Trigonometry - Study Mode
[#251] If secA + tanA = a, then the value of cosA is?
Correct Answer
(B) $$frac{{2a}}{{{a^2} + 1}}$$
Explanation
Solution: $$eqalign{
& { ext{secA}} + { ext{tanA}} = a cr
& { ext{we know that}} cr
& Rightarrow { ext{se}}{{ ext{c}}^2}{ ext{A}} - { ext{ta}}{{ ext{n}}^2}{ ext{A}} = 1 cr
& Rightarrow left( {{ ext{secA}} - { ext{tanA}}}
ight)left( {{ ext{secA}} + { ext{tanA}}}
ight) = 1 cr
& Rightarrow { ext{secA}} - { ext{tanA}} = frac{1}{a} cr
& Rightarrow { ext{secA}} + { ext{tanA}} = a cr
& Rightarrow { ext{2secA}} = a + frac{1}{a} cr
& Rightarrow { ext{2secA}} = frac{{{a^2} + 1}}{a} cr
& Rightarrow sec heta = frac{{{a^2} + 1}}{{2a}} cr
& { ext{So, }}cos heta = frac{{2a}}{{{a^2} + 1}} cr} $$
[#252] If sinP + cosecP = 2, then the value of sin 7 P + cosec 7 P is?
Correct Answer
(B) 2
Explanation
Solution: $$eqalign{
& { ext{sin P}} + { ext{cosec P}} = 2 cr
& { ext{For P}} = {90^ circ } cr
& Rightarrow { ext{sin }}{90^ circ } + { ext{cosec }}{90^ circ } = 2 cr
& Rightarrow 1 + 1 = 2 cr
& Rightarrow 2 = 2left( {{ ext{satisfy}}}
ight) cr
& { ext{So, }}{sin ^7}{ ext{P}} + { ext{cose}}{{ ext{c}}^7}{ ext{P}} cr
& Rightarrow {sin ^7}{90^ circ } + { ext{cose}}{{ ext{c}}^7}{90^ circ } cr
& Rightarrow {1^7} + {1^7} cr
& Rightarrow 2 cr} $$
[#253] The value of the expression 2(sin 6 θ + cos 6 θ) - 3(sin 4 θ + cos 4 θ) + 1 is?
Correct Answer
(B) 0
Explanation
Solution: $${ ext{ 2}}left( {{{sin }^6} heta + { ext{co}}{{ ext{s}}^6} heta }
ight) - { ext{3}}left( {{{sin }^4} heta + { ext{co}}{{ ext{s}}^4} heta }
ight){ ext{ + 1}}$$ $$ Rightarrow { ext{2}}left( {1 - 3{{sin }^2} heta { ext{co}}{{ ext{s}}^2} heta }
ight) - $$ xa0 xa0 $${ ext{3}}left( {1 - 2{{sin }^2} heta c{ ext{o}}{{ ext{s}}^2} heta }
ight){ ext{ + }}$$ xa0 xa0 $${ ext{1}}$$ $$eqalign{
& Rightarrow { ext{ 2}} - 6{sin ^2} heta .{ ext{co}}{{ ext{s}}^2} heta - 3 + 6{sin ^2} heta .{ ext{co}}{{ ext{s}}^2} heta { ext{ + 1}} cr
& Rightarrow 2 - 3 + 1 cr
& Rightarrow 0 cr} $$
[#254] If $${ ext{cos}} heta = frac{{{x^2} - {y^2}}}{{{x^2} + {y^2}}}$$ xa0xa0 then the value of$${ ext{cot}} heta $$xa0 is equal to $$left[ {{ ext{if }}{0^ circ } leqslant heta leqslant {{90}^ circ }}
ight]$$
Correct Answer
(D) $$frac{{{x^2} - {y^2}}}{{2xy}}$$
Explanation
Solution: $${ ext{cos}} heta = frac{{{x^2} - {y^2}}}{{{x^2} + {y^2}}}$$ AC 2 = (x 2 + y 2 ) 2 - (x 2 - y 2 ) 2 = x 4 + y 4 + 2x 2 y 2 - x 4 - y 4 + 2x 2 y 2 = 4x 2 y 2 ⇒ AC = 2xy ⇒ cotθ = $$frac{{{x^2} - {y^2}}}{{2xy}}$$
[#255] If x = cosecθ - sinθ and y = secθ - cosθ, then the relation between x and y is?
Correct Answer
(B) x 2 y 2 (x 2 + y 2 + 3) = 1
Explanation
Solution: $$eqalign{
& x = { ext{cosec}} heta - sin heta { ext{ }} cr
& y = sec heta - cos heta cr
& { ext{Put }} heta = {45^ circ } cr
& x = sqrt 2 - frac{1}{{sqrt 2 }} = frac{1}{{sqrt 2 }} cr
& y = sqrt 2 - frac{1}{{sqrt 2 }} = frac{1}{{sqrt 2 }} cr
& { ext{by options (B) }}{x^2}{y^2}left( {{x^2} + {y^2} + 3}
ight) cr
& = {left( {frac{1}{{sqrt 2 }}}
ight)^2} imes {left( {frac{1}{{sqrt 2 }}}
ight)^2} cr
& left[ {{{left( {frac{1}{{sqrt 2 }}}
ight)}^2} imes {{left( {frac{1}{{sqrt 2 }}}
ight)}^2} + 3}
ight] cr
& = frac{1}{2} imes frac{1}{2}left( {frac{1}{2} + frac{1}{2} + 3}
ight) cr
& = frac{1}{4}left( {1 + 3}
ight) cr
& = 1{ ext{ }}left( {{ ext{satisfy}}}
ight) cr} $$