Trigonometry - Study Mode

[#106] If $$ heta + phi = frac{pi }{2}$$ xa0 and $$sin heta = frac{1}{2},$$ xa0 then the value of $${ ext{sin}}phi $$ xa0is?
Correct Answer

(D) $$frac{{sqrt 3 }}{2}$$

Explanation

Solution: $$eqalign{
& heta + phi = frac{pi }{2} cr
& heta + phi = {90^ circ },......(i) cr
& sin heta = frac{1}{2} cr
& sin heta = { ext{sin 3}}{0^ circ } = frac{1}{2},......(ii) cr
& { ext{Put }} heta = {30^ circ }{ ext{ in equation (i)}} cr
& {30^ circ } + phi = {90^ circ } cr
& phi = {60^ circ } cr
& sin phi = sin {60^ circ } = frac{{sqrt 3 }}{2} cr} $$

[#107] Find the value of the following 3(sin 4 θ + cos 4 θ) + 2(sin 6 θ + cos 6 θ) + 12sin 2 θ.cos 2 θ = ?
Correct Answer

(D) 5

Explanation

Solution: $$eqalign{
& { ext{The value of }} cr
& { ext{3}}left( {{{sin }^4} heta + { ext{co}}{{ ext{s}}^4} heta }
ight) + 2left( {{{sin }^6} heta + { ext{co}}{{ ext{s}}^6} heta }
ight) + 12{sin ^2} heta .{ ext{co}}{{ ext{s}}^2} heta cr
& { ext{Using }} heta = {0^ circ } cr
& x08ecause sin {0^ circ } = {0^ circ } cr
& cos {0^ circ } = 1 cr
& Rightarrow 3left( {0 + {1^4}}
ight) + 2left( {0 + {1^6}}
ight) + 12 imes 0 imes 1 cr
& Rightarrow 3 + 2 cr
& Rightarrow 5 cr} $$

[#108] The numerical value of $$frac{{{ ext{co}}{{ ext{s}}^2}{{45}^ circ }}}{{{{sin }^2}{{60}^ circ }}}$$ xa0+ $$frac{{{ ext{co}}{{ ext{s}}^2}{{60}^ circ }}}{{{{sin }^2}{{45}^ circ }}}$$ xa0- $$frac{{{ ext{ta}}{{ ext{n}}^2}{{30}^ circ }}}{{{ ext{co}}{{ ext{t}}^2}{{45}^ circ }}}$$ xa0- $$frac{{{{sin }^2}{{30}^ circ }}}{{{ ext{co}}{{ ext{t}}^2}{{30}^ circ }}}$$ xa0is?
Correct Answer

(A) $$frac{3}{4}$$

Explanation

Solution: $$eqalign{
& frac{{{ ext{co}}{{ ext{s}}^2}{{45}^ circ }}}{{{{sin }^2}{{60}^ circ }}}{ ext{ + }}frac{{{ ext{co}}{{ ext{s}}^2}{{60}^ circ }}}{{{{sin }^2}{{45}^ circ }}} - frac{{{ ext{ta}}{{ ext{n}}^2}{{30}^ circ }}}{{{ ext{co}}{{ ext{t}}^2}{{45}^ circ }}} - frac{{{{sin }^2}{{30}^ circ }}}{{{ ext{co}}{{ ext{t}}^2}{{30}^ circ }}} cr
& Rightarrow frac{{{{left( {frac{1}{{sqrt 2 }}}
ight)}^2}}}{{{{left( {frac{{sqrt 3 }}{2}}
ight)}^2}}} + frac{{{{left( {frac{1}{2}}
ight)}^2}}}{{{{left( {frac{1}{{sqrt 2 }}}
ight)}^2}}} - frac{{{{left( {frac{1}{{sqrt 3 }}}
ight)}^2}}}{{{{left( 1
ight)}^2}}} - frac{{{{left( {frac{1}{2}}
ight)}^2}}}{{{{left( {sqrt 3 }
ight)}^2}}} cr
& Rightarrow left( {frac{1}{2} imes frac{4}{3}}
ight) + left( {frac{1}{4} imes frac{2}{1}}
ight) - left( {frac{1}{3} imes 1}
ight) - left( {frac{1}{4} imes frac{1}{3}}
ight) cr
& Rightarrow frac{2}{3} + frac{1}{2} - frac{1}{3} - frac{1}{{12}} cr
& Rightarrow frac{1}{3} + frac{1}{2} - frac{1}{{12}} cr
& Rightarrow frac{{4 + 6 - 1}}{{12}} cr
& Rightarrow frac{9}{{12}} cr
& Rightarrow frac{3}{4} cr} $$

[#109] If θ be a positive acute angle satisfying cos 2 θ + cos 4 θ = 1, then the value of tan 2 θ + tan 4 θ is?
Correct Answer

(B) 1

Explanation

Solution: $$eqalign{
& { ext{co}}{{ ext{s}}^2} heta + { ext{co}}{{ ext{s}}^4} heta = 1 cr
& Rightarrow { ext{co}}{{ ext{s}}^4} heta = 1 - {cos ^2} heta cr
& Rightarrow { ext{co}}{{ ext{s}}^4} heta = {sin ^2} heta cr
& Rightarrow {cos ^2} heta .{cos ^2} heta = {sin ^2} heta cr
& Rightarrow {cos ^2} heta = frac{{{{sin }^2} heta }}{{{{cos }^2} heta }} cr
& Rightarrow { ext{co}}{{ ext{s}}^2} heta = { ext{ta}}{{ ext{n}}^2} heta cr
& Rightarrow { ext{ta}}{{ ext{n}}^2} heta + { ext{ta}}{{ ext{n}}^4} heta cr
& Rightarrow { ext{co}}{{ ext{s}}^2} heta + { ext{co}}{{ ext{s}}^4} heta = 1 cr} $$

[#110] If $${ ext{tan}} heta = frac{4}{3}{ ext{,}}$$ xa0 then the value of $$frac{{3sin heta + 2{ ext{cos}} heta }}{{3sin heta - 2{ ext{cos}} heta }}$$ xa0 is?
Correct Answer

(C) 3.0

Explanation

Solution: $$frac{{3sin heta + 2{ ext{cos}} heta }}{{3sin heta - 2{ ext{cos}} heta }}$$ Divide numerator & denominator by cosθ $$eqalign{
& = frac{{frac{{3sin heta }}{{cos heta }} + frac{{2cos heta }}{{cos heta }}}}{{frac{{3sin heta }}{{cos heta }} - frac{{2cos heta }}{{cos heta }}}}left[ {frac{{sin heta }}{{cos heta }} = an heta }
ight] cr
& = frac{{3 an heta + 2}}{{3 an heta - 2}} cr
& { ext{Put value of tan}} heta cr
& = frac{{3 imes frac{4}{3} + 2}}{{3 imes frac{4}{3} - 2}} cr
& = frac{6}{2} cr
& = 3 cr} $$