Trigonometry - Study Mode
[#241] If $$sin left( { heta + {{30}^ circ }}
ight) = frac{3}{{sqrt {12} }}{ ext{,}}$$ xa0 xa0 then find $${ ext{co}}{{ ext{s}}^2} heta ?$$
Correct Answer
(B) $$frac{3}{4}$$
Explanation
Solution: $$eqalign{
& sin left( { heta + {{30}^ circ }}
ight) = frac{3}{{sqrt {12} }} = frac{3}{{2sqrt 3 }} cr
& = frac{{sqrt 3 }}{2} = sin left( { heta + {{30}^ circ }}
ight) = sin {60^ circ } cr
& herefore heta = {30^ circ } cr
& { ext{co}}{{ ext{s}}^2} heta = { ext{co}}{{ ext{s}}^2}{30^ circ } cr
& = {left( {frac{{sqrt 3 }}{2}}
ight)^2} cr
& = frac{3}{4} cr} $$
[#242] If $${ ext{0}} leqslant heta leqslant {90^ circ }$$ xa0 and $$4{cos ^2} heta $$ xa0 - $$4sqrt 3 cos heta $$ xa0 + 3 = 0 then the value of $$ heta $$ is?
Correct Answer
(A) 30°
Explanation
Solution: $$eqalign{
& 4{cos ^2} heta - 4sqrt 3 cos heta + 3 = 0 cr
& { ext{Hit and Trial method}} cr
& { ext{Put }} heta = {30^ circ }{ ext{option A}} cr
& 4{cos ^2}{30^ circ } - 4sqrt 3 cos {30^ circ } + 3 = 0 cr
& Rightarrow 4left( {frac{3}{4}}
ight) - 4sqrt 3 left( {frac{{sqrt 3 }}{2}}
ight) + 3 = 0 cr
& Rightarrow 0 = 0 cr} $$
[#243] The value of sec 4 A(1 - sin 4 A) - 2tan 2 A is?
Correct Answer
(D) 1
Explanation
Solution: $$eqalign{
& { ext{According to the question ,}} cr
& { ext{ se}}{{ ext{c}}^4}{ ext{A}}left( {1 - {{sin }^4}{ ext{A}}}
ight) - 2{ ext{ta}}{{ ext{n}}^2}{ ext{A}} cr
& { ext{Put A}} = {45^ circ } cr
& Rightarrow { ext{ se}}{{ ext{c}}^4}{45^ circ }left( {1 - {{sin }^4}{{45}^ circ }}
ight) - 2{ ext{ta}}{{ ext{n}}^2}{45^ circ } cr
& Rightarrow 4left( {1 - frac{1}{4}}
ight) - 2 cr
& Rightarrow 4 imes frac{3}{4} - { ext{2}} cr
& Rightarrow 3 - 2 cr
& Rightarrow 1 cr} $$
[#244] If $${ ext{co}}{{ ext{s}}^4} heta - {sin ^4} heta = frac{2}{3}{ ext{,}}$$ xa0 xa0 then the value of $${ ext{2co}}{{ ext{s}}^2} heta - 1$$ xa0 is?
Correct Answer
(C) $$frac{2}{3}$$
Explanation
Solution: $$eqalign{
& { ext{co}}{{ ext{s}}^4} heta - {sin ^4} heta = frac{2}{3} cr
& left[ {{a^4} - {b^4} = left( {{a^2} - {b^2}}
ight)left( {{a^2} + {b^2}}
ight)}
ight] cr
& Rightarrow left( {{ ext{co}}{{ ext{s}}^2} heta + {{sin }^2} heta }
ight)left( {{ ext{co}}{{ ext{s}}^2} heta - {{sin }^2} heta }
ight) = frac{2}{3} cr
& left[ {{a^2} - {b^2} = left( {a - b}
ight)left( {a + b}
ight)}
ight] cr
& Rightarrow 1 imes left( {{ ext{co}}{{ ext{s}}^2} heta - {{sin }^2} heta }
ight) = frac{2}{3} cr
& Rightarrow { ext{co}}{{ ext{s}}^2} heta - left( {1 - { ext{co}}{{ ext{s}}^2} heta }
ight) = frac{2}{3} cr
& left[ {{{sin }^2} heta = 1 - { ext{co}}{{ ext{s}}^2} heta }
ight] cr
& Rightarrow 2{ ext{co}}{{ ext{s}}^2} heta - 1 = frac{2}{3} cr} $$
[#245] If $$sin heta - cos heta = frac{7}{{13}}$$ xa0xa0 and $${0^ circ }{ ext{ < }} heta { ext{ < }}{90^ circ }{ ext{,}}$$ xa0 then the value of $$sin heta $$ xa0+ $${ ext{cos}} heta $$xa0 is?
Correct Answer
(A) $$frac{{17}}{{13}}$$
Explanation
Solution: $$eqalign{
& sin heta - cos heta = frac{7}{{13}} = alpha cr
& { ext{When, }} cr
& ax + by = m,......(i) cr
& bx - ay = n,......(ii) cr} $$ By adding these two equations after making square on both sides we get, $$eqalign{
& left( {{a^2} + {b^2}}
ight)left( {{x^2} + {y^2}}
ight) = {m^2} + {n^2} cr
& { ext{In the same process}} cr
& sin heta pm cos heta = a cr
& { ext{Then,}}sin heta pm cos heta = sqrt {2 - {a^2}} cr
& Rightarrow sin heta + cos heta = sqrt {2 - {{left( {frac{7}{{13}}}
ight)}^2}} cr
& Rightarrow sin heta + cos heta = sqrt {2 - left( {frac{{49}}{{169}}}
ight)} cr
& Rightarrow sin heta + cos heta = sqrt {frac{{289}}{{169}}} cr
& Rightarrow sin heta + cos heta = frac{{17}}{{13}} cr} $$