Trigonometry - Study Mode

[#91] The value of $$1 + sqrt {frac{{cot heta + cos heta }}{{cot heta - cos heta }}} ,$$ xa0 xa0if 0° < θ < 90°, is equal to:
Correct Answer

(D) 1 + secθ + tanθ

Explanation

Solution: $$eqalign{
& 1 + sqrt {frac{{cot heta + cos heta }}{{cot heta - cos heta }}} cr
& = 1 + sqrt {frac{{frac{{cos heta }}{{sin heta }} + cos heta }}{{frac{{cos heta }}{{sin heta }} - cos heta }}} cr
& = 1 + sqrt {frac{{cot heta + sin heta cos heta }}{{cot heta - cos heta sin heta }}} cr
& = 1 + sqrt {frac{{cos heta left( {1 + sin heta }
ight)}}{{cos heta left( {1 - sin heta }
ight)}}} cr
& = 1 + sqrt {frac{{1 + sin heta }}{{1 - sin heta }}} cr
& { ext{Rationalise,}} cr
& = 1 + sqrt {frac{{1 + sin heta }}{{1 - sin heta }} imes frac{{1 + sin heta }}{{1 + sin heta }}} cr
& = 1 + sqrt {frac{{{{left( {1 + sin heta }
ight)}^2}}}{{1 - {{sin }^2} heta }}} cr
& = 1 + sqrt {{{left( {1 + sin heta }
ight)}^2} imes {{sec }^2} heta } cr
& = 1 + left( {1 + sin heta }
ight)sec heta cr
& = sec heta + sin heta sec heta cr
& = 1 + sec heta + an heta cr} $$

[#92] What is the value of 5sin 2 60° + 7sin 2 45°+ 8cos 2 45°?
Correct Answer

(C) $$frac{{45}}{4}$$

Explanation

Solution: $$eqalign{
& 5{sin ^2}{60^ circ } + 7{sin ^2}{45^ circ } + 8{cos ^2}{45^ circ } cr
& = 5{left( {frac{{sqrt 3 }}{2}}
ight)^2} + 7{left( {frac{1}{{sqrt 2 }}}
ight)^2} + 8{left( {frac{1}{{sqrt 2 }}}
ight)^2} cr
& = frac{{15}}{4} + frac{7}{2} + frac{8}{2} cr
& = frac{{15 + 14 + 16}}{4} cr
& = frac{{45}}{4} cr} $$

[#93] What is the value of $$frac{{cos {{40}^ circ } - cos {{140}^ circ }}}{{sin {{80}^ circ } + sin {{20}^ circ }}}?$$
Correct Answer

(B) $$frac{2}{{sqrt 3 }}$$

Explanation

Solution: $$eqalign{
& frac{{cos {{40}^ circ } - cos {{140}^ circ }}}{{sin {{80}^ circ } + sin {{20}^ circ }}} cr
& = frac{{ - 2sin left( {frac{{{{40}^ circ } + {{140}^ circ }}}{2}}
ight)sin left( {frac{{{{40}^ circ } - {{140}^ circ }}}{2}}
ight)}}{{2sin left( {frac{{{{80}^ circ } + {{20}^ circ }}}{2}}
ight)cos left( {frac{{{{80}^ circ } - {{20}^ circ }}}{2}}
ight)}} cr
& = frac{{ - 2sin left( {frac{{{{180}^ circ }}}{2}}
ight)sin left( { - frac{{{{100}^ circ }}}{2}}
ight)}}{{2sin left( {frac{{{{100}^ circ }}}{2}}
ight)cos left( {frac{{{{60}^ circ }}}{2}}
ight)}} cr
& = frac{{ - 2sin {{90}^ circ } imes sin left( { - {{50}^ circ }}
ight)}}{{2sin {{50}^ circ } imes cos {{30}^ circ }}} cr
& = frac{{ - 2sin {{90}^ circ } imes left( { - sin {{50}^ circ }}
ight)}}{{2sin {{50}^ circ } imes cos {{30}^ circ }}} cr
& = frac{{2sin {{90}^ circ } imes sin {{50}^ circ }}}{{2sin {{50}^ circ } imes cos {{30}^ circ }}} cr
& = frac{{sin {{90}^ circ }}}{{cos {{30}^ circ }}} cr
& = frac{1}{{frac{{sqrt 3 }}{2}}} cr
& = frac{2}{{sqrt 3 }} cr} $$

[#94] (sinθ + cosecθ) 2 + (cosθ + secθ) 2 - 1 = . . . . . . . .
Correct Answer

(A) 6 + tan 2 θ + cot 2 θ

Explanation

Solution: (sinθ + cosecθ) 2 + (cosθ + secθ) 2 - 1 = sin 2 θ + cosec 2 θ + 2 + cos 2 θ + sec 2 θ + 2 - 1 = (sin 2 θ + cos 2 θ) + 1 + cot 2 θ + 2 + 1 + tan 2 θ + 2 - 1 = 6 + tan 2 θ + cot 2 θ

[#95] If sec4θ = cosec(θ + 20°), then θ is equal to:
Correct Answer

(C) 14°

Explanation

Solution: sec4θ = cosec(θ + 20°) sec4θ = sec[90° - (θ + 20°)] sec4θ = sec(70° - θ) 4θ = 70° - θ 5θ = 70° θ = 14°