Trigonometry - Study Mode
[#276] Which of the following is equal to $$frac{1}{{ an heta }} + an heta ?$$
Correct Answer
(B) $${ ext{cosec}}, heta cdot sec heta $$
Explanation
Solution: $$eqalign{
& frac{1}{{ an heta }} + an heta cr
& { ext{put }} heta = {45^ circ } cr
& frac{1}{1} + 1 = 2 cr
& { ext{In option}} cr
& Rightarrow left( { ext{A}}
ight)frac{{{ ext{cosec}}, heta }}{{sec heta }} = frac{{{ ext{cosec}},{{45}^ circ }}}{{sec {{45}^ circ }}} = frac{{sqrt 2 }}{{sqrt 2 }} = 1 cr
& Rightarrow left( { ext{B}}
ight)sec heta imes { ext{cosec}}, heta cr
& = sec {45^ circ } imes { ext{cosec}},{45^ circ } cr
& = sqrt 2 imes sqrt 2 = 2 cr
& Rightarrow left( { ext{C}}
ight),1 cr
& Rightarrow left( { ext{D}}
ight){ an ^2} heta = { an ^2}{45^ circ } = 1 cr} $$
[#277] Which of the following is equal to $$left[ {frac{{ an heta + sec heta - 1}}{{ an heta - sec heta + 1}}}
ight]?$$
Correct Answer
(A) $$frac{{1 + sin heta }}{{cos heta }}$$
Explanation
Solution: $$eqalign{
& left[ {frac{{ an heta + sec heta - 1}}{{ an heta - sec heta + 1}}}
ight] cr
& = left[ {frac{{ an heta + sec heta - left( {{{sec }^2} heta - {{ an }^2} heta }
ight)}}{{left( { an heta - sec heta + 1}
ight)}}}
ight] cr
& = frac{{left( {sec heta + an heta }
ight)left[ {1 - sec heta + an heta }
ight]}}{{left( { an heta - sec heta + 1}
ight)}} cr
& = sec heta + an heta cr
& = frac{1}{{cos heta }} + frac{{sin heta }}{{cos heta }} cr
& = frac{{1 + sin heta }}{{cos heta }} cr} $$
[#278] If $$cos heta = frac{{2p}}{{1 + {p^2}}},$$ xa0 then tanθ is equal to:
Correct Answer
(D) $$frac{{1 - {p^2}}}{{2p}}$$
Explanation
Solution: $$eqalign{
& cos heta = frac{{2p}}{{1 + {p^2}}} cr
& A{B^2} = {left( {1 + {p^2}}
ight)^2} - {left( {2p}
ight)^2} cr
& AB = 1 - {p^2} cr
& an heta = frac{{1 - {p^2}}}{{2p}} cr} $$
[#279] Which of the following will satisfy a 2 = b 2 + (ab) 2 for the values a and b?
Correct Answer
(C) a = cotx, b = cosx
Explanation
Solution: We need to find which option makes the equation a 2 = b 2 + (ab) 2 true. Remember, we are dealing with trigonometric functions: sin(x), cos(x), tan(x), and cot(x) . Let's look at each option: Option A: a = sin(x), b = cot(x) This means our equation becomes: sin 2 (x) = cot 2 (x) + (sin(x) * cot(x)) 2 Remember that cot(x) = cos(x) / sin(x) . So, we can rewrite the equation. sin 2 (x) = (cos 2 (x) / sin 2 (x)) + (sin(x) * (cos(x) / sin(x))) 2 sin 2 (x) = (cos 2 (x) / sin 2 (x)) + cos 2 (x) This looks complicated, and it's unlikely to simplify easily to something that's always true. Option B: a = cos(x), b = tan(x) This means our equation becomes: cos 2 (x) = tan 2 (x) + (cos(x) * tan(x)) 2 Remember that tan(x) = sin(x) / cos(x) . Substitute that in. cos 2 (x) = (sin 2 (x) / cos 2 (x)) + (cos(x) * (sin(x) / cos(x))) 2 cos 2 (x) = (sin 2 (x) / cos 2 (x)) + sin 2 (x) Again, this doesn't look like it will easily simplify to a true statement. Option C: a = cot(x), b = cos(x) This means our equation becomes: cot 2 (x) = cos 2 (x) + (cot(x) * cos(x)) 2 Substitute cot(x) = cos(x) / sin(x) . (cos 2 (x) / sin 2 (x)) = cos 2 (x) + ((cos(x) / sin(x)) * cos(x)) 2 (cos 2 (x) / sin 2 (x)) = cos 2 (x) + (cos 2 (x) / sin 2 (x)) * cos 2 (x) (cos 2 (x) / sin 2 (x)) = cos 2 (x) + (cos 4 (x) / sin 2 (x)) Let's try to manipulate this a bit. Multiply both sides by sin 2 (x): cos 2 (x) = cos 2 (x)sin 2 (x) + cos 4 (x) cos 2 (x) = cos 2 (x)(sin 2 (x) + cos 2 (x)) Remember the fundamental trigonometric identity: sin 2 (x) + cos 2 (x) = 1 cos 2 (x) = cos 2 (x) * 1 cos 2 (x) = cos 2 (x) . This is always true! Option D: a = sin(x), b = tan(x) This means our equation becomes: sin 2 (x) = tan 2 (x) + (sin(x) * tan(x)) 2 Substitute tan(x) = sin(x) / cos(x) . sin 2 (x) = (sin 2 (x) / cos 2 (x)) + (sin(x) * (sin(x) / cos(x))) 2 sin 2 (x) = (sin 2 (x) / cos 2 (x)) + (sin 4 (x) / cos 2 (x)) This also doesn't look like it will simplify easily. Therefore, Option C is the correct answer. It's the only one that simplifies to a true identity.
[#280] What is the value of tan240°
Correct Answer
(C) √3
Explanation
Solution: tan240° = tan(180° + 60°) = tan(180° + θ) = tanθ = tan60° =√3