Trigonometry - Study Mode
[#176] What is the value of $$frac{{{{left[ {1 - an left( {{{90}^ circ } - heta }
ight)}
ight]}^2}}}{{left[ {{{cos }^2}left( {{{90}^ circ } - heta }
ight)}
ight]}} - 1 = ?$$
Correct Answer
(A) -sin2θ
Explanation
Solution: $$eqalign{
& frac{{{{left[ {1 - an left( {{{90}^ circ } - heta }
ight)}
ight]}^2}}}{{left[ {{{cos }^2}left( {{{90}^ circ } - heta }
ight)}
ight]}} - 1 cr
& { ext{By putting }} heta = {45^ circ } cr
& Rightarrow frac{{{{left[ {1 - an left( {{{90}^ circ } - {{45}^ circ }}
ight)}
ight]}^2}}}{{left[ {{{cos }^2}left( {{{90}^ circ } - {{45}^ circ }}
ight)}
ight]}} - 1 cr
& Rightarrow frac{{{{left[ {1 - an {{45}^ circ }}
ight]}^2}}}{{{{cos }^2}{{45}^ circ }}} - 1 cr
& Rightarrow 0 - 1 cr
& Rightarrow - 1 cr
& { ext{By satisfying in option A}} cr
& Rightarrow - sin 2 heta cr
& Rightarrow - sin {90^ circ } cr
& Rightarrow - 1 cr} $$
[#177] What is the value of cos15° - cos165°?
Correct Answer
(C) $$frac{{sqrt 3 + 1}}{{sqrt 2 }}$$
Explanation
Solution: [x08egin{array}{l}
cos {15^ circ } - cos {165^ circ }\
= cos {15^ circ } - cos left( {{{180}^ circ } - {{15}^ circ }}
ight)\
= cos {15^ circ } + cos {15^ circ }\
= 2cos {15^ circ }\
= 2left( {frac{{sqrt 3 + 1}}{{2sqrt 2 }}}
ight)\
= frac{{sqrt 3 + 1}}{{sqrt 2 }}\
{x08f{Note:}}\
left[ x08egin{array}{l}
cos {15^ circ } = cos left( {{{45}^ circ } - {{30}^ circ }}
ight)\
= cos {45^ circ }cos {30^ circ } + sin {45^ circ }sin {30^ circ }\
= frac{1}{{sqrt 2 }} imes frac{{sqrt 3 }}{2} + frac{1}{{sqrt 2 }} imes frac{1}{2}\
= frac{{sqrt 3 + 1}}{{2sqrt 2 }}
end{array}
ight]
end{array}]
[#178] If 4 - 2sin 2 θ - 5cosθ = 0, 0° < θ < 90°, then the value of sinθ + tanθ is:
Correct Answer
(B) $$frac{{3sqrt 3 }}{2}$$
Explanation
Solution: $$eqalign{
& 4 - 2{sin ^2} heta - 5cos heta = 0 cr
& { ext{Let }} heta = {60^ circ } cr
& 4 - 2{sin ^2}{60^ circ } - 5cos {60^ circ } = 0 cr
& 4 - 2 imes frac{3}{4} - 5 imes frac{1}{2} = 0 cr
& 4 - 4 = 0 cr
& sin heta + an heta = sin {60^ circ } + an {60^ circ } cr
& = frac{{sqrt 3 }}{2} + sqrt 3 cr
& = frac{{3sqrt 3 }}{2} cr} $$
[#179] The value of $$left( {frac{{sin A}}{{1 - cos A}} + frac{{1 - cos A}}{{sin A}}}
ight) div left( {frac{{{{cot }^2}A}}{{1 + { ext{cosec}},A}} + 1}
ight){ ext{is:}}$$
Correct Answer
(D) 2
Explanation
Solution: $$eqalign{
& left( {frac{{sin A}}{{1 - cos A}} + frac{{1 - cos A}}{{sin A}}}
ight) div left( {frac{{{{cot }^2}A}}{{1 + { ext{cosec}},A}} + 1}
ight) cr
& = left( {frac{{{{sin }^2}A + {{left( {1 + cos A}
ight)}^2}}}{{sin Aleft( {1 - cos A}
ight)}}}
ight) div left( {frac{{frac{{{{cos }^2}A}}{{{{sin }^2}A}} imes sin A}}{{1 + sin A}} + 1}
ight) cr
& = frac{{{{sin }^2}A + 1 + {{cos }^2}A - 2cos A}}{{sin Aleft( {1 - cos A}
ight)}} div frac{{{{cos }^2}A + sin A + {{sin }^2}A}}{{sin Aleft( {1 + sin A}
ight)}} cr
& = frac{{2left( {1 - cos A}
ight)}}{{sin Aleft( {1 - cos A}
ight)}} imes frac{{sin Aleft( {1 + sin A}
ight)}}{{left( {1 + sin A}
ight)}} cr
& = 2 cr} $$
[#180] Evaluate the following expression in terms of trigonometric ratios. $$frac{{{{cot }^2}Aleft( {sec A - 1}
ight)}}{{1 + sin A}}$$
Correct Answer
(B) $$frac{{{{sec }^2}Aleft( {1 - sin A}
ight)}}{{1 + sec A}}$$
Explanation
Solution: $$eqalign{
& frac{{{{cot }^2}Aleft( {sec A - 1}
ight)}}{{left( {1 + sin A}
ight)}} cr
& = frac{{{{cos }^2}Aleft( {1 - cos A}
ight)}}{{{{sin }^2}Aleft( {1 + sin A}
ight).cos A}} cr
& = frac{{left( {1 - {{sin }^2}A}
ight)}}{{left( {1 - {{cos }^2}A}
ight)}} imes frac{{left( {1 - cos A}
ight)}}{{left( {1 + sin A}
ight).cos A}} cr
& = frac{{left( {1 + sin A}
ight)left( {1 - sin A}
ight).left( {1 - cos A}
ight)}}{{left( {1 + cos A}
ight)left( {1 - cos A}
ight).left( {1 + sin A}
ight).cos A}} cr
& = frac{{frac{{sec Aleft( {1 - sin A}
ight)}}{{left( {sec A + 1}
ight)}}}}{{sec A}} cr
& = frac{{{{sec }^2}Aleft( {1 - sin A}
ight)}}{{left( {1 + sec A}
ight)}} cr} $$