrt 3 }}{2}}
ight)^2} - frac{3}{2} imes { ext{2}} imes {left( {frac{1}{{sqrt 3 }}}
ight)^2}{ ext{ + }}$$ xa0 xa0 xa0 $$frac{4}{5}{left( {frac{1}{{sqrt 2 }}}
ight)^2} imes $$ xa0 $${left( {sqrt 3 }
ight)^2} = 0$$ $$eqalign{
& Rightarrow frac{{3x}}{4} - frac{3}{2} imes 2 imes frac{1}{3} + frac{4}{5} imes frac{1}{2} imes 3 = 0 cr
& Rightarrow frac{{3x}}{4} - 1 + frac{6}{5} = 0 cr
& Rightarrow frac{{3x}}{4} = 1 - frac{6}{5} cr
& Rightarrow frac{{5 - 6}}{5} cr
& Rightarrow frac{{ - 1}}{5} cr
& herefore x = - frac{1}{5} imes frac{4}{3} cr
& ,,,,,,,,,, = - frac{4}{{15}} cr} $$

[#102] If 0° < A < 90°, then the value of tan 2 A + cot 2 A - sec 2 A cosec 2 A is?
Correct Answer

(D) -2

Explanation

Solution: $$eqalign{
& { ext{ta}}{{ ext{n}}^2}{ ext{A}} + { ext{co}}{{ ext{t}}^2}{ ext{A}} - {sec ^2}{ ext{A}}{ ext{.cose}}{{ ext{c}}^2}{ ext{A}} cr
& {x08f{Shortcut,, method:}} cr
& { ext{Put A}} = {45^ circ } cr
& Rightarrow { ext{ta}}{{ ext{n}}^2}{45^ circ } + { ext{co}}{{ ext{t}}^2}{45^ circ } - {sec ^2}{45^ circ }{ ext{.cose}}{{ ext{c}}^2}{45^ circ } cr
& Rightarrow 1 + 1 - {left( {sqrt 2 }
ight)^2}{left( {sqrt 2 }
ight)^2} cr
& Rightarrow - 2 cr} $$

[#103] If α and β are positive acute angles, sin(4α - β) = 1 and cos(2α + β) = $$frac{1}{2}{ ext{,}}$$ then the value of sin(α + 2β) is?
Correct Answer

(D) $$frac{1}{{sqrt 2 }}$$

Explanation

Solution: $$eqalign{
& { ext{sin}}left( {4alpha - x08eta }
ight) = 1 = sin {90^ circ } cr
& 4alpha - x08eta = {90^ circ }{ ext{ }} cr
& { ext{cos}}left( {2alpha + x08eta }
ight) = frac{1}{2} = cos {60^ circ } cr
& 2alpha + x08eta = {60^ circ } cr
& { ext{Adding }}6alpha = {150^ circ } cr
& alpha = {25^ circ } cr
& x08eta = {10^ circ } cr
& Rightarrow { ext{sin}}left( {alpha + 2x08eta }
ight) cr
& Rightarrow { ext{sin}}left( {{{25}^ circ } + 2 imes {{10}^ circ }}
ight) cr
& Rightarrow { ext{sin}}{45^ circ } cr
& Rightarrow frac{1}{{sqrt 2 }} cr} $$

[#104] If θ is a positive acute angle and 4cos 2 θ - 1 = 0, then the value of tan(θ - 15°) is equal to?
Correct Answer

(B) 1

Explanation

Solution: $$eqalign{
& { ext{4co}}{{ ext{s}}^2} heta - 1 = 0 cr
& Rightarrow { ext{4co}}{{ ext{s}}^2} heta = 1 cr
& Rightarrow { ext{co}}{{ ext{s}}^2} heta = frac{1}{4} cr
& Rightarrow cos heta = frac{1}{2} = cos {60^ circ } cr
& heta = {60^ circ } cr
& Rightarrow an left( { heta - {{15}^ circ }}
ight) cr
& Rightarrow an left( {{{60}^ circ } - {{15}^ circ }}
ight) cr
& Rightarrow an {45^ circ } cr
& Rightarrow 1 cr} $$

[#105] The value of $$frac{{sin {{25}^ circ }.cos {{65}^ circ } + cos {{25}^ circ }.sin {{65}^ circ }}}{{{{ an }^2}{{70}^ circ } - { ext{cose}}{{ ext{c}}^2}{{20}^ circ }}}$$ xa0 xa0 xa0 is?
Correct Answer

(A) -1

Explanation

Solution: $$frac{{sin {{25}^ circ }.cos {{65}^ circ } + cos {{25}^ circ }.sin {{65}^ circ }}}{{{{ an }^2}{{70}^ circ } - cos e{c^2}{{20}^ circ }}}$$ $$ = frac{{sin {{25}^ circ }.cos left( {{{90}^ circ } - {{25}^ circ }}
ight) + ,cos {{25}^ circ }.,sin left( {{{90}^ circ } - {{25}^ circ }}
ight)}}{{{{ an }^2}{{70}^ circ } - { ext{cose}}{{ ext{c}}^2}left( {{{90}^ circ } - {{70}^ circ }}
ight)}}$$ $$eqalign{
& = frac{{{{sin }^2}{{25}^ circ } + {{cos }^2}{{25}^ circ }}}{{{{ an }^2}{{70}^ circ } - se{c^2}{{70}^ circ }}} cr
& = frac{1}{{ - 1}} cr
& = - 1 cr} $$