Trigonometry - Study Mode

[#181] If $$frac{{{{sin }^2} heta - 3sin heta + 2}}{{{{cos }^2} heta }} = 1,$$ xa0 xa0 where 0° < θ < 90°, then what is the value of (cos2θ + sin3θ + cosec2θ)?
Correct Answer

(C) $$frac{{9 + 4sqrt 3 }}{6}$$

Explanation

Solution: $$eqalign{
& frac{{{{sin }^2} heta - 3sin heta + 2}}{{{{cos }^2} heta }} = 1 cr
& { ext{Put }} heta = {30^ circ } cr
& frac{{{{sin }^2}{{30}^ circ } - 3sin {{30}^ circ } + 2}}{{{{cos }^2}{{30}^ circ }}} = 1 cr
& frac{1}{4} - frac{3}{2} + 2 = frac{3}{4} cr
& frac{3}{4} = frac{3}{4} cr
& cos 2 heta + sin 3 heta + { ext{cosec}},2 heta cr
& = cos {60^ circ } + sin {90^ circ } + { ext{cosec}},{60^ circ } cr
& = 1 + frac{1}{2} + frac{2}{{sqrt 3 }} cr
& = frac{{9 + 4sqrt 3 }}{6} cr} $$

[#182] The value of $$sqrt {{{sec }^2} heta + { ext{cose}}{{ ext{c}}^2} heta } imes sqrt {{{ an }^2} heta - {{sin }^2} heta } $$ xa0 xa0 xa0 is equal to:
Correct Answer

(B) sinθsec 2 θ

Explanation

Solution: $$eqalign{
& sqrt {{{sec }^2} heta + { ext{cose}}{{ ext{c}}^2} heta } imes sqrt {{{ an }^2} heta - {{sin }^2} heta } cr
& = sqrt {1 + {{ an }^2} heta + 1 + {{cot }^2} heta } imes sqrt {{{sin }^2} heta left( {frac{1}{{{{cos }^2} heta }} - 1}
ight)} cr
& = sqrt {2 + {{ an }^2} heta + {{cot }^2} heta } imes sqrt {frac{{{{sin }^2} heta .{{sin }^2} heta }}{{{{cos }^2} heta }}} cr
& = left( { an heta + cot heta }
ight) imes left( {frac{{{{sin }^2} heta }}{{cos heta }}}
ight) cr
& = left( {frac{{sin heta }}{{cos heta }} + frac{{cos heta }}{{sin heta }}}
ight) imes left( {frac{{{{sin }^2} heta }}{{cos heta }}}
ight) cr
& = frac{1}{{sin heta .cos heta }} imes frac{{{{sin }^2} heta }}{{cos heta }} cr
& = sin heta .{sec ^2} heta cr} $$

[#183] If tanθ + secθ = 7, θ being acute, then the value of 5sinθ is:
Correct Answer

(D) $$frac{{24}}{5}$$

Explanation

Solution: $$eqalign{
& sec heta + an heta = 7........left( { ext{i}}
ight) cr
& sec heta - an heta = frac{1}{7}........left( {{ ext{ii}}}
ight) cr
& { ext{From equation }}left( { ext{i}}
ight){ ext{ and }}left( {{ ext{ii}}}
ight) cr
& 2sec heta = 7 + frac{1}{7} = frac{{50}}{7} cr
& sec heta = frac{{25 o h}}{{7 o b}},,,,,p = 24 cr
& sin heta = frac{{24}}{{25}} cr
& 5sin heta = 5 imes frac{{24}}{{25}} = frac{{24}}{5} cr} $$

[#184] Evaluate the following: $$frac{{cos 2 heta cdot cos 3 heta - cos 2 heta cdot cos 7 heta + cos heta cdot cos 10 heta }}{{sin 4 heta cdot sin 3 heta - sin 2 heta cdot sin 5 heta + sin 4 heta cdot sin 7 heta }}$$
Correct Answer

(A) cot6θ.cot5θ

[#185] If sec 2 θ + tan 2 θ = $$3frac{1}{2},$$ xa00° < θ < 90°, then (cosθ + sinθ) is equal to
Correct Answer

(B) $$frac{{2 + sqrt 5 }}{3}$$

Explanation

Solution: $$eqalign{
& {sec ^2} heta + { an ^2} heta = 3frac{1}{2} cr
& Rightarrow 1 + { an ^2} heta + { an ^2} heta = frac{7}{2} cr
& Rightarrow 2{ an ^2} heta = frac{7}{2} - 1 cr
& Rightarrow 2{ an ^2} heta = frac{5}{2} cr
& Rightarrow 3{ an ^2} heta = frac{5}{4} cr
& Rightarrow an heta = frac{{sqrt 5 o P}}{{2 o B}} cr
& H = sqrt {5 + 4} = 3 cr
& herefore ,cos heta + sin heta cr
& = frac{2}{3} + frac{{sqrt 5 }}{3} cr
& = frac{{2 + sqrt 5 }}{3} cr} $$