Trigonometry - Study Mode
[#186] The value of the expression:
sin 2 1° + sin 2 11° + sin 2 21° + sin 2 31° + sin 2 41° + sin 2 45° + sin 2 49° + sin 2 59° + sin 2 69° + sin 2 79° + sin 2 89° is?
Correct Answer
(B) $${ ext{5}}frac{1}{2}$$
Explanation
Solution: sin 2 1° + sin 2 11° + sin 2 21° + sin 2 31° + sin 2 41° + sin 2 45° + sin 2 49° + sin 2 59° + sin 2 69° + sin 2 79° + sin 2 89° = (sin 2 1° + sin 2 89°) + (sin 2 11° + sin 2 79°) + (sin 2 21° + sin 2 69°) + (sin 2 31° + sin 2 59°) + (sin 2 41° + sin 2 49°) + sin 2 45° = 1 + 1 + 1 + 1 + 1 + $$frac{1}{2}$$ xa0 [sin 2 A + sin 2 B = 1. If, A + B = 90°] = $$5frac{1}{2}$$
[#187] If cos20° = m and cos70° =n, then the value of m 2 + n 2 is?
Correct Answer
(B) 1
Explanation
Solution: $$eqalign{
& cos {20^ circ } = m{ ext{ }} cr
& cos {70^ circ } = n cr
& { ext{So,}} cr
& Leftrightarrow {m^2} + {n^2} = { ext{co}}{{ ext{s}}^2}{20^ circ } + { ext{co}}{{ ext{s}}^2}{70^ circ } cr
& left[ {{ ext{If co}}{{ ext{s}}^2}{ ext{A + co}}{{ ext{s}}^2}{ ext{B}} = { ext{1}}}
ight] cr
& ({ ext{If, A}} + { ext{B}} = {90^ circ }) cr
& Leftrightarrow 1 cr} $$
[#188] If $${ ext{sin}}left( {{{90}^ circ } - heta }
ight)$$ xa0 + $${ ext{cos}} heta $$xa0 = $$sqrt 2 { ext{cos}}left( {{{90}^ circ } - heta }
ight){ ext{,}}$$ xa0xa0 then the value of $${ ext{cosec}} heta $$ xa0 is?
Correct Answer
(C) $$sqrt {frac{3}{2}} $$
Explanation
Solution: $$eqalign{
& { ext{sin}}left( {{{90}^ circ } - heta }
ight) + { ext{cos}} heta = sqrt 2 { ext{cos}}left( {{{90}^ circ } - heta }
ight) cr
& Rightarrow { ext{cos}} heta + { ext{cos}} heta = sqrt 2 sin heta cr
& Rightarrow frac{{2cos heta }}{{sin heta }} = sqrt 2 cr
& Rightarrow cot heta = frac{{1 o { ext{B}}}}{{sqrt 2 o { ext{P}}}} cr
& { ext{So, H}} o ext{alignment} cr
& herefore { ext{cosec}} heta = frac{{ ext{H}}}{{ ext{P}}} = frac{{sqrt 3 }}{{sqrt 2 }} cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = sqrt {frac{3}{2}} cr} $$
[#189] If $${ ext{sin A}} - cos { ext{A}}$$ xa0 = $$frac{{sqrt 3 - 1}}{2}{ ext{,}}$$ xa0 then the value of $${ ext{sin A}}.{ ext{cosA}}$$ xa0 is?
Correct Answer
(D) $$frac{{sqrt 3 }}{4}$$
Explanation
Solution: $$eqalign{
& { ext{sin A}} - cos { ext{A}} = frac{{sqrt 3 - 1}}{2} cr
& {x08f{Shortcut ,,method:}} cr
& { ext{Put, }} heta = {60^ circ } cr
& Rightarrow { ext{sin A}} - cos { ext{A}} = frac{{sqrt 3 - 1}}{2} cr
& Rightarrow { ext{sin }}{60^ circ } - cos {60^ circ } = frac{{sqrt 3 - 1}}{2} cr
& Rightarrow frac{{sqrt 3 }}{2} - frac{1}{2} = frac{{sqrt 3 - 1}}{2} cr
& Rightarrow frac{{sqrt 3 - 1}}{2} = frac{{sqrt 3 - 1}}{2}({ ext{Matched}}) cr
& Hence, cr
& { ext{sin A}}.cos{ ext{A}} cr
& Rightarrow frac{{sqrt 3 }}{2} imes frac{1}{2} cr
& Rightarrow frac{{sqrt 3 }}{4} cr
& cr
& {x08f{Alternate:}} cr
& { ext{sin A}} - cos { ext{A}} = frac{{sqrt 3 - 1}}{2} cr
& { ext{Squaring both side,}} cr
& Rightarrow { ext{ si}}{{ ext{n}}^2}{ ext{ A + }}{cos ^2}{ ext{A}} - { ext{2}}{ ext{.sin A}}.cos{ ext{A}} = {left( {frac{{sqrt 3 - 1}}{2}}
ight)^2} cr
& Rightarrow 1 - 2{ ext{sin A}}.cos{ ext{A = }}frac{{3 + 1 - 2sqrt 3 }}{4} cr
& Rightarrow 2{ ext{sin A}}.cos{ ext{A}} = 1 - 2frac{{left( {2 - sqrt 3 }
ight)}}{4} cr
& Rightarrow 2{ ext{sin A}}.cos{ ext{A}} = frac{{2 - 2 + sqrt 3 }}{2} cr
& Rightarrow { ext{sin A}}.cos{ ext{A}} = frac{{sqrt 3 }}{4} cr} $$
[#190] If $$frac{{{{sec }^2}{{70}^ circ } - { ext{co}}{{ ext{t}}^2}{{20}^ circ }}}{{2left( {{ ext{cose}}{{ ext{c}}^2}{{59}^ circ } - {{ an }^2}{{31}^ circ }}
ight)}}$$ xa0 xa0 = $$frac{2}{m}{ ext{,}}$$ xa0then m is equal to?
Correct Answer
(C) 4
Explanation
Solution: $$eqalign{
& frac{{{{sec }^2}{{70}^ circ } - { ext{co}}{{ ext{t}}^2}{{20}^ circ }}}{{2left( {{ ext{cose}}{{ ext{c}}^2}{{59}^ circ } - {{ an }^2}{{31}^ circ }}
ight)}} = frac{2}{m} cr
& Rightarrow frac{{{{sec }^2}{{70}^ circ } - { ext{co}}{{ ext{t}}^2}left( {{{90}^ circ } - {{70}^ circ }}
ight)}}{{2left( {{ ext{cose}}{{ ext{c}}^2}{{59}^ circ } - {{ an }^2}left( {{{90}^ circ } - {{59}^ circ }}
ight)}
ight)}} = frac{2}{m} cr
& Rightarrow frac{{{{sec }^2}{{70}^ circ } - { ext{ta}}{{ ext{n}}^2}{{70}^ circ }}}{{2left( {{ ext{cose}}{{ ext{c}}^2}{{59}^ circ } - {{cot }^2}{{59}^ circ }}
ight)}} = frac{2}{m} cr
& Rightarrow frac{1}{2} = frac{2}{m}left[ {{{sec }^2} heta - { ext{ta}}{{ ext{n}}^2} heta = 1}
ight] cr
& (cose{c^2} heta - {cot ^2} heta = 1) cr
& Rightarrow m = 2 imes 2 cr
& Rightarrow m = 4 cr} $$