Trigonometry - Study Mode

[#146] If θ be acute angle and tan(4θ - 50°) = cot(50° - θ), then the value of θ in degrees is?
Correct Answer

(A) 30°

Explanation

Solution: $$eqalign{
& { ext{We know that }} cr
& { ext{tan}}left( {{{90}^ circ } - heta }
ight) = { ext{cot}} heta cr
& { ext{and, cot}}left( {{{90}^ circ } - heta }
ight) = { ext{tan}} heta cr
& Rightarrow { ext{tan}}left( {4 heta - {{50}^ circ }}
ight) = { ext{cot}}left( {{{50}^ circ } - heta }
ight) cr
& Rightarrow cot left[ {{{90}^ circ } - left( {4 heta - {{50}^ circ }}
ight)}
ight] = { ext{cot}}left( {{{50}^ circ } - heta }
ight) cr
& Rightarrow {90^ circ } - left( {4 heta - {{50}^ circ }}
ight) = left( {{{50}^ circ } - heta }
ight) cr
& Rightarrow {90^ circ } - 4 heta + {50^ circ } = {50^ circ } - heta cr
& Rightarrow {90^ circ } = 3 heta cr
& { ext{then}}, heta = {30^ circ } cr} $$

[#147] The value of the following is : $$frac{{{{left( { an {{20}^ circ }}
ight)}^2}}}{{{{left( {{ ext{cosec 7}}{0^ circ }}
ight)}^2}}}$$ xa0 $$ + $$ $$frac{{{{left( {cot {{20}^ circ }}
ight)}^2}}}{{{{left( {{ ext{sec 7}}{0^ circ }}
ight)}^2}}}$$ xa0 $$ + $$ $$2 an {15^ circ }$$ . $$ an {45^ circ }$$ . $$ an {75^ circ }$$
Correct Answer

(D) 3

Explanation

Solution: $$frac{{{{left( { an {{20}^ circ }}
ight)}^2}}}{{{{left( {{ ext{cosec 7}}{0^ circ }}
ight)}^2}}}$$ xa0 $$ + $$ $$frac{{{{left( {cot {{20}^ circ }}
ight)}^2}}}{{{{left( {{ ext{sec 7}}{0^ circ }}
ight)}^2}}}$$ xa0 $$ + $$ $$2 an {15^ circ }$$ . $$ an {45^ circ }$$ . $$ an {75^ circ }$$ $$eqalign{
& Rightarrow frac{{{{left( { an {{20}^ circ }}
ight)}^2}}}{{{ ext{se}}{{ ext{c}}^2}{{20}^ circ }}} + frac{{{{left( {cot {{20}^ circ }}
ight)}^2}}}{{{ ext{cose}}{{ ext{c}}^2}{{20}^ circ }}} + 2 an {15^ circ }. an {75^ circ } cr
& left[ {{ ext{tan 1}}{5^ circ }{ ext{.tan 7}}{5^ circ } = { ext{1}}{ ext{. If, A}} + { ext{B}} = {{90}^ circ }}
ight] cr
& Rightarrow left( {{{sin }^2}{{20}^ circ } + { ext{co}}{{ ext{s}}^2}{{20}^ circ }}
ight) + 2 cr
& Rightarrow 1 + 2 cr
& Rightarrow 3 cr} $$

[#148] The value of the following is : $${left( {frac{{{ ext{sin 4}}{{ ext{7}}^ circ }}}{{cos {{43}^ circ }}}}
ight)^2}$$ xa0 + $${left( {frac{{cos {{43}^ circ }}}{{{ ext{sin }}{{47}^ circ }}}}
ight)^2}$$ xa0 - $$4{ ext{co}}{{ ext{s}}^2}{45^ circ }$$ xa0 = ?
Correct Answer

(D) 0

Explanation

Solution: $$eqalign{
& {left( {frac{{{ ext{sin 4}}{{ ext{7}}^ circ }}}{{cos {{43}^ circ }}}}
ight)^2} + {left( {frac{{cos {{43}^ circ }}}{{{ ext{sin }}{{47}^ circ }}}}
ight)^2} - 4{ ext{co}}{{ ext{s}}^2}{45^ circ } cr
& = {left( {frac{{{ ext{cos 4}}{{ ext{3}}^ circ }}}{{cos {{43}^ circ }}}}
ight)^2} + {left( {frac{{sin{{47}^ circ }}}{{{ ext{sin }}{{47}^ circ }}}}
ight)^2} - 4 imes frac{1}{2} cr
& = 1 + 1 - 2 cr
& = 0 cr
& {x08f{Note:}} cr
& left( {sin left( {{{90}^ circ } - heta }
ight)}
ight) = cos heta cr} $$

[#149] In ΔABC, right-angled at B, AB = 7 cm and AC - BC = 1 cm. Find the value of sinC.
Correct Answer

(D) $$frac{7}{{25}}$$

Explanation

Solution: As we know triplet 7, 24, 25 AC - BC = 25 - 24 = 1 sinC = $$frac{7}{{25}}$$

[#150] The value of sin 2 30°.cos 2 45° + 2tan 2 30° - sec 2 60° is equal to:
Correct Answer

(B) $$ - frac{{77}}{{24}}$$

Explanation

Solution: $$eqalign{
& {sin ^2}{30^ circ }.{cos ^2}{45^ circ } + 2{ an ^2}{30^ circ } - {sec ^2}{60^ circ } cr
& = {left( {frac{1}{2}}
ight)^2}.{left( {frac{1}{{sqrt 2 }}}
ight)^2} + 2{left( {frac{1}{{sqrt 3 }}}
ight)^2} - {left( 2
ight)^2} cr
& = frac{1}{4} imes frac{1}{2} + 2{left( {frac{1}{{sqrt 3 }}}
ight)^2} - {left( 2
ight)^2} cr
& = frac{1}{8} + frac{2}{3} - 4 cr
& = frac{{3 + 16 - 96}}{{24}} cr
& = - frac{{77}}{{24}} cr} $$