Trigonometry - Study Mode
[#221] Find the value of, 8cos10°. cos20°. cos40° = ?
Correct Answer
(D) cot10°
Explanation
Solution: Let x = 8cos10°. cos20°. cos40° Multiply on both side by sin10° and applying formula (2sinθ. cosθ = sin 2 θ) ⇒ x sin10° = 4 × 2sin10° cos10°. cos20°. cos40° ⇒ x sin10° = 2 × 2sin20°.cos20°. cos40° ⇒ x sin10° = 2 × sin40°. cos40° ⇒ x sin10° = sin80° ⇒ x sin10° = sin(90° - 10°) ⇒ x sin10° = cos10° then, x = $$frac{{cos {{10}^ circ }}}{{sin {{10}^ circ }}}$$ x = cot10°
[#222] The value of $${ ext{se}}{{ ext{c}}^2}{17^ circ }$$ xa0- $$frac{1}{{{ ext{ta}}{{ ext{n}}^2}{{73}^ circ }}}$$ xa0- $$sin {17^ circ }$$ $$sec {73^ circ }$$ xa0is?
Correct Answer
(B) 0
Explanation
Solution: $$eqalign{
& = {sec ^2}{17^ circ } - frac{1}{{{{ an }^2}{{73}^ circ }}} - sin {17^ circ }sec {73^ circ } cr
& = {sec ^2}{17^ circ } - {cot ^2}{73^ circ } - sin {17^ circ }sec left( {{{90}^ circ } - {{17}^ circ }}
ight) cr
& = {sec ^2}{17^ circ } - {cot ^2}left( {{{90}^ circ } - {{17}^ circ }}
ight) - sin {17^ circ }cos ec{17^ circ } cr
& = {sec ^2}{17^ circ } - { an ^2}{17^ circ } - 1 cr
& = 1 - 1left[ {x08ecause {{sec }^2} heta - {{ an }^2} heta = 1}
ight] cr
& = 0 cr} $$
[#223] The value of coses 2 60° + sec 2 60° - cot 2 60° + tan 2 30° will be?
Correct Answer
(D) $${ ext{5}}frac{1}{3}$$
Explanation
Solution: $$eqalign{
& { ext{cose}}{{ ext{c}}^2}{60^ circ } + { ext{se}}{{ ext{c}}^2}{60^ circ } - { ext{co}}{{ ext{t}}^2}{60^ circ } + { ext{ta}}{{ ext{n}}^2}{30^ circ } cr
& = {left( {frac{2}{{sqrt 3 }}}
ight)^2} + {left( 2
ight)^2} - {left( {frac{1}{{sqrt 3 }}}
ight)^2} + {left( {frac{1}{{sqrt 3 }}}
ight)^2} cr
& = frac{4}{3} + 4 - frac{1}{3} + frac{1}{3} cr
& = frac{{16}}{3} cr
& = 5frac{1}{3} cr} $$
[#224] In a ΔABC, if 4∠A = 3∠B = 12∠C, find ∠A?
Correct Answer
(C) 67.5°
Explanation
Solution: $$eqalign{
& { ext{4}}angle { ext{A}} = { ext{3}}angle { ext{B}} = { ext{12}}angle { ext{C}} cr
& { ext{A}}:{ ext{B}}:{ ext{C}} = frac{1}{4}:frac{1}{3}:frac{1}{{12}} cr
& { ext{A}}:{ ext{B}}:{ ext{C}} = 3:4:1 cr
& { ext{Now,}} cr
& 3x + 4x + x = {180^ circ } cr
& 8x = {180^ circ } cr
& x = frac{{{{180}^ circ }}}{8} cr
& angle { ext{A}} = 3x = frac{{{{180}^ circ }}}{8} imes 3 cr
& ,,,,,,,,,,,,,,,,,,,,,,, = {67.5^ circ } cr} $$
[#225] If θ is a acute angle and sin(θ + 18°) = $$frac{1}{2}{ ext{,}}$$ then the value of θ in circular measure is?
Correct Answer
(B) $$frac{pi }{{15}}$$ Radians
Explanation
Solution: $$eqalign{
& { ext{sin}}left( { heta + {{18}^ circ }}
ight){ ext{ = }}frac{1}{2} cr
& { ext{sin}}left( { heta + {{18}^ circ }}
ight) = { ext{sin }}{30^ circ } cr
& heta + {18^ circ } = {30^ circ } cr
& herefore heta = {12^ circ } cr
& { ext{We know that,}} cr
& {180^ circ } = pi cr
& {12^ circ } = frac{pi }{{{{180}^ circ }}} imes 12 = frac{pi }{{15}} cr} $$