Trigonometry - Study Mode

[#131] If sin(A + B) = cos(A + B), what is the value of tanA?
Correct Answer

(A) $$frac{{1 - an B}}{{1 + an B}}$$

Explanation

Solution: $$eqalign{
& sin left( {A + B}
ight) = cos left( {A + B}
ight) cr
& frac{{sin left( {A + B}
ight)}}{{cos left( {A + B}
ight)}} = 1 cr
& an left( {A + B}
ight) = 1 cr
& an left( {A + B}
ight) = an {45^ circ } cr
& A + B = {45^ circ } cr
& A = {45^ circ } - B cr
& an A = an {45^ circ } - an B cr
& an A = frac{{ an {{45}^ circ } - an B}}{{1 + an {{45}^ circ } an B}} cr
& an A = frac{{1 - an B}}{{1 + an B}} cr} $$

[#132] $$frac{{{{left( {1 + cos heta }
ight)}^2} + {{sin }^2} heta }}{{left( {{ ext{cose}}{{ ext{c}}^2} heta - 1}
ight){{sin }^2} heta }} = ?$$
Correct Answer

(D) 2secθ(1 + secθ)

Explanation

Solution: $$eqalign{
& frac{{{{left( {1 + cos heta }
ight)}^2} + {{sin }^2} heta }}{{left( {{ ext{cose}}{{ ext{c}}^2} heta - 1}
ight){{sin }^2} heta }} cr
& = frac{{1 + {{cos }^2} heta + 2cos heta + {{sin }^2} heta }}{{left( {{ ext{cose}}{{ ext{c}}^2} heta - 1}
ight){{sin }^2} heta }} cr
& = frac{{2left( {1 + cos heta }
ight)}}{{frac{{left( {1 - {{sin }^2} heta }
ight)}}{{{{sin }^2} heta }}.{{sin }^2} heta }} cr
& = frac{{2left( {cos heta + 1}
ight)}}{{{{cos }^2} heta }} cr
& = 2sec heta left( {frac{{cos heta }}{{cos heta }} + frac{1}{{cos heta }}}
ight) cr
& = 2sec heta left( {1 + sec heta }
ight) cr} $$

[#133] If cos(A - B) = $$frac{{sqrt 3 }}{2}$$ and sec A = 2, 0° ≤ A ≤ 90°, 0° ≤ B ≤ 90° then what is the measure of B?
Correct Answer

(C) 30°

Explanation

Solution: $$eqalign{
& cos left( {A - B}
ight) = frac{{sqrt 3 }}{2} cr
& cos left( {A - B}
ight) = cos {30^ circ } cr
& A - B = {30^ circ }........left( { ext{i}}
ight) cr
& sec A = 2 cr
& cos A = frac{1}{2} = cos {60^ circ } cr
& A = {60^ circ }........left( {{ ext{ii}}}
ight) cr
& { ext{From equation }}left( { ext{i}}
ight){ ext{ and }}left( {{ ext{ii}}}
ight) cr
& A = {60^ circ },& ,B = {30^ circ } cr} $$

[#134] The value of $$frac{{2{{cos }^3} heta - cos heta }}{{sin heta - 2{{sin }^3} heta }}:$$
Correct Answer

(C) cotθ

Explanation

Solution: $$eqalign{
& frac{{2{{cos }^3} heta - cos heta }}{{sin heta - 2{{sin }^3} heta }} cr
& = frac{{cos heta left[ {2{{cos }^2} heta - 1}
ight]}}{{sin heta left[ {1 - 2{{sin }^2} heta }
ight]}} cr
& = frac{{cos heta imes cos 2 heta }}{{sin heta imes cos 2 heta }} cr
& = cot heta cr} $$

[#135] The value of $$frac{{sec heta left( {sin heta - 2{{sin }^3} heta }
ight)}}{{2{{cos }^3} heta - cos heta }}$$ xa0 xa0is:
Correct Answer

(C) secθ.tanθ

Explanation

Solution: $$eqalign{
& frac{{sec heta left( {sin heta - 2{{sin }^3} heta }
ight)}}{{2{{cos }^3} heta - cos heta }} cr
& frac{{sec heta .sin heta left( {1 - 2{{sin }^2} heta }
ight)}}{{cos heta left( {2{{cos }^2} heta - 1}
ight)}} cr
& frac{{sec heta .sin heta imes cos 2 heta }}{{cos heta imes cos 2 heta }} cr
& sec heta . an heta cr} $$