Trigonometry - Study Mode

[#246] If $$cot heta = 4{ ext{,}}$$ xa0 then the value of $$frac{{5sin heta + 3cos heta }}{{5sin heta - 3cos heta }}$$ xa0 is?
Correct Answer

(A) $$ - frac{{17}}{7}$$

Explanation

Solution: $$eqalign{
& cot heta = 4 cr
& herefore frac{{5sin heta + 3cos heta }}{{5sin heta - 3cos heta }} cr
& = frac{{5 + 3cot heta }}{{5 - 3cot heta }} cr
& = frac{{5 + 3 imes 4}}{{5 - 3 imes 4}} cr
& = - frac{{17}}{7} cr} $$

[#247] The value of cos 2 20° + cos 2 70° is?
Correct Answer

(D) 1

Explanation

Solution: $$eqalign{
& { ext{co}}{{ ext{s}}^2}{20^ circ } + { ext{co}}{{ ext{s}}^2}{70^ circ } cr
& = { ext{co}}{{ ext{s}}^2}left( {{{90}^ circ } - {{70}^ circ }}
ight) + { ext{co}}{{ ext{s}}^2}{70^ circ } cr
& = {sin ^2}{70^ circ } + { ext{co}}{{ ext{s}}^2}{70^ circ } cr
& = 1 cr} $$

[#248] If rsinθ = 1, rcosθ = $$sqrt 3 { ext{,}}$$ xa0then the value of r 2 tanθ is?
Correct Answer

(C) $$frac{4}{{sqrt 3 }}$$

Explanation

Solution: $$eqalign{
& rsin heta = 1,{ ext{ }}rcos heta = sqrt 3 cr
& { ext{Put }} heta = {30^ circ } cr
& r = 2 cr
& { ext{So}},{ ext{ }}{r^2} an heta cr
& = {left( 2
ight)^2} imes { ext{tan}}{30^ circ } cr
& = 4 imes frac{1}{{sqrt 3 }} cr
& = frac{4}{{sqrt 3 }} cr} $$

[#249] If secθ + tanθ = m(>1), then the value of sinθ is (0° < θ < 90°)
Correct Answer

(B) $$frac{{{m^2} - 1}}{{{m^2} + 1}}$$

Explanation

Solution: $$eqalign{
& sec heta + an heta = m,.......(i) cr
& { ext{then, }}sec heta - an heta = frac{1}{m},......(ii) cr
& Because{ ext{ }}{sec ^2} heta - { ext{ta}}{{ ext{n}}^2} heta = 1 cr
& { ext{From equation (i)}} - { ext{(ii)}} cr
& 2 an heta = m - frac{1}{m} cr} $$ $$eqalign{
& { ext{tan}} heta = frac{{{m^2} - 1}}{{2m}} cr
& sin heta = frac{{{m^2} - 1}}{{{m^2} + 1}} cr} $$

[#250] The expression of $$frac{{cot heta + operatorname{cosec} heta - 1}}{{cot heta + operatorname{cosec} heta + 1}}$$ xa0xa0 is equal to?
Correct Answer

(B) $$frac{{1 - cos heta }}{{sin heta }}$$

Explanation

Solution: $$eqalign{
& frac{{cot heta + operatorname{cosec} heta - 1}}{{cot heta + operatorname{cosec} heta + 1}} cr
& { ext{Put }} heta = {45^ circ } cr
& = frac{{1 + sqrt 2 - 1}}{{1 + sqrt 2 + 1}} cr
& = frac{{sqrt 2 }}{{2 + sqrt 2 }} cr
& = frac{{sqrt 2 }}{{sqrt 2 left( {sqrt 2 + 1}
ight)}} cr
& = frac{1}{{sqrt 2 + 1}} cr
& = sqrt 2 - 1 cr
& { ext{Now option B}} cr
& frac{{1 - cos heta }}{{sin heta }} cr
& = frac{{1 - frac{1}{{sqrt 2 }}}}{{frac{1}{{sqrt 2 }}}} cr
& = sqrt 2 - 1{ ext{ }}left( {{ ext{Satisfy}}}
ight) cr} $$