Trigonometry - Study Mode
[#231] If $$frac{{ an heta + sin heta }}{{ an heta - sin heta }} = frac{{{ ext{k}} + 1}}{{{ ext{k}} - 1}},$$ xa0 xa0 then k = ?
Correct Answer
(B) secθ
Explanation
Solution: $$eqalign{
& frac{{ an heta + sin heta }}{{ an heta - sin heta }} = frac{{{ ext{k}} + 1}}{{{ ext{k}} - 1}} cr
& frac{{ an heta }}{{sin heta }} = { ext{k}} cr
& { ext{k}} = sec heta cr} $$
[#232] If $$sin heta = frac{{{p^2} - 1}}{{{p^2} + 1}},$$ xa0 then cosθ is equal to:
Correct Answer
(A) $$frac{{2p}}{{1 + {p^2}}}$$
Explanation
Solution: $$eqalign{
& sin heta = frac{{{p^2} - 1}}{{{p^2} + 1}} cr
& { ext{Let }}p = 2 cr} $$ $$eqalign{
& { ext{Now from option A}} cr
& frac{{2 imes 2}}{{1 + 4}} = frac{4}{5} cr} $$
[#233] If $$left( {frac{{ an heta - sec heta + 1}}{{ an heta + sec heta - 1}}}
ight)sec heta = frac{1}{{ ext{k}}},$$ xa0 xa0 xa0then k = . . . . . . . .
Correct Answer
(A) 1 + sinθ
Explanation
Solution: $$eqalign{
& left( {frac{{ an heta - sec heta + 1}}{{ an heta + sec heta - 1}}}
ight)sec heta = frac{1}{k} cr
& left( {frac{{ an heta - sec heta + 1}}{{frac{1}{{sec heta - an heta }} - 1}}}
ight)sec heta = frac{1}{k} cr
& frac{1}{k} = left( {frac{{ an heta - sec heta + 1}}{{1 - sec heta + an heta }}}
ight)left( {sec heta - an heta }
ight)sec heta cr
& frac{1}{k} = {sec ^2} heta - an heta sec heta cr
& frac{1}{k} = frac{1}{{{{cos }^2} heta }} - frac{{sin heta }}{{{{cos }^2} heta }} cr
& k = frac{{{{cos }^2} heta }}{{left( {1 - sin heta }
ight)}}frac{{left( {1 + sin heta }
ight)}}{{left( {1 + sin heta }
ight)}} cr
& k = frac{{{{cos }^2} heta left( {1 + sin heta }
ight)}}{{{{cos }^2} heta }} cr
& k = 1 + sin heta cr} $$
[#234] The value of m[sinθ + 2cos 2 ∅ + 3sinθ + 4cos 2 ∅ + . . . . . . . . + 18cos 2 ∅] is a perfect square of an integer, θ = 30°, ∅ = 45° and 150 ≤ m ≤ 180. Find the value of m.
Correct Answer
(C) 152
Explanation
Solution: $$eqalign{
& mleft[ {sin heta + 2{{cos }^2}phi + 3sin heta + 4{{cos }^2}phi + ,.....,18{{cos }^2}phi }
ight] cr
& heta = {30^ circ },,,phi = {45^ circ } cr
& = mleft[ {left( {sin heta + 3sin heta + 5sin heta + ,.....,17sin heta }
ight) + left( {2{{cos }^2}phi + 4{{cos }^2}phi + ,.....,18{{cos }^2}phi }
ight)}
ight] cr
& Rightarrow { ext{Sum of odd number}} = {left( 9
ight)^2}, cr
& { ext{Sum of even number}} = 9 imes 10 cr
& = mleft[ {81sin heta + 90{{cos }^2}phi }
ight] cr
& = mleft[ {81 imes sin 30 + 90 imes {{cos }^2}45}
ight] cr
& = mleft[ {81 imes frac{1}{2} + 90 imes frac{1}{2}}
ight] cr
& = m imes frac{{171}}{2} cr
& { ext{Now, we will check through the option,}} cr
& { ext{Putting }}m = 152 cr
& = 152 imes frac{{171}}{2} cr
& = 4 imes 19 imes 19 imes 9 cr
& = {left( {2 imes 19 imes 3}
ight)^2} cr
& = { ext{ perfect square}}{ ext{. Ans}}{ ext{.}} cr} $$
[#235] If 1 + sin 2 θ - 3sinθcosθ = 0, then the value of cotθ is:
Correct Answer
(A) 2
Explanation
Solution: 1 + sin 2 θ - 3sinθ.cosθ = 0 ⇒ 1 - 2sinθ.cosθ = sinθ.cosθ - sin 2 θ ⇒ sin 2 θ + cos 2 θ - 2sinθ.cosθ = sinθ(cosθ - sinθ) ⇒ (cosθ - sinθ) 2 = sinθ(cosθ - sinθ) ⇒ cosθ - sinθ = sinθ ⇒ cosθ = 2sinθ ⇒ cotθ = 2