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Published on: 04/09/2019
Trigonometry
Download Tamil Nadu 10th Standard Maths question papers, model tests, one-mark questions, important questions, and public exam papers in PDF format. Free study materials and answer keys for TN State Board students.
Questions + Answers key
Take MCQ Maths Test1.
A tower stands vertically on the ground. from a point on the ground, which is 48m away from the foot of the tower, the angel of elevation of the top of the tower is 30°.find the height of the tower.
2.
Prove that tan2\(\theta \)-sin2 \(\theta \) = tan2 \(\theta \) sin2 \(\theta \)
3.
As shown in the figure, Two trees are standing on the flat ground. the angel of elevation of the top of both the trees from a point x on the ground is 40° .if the horizontal distance between x and the smaller tree is 8m and the distance of the top of the trees is 20m, calculate, the distance between the point x and the top of the smaller tree.
4.
prove the following identities.
\(\frac { sinA-sinB }{ cosA+cosB } +\frac { cosA-cosB }{ sinA+sinB } =0\)
5.
If \(\frac { co{ s }^{ 2 }\theta }{ sin\theta } \) = p and \(\frac { sin^{ 2 }\theta }{ cos\theta } \) = q, then prove that p2q2(p2 + q2 + 3) = 1
6.
prove that\(\left( \frac { co{ s }^{ 3 }A-si{ n }^{ 3 }A }{ cosA-sinA } \right) -\left( \frac { co{ s }^{ 3 }A+si{ n }^{ 3 }A }{ cosA+sinA } \right) =2sinAcosA\)
7.
Two persons are standing ‘x’ metres apart from each other and the height of the first person is double that of the other. If from the middle point of the line joining their feet an observer finds the angular elevations of their tops to be complementary, then the height of the shorter person (in metres) is
\(\sqrt { 2 } \) x
\(\frac { x }{ 2\sqrt { 2 } } \)
\(\frac { x }{ \sqrt { 2 } } \)
2x
8.
A tower is 60 m height. Its shadow is x metres shorter when the sun’s altitude is 45° than when it has been 30°, then x is equal to
41.92 m
43.92 m
43 m
45.6 m
9.
The electric pole subtends an angle of 30° at a point on the same level as its foot. At a second point ‘b’ metres above the first, the depression of the foot of the pole is 60°. The height of the pole (in metres) is equal to
\(\sqrt { 3 } \) b
\(\frac { b }{ 3 } \)
\(\frac { b }{ 2 } \)
\(\frac { b }{ \sqrt { 3 } } \)
10.
(1 + tan \(\theta \) + sec\(\theta \)) (1 + cot\(\theta \) - cosec\(\theta \)) is equal to
0
1
2
-1
11.
If sin \(\theta \) + cos\(\theta \) = a and sec \(\theta \) + cosec \(\theta \) = b, then the value of b(a2 - 1) is equal to
2a
3a
0
2ab
12.
The value of \(si{ n }^{ 2 }\theta +\frac { 1 }{ 1+ta{ n }^{ 2 }\theta } \) is equal to
\(ta{ n }^{ 2 }\theta \)
1
\(cot^{ 2 }\theta \)
0
1.
Let PQ the height of the tower.
Take PQ = h and QR is the distance between the tower and the point R.in right triangle PQR,\(\angle \)PRQ=30°
tan\(\theta =\frac { PQ }{ QR } \)
tan30° = \(\frac { h }{ 48 } \) gives,\(\frac { 1 }{ \sqrt { 3 } } =\frac { h }{ 48 } \) so, h =\(16\sqrt { 3 } \)
Therefore the height of the tower \(16\sqrt { 3 } \) m
2.
tan2 \(\theta \) - sin 2\(\theta \) = tan2 \(\theta \) -\(\frac { si{ n }^{ 2 }\theta }{ co{ s }^{ 2 }\theta } \),cos2\(\theta \)
= tan2 \(\theta \) (1-cos2 \(\theta \) ) = tan2 \(\theta \) sin2 \(\theta \)
3.
Let AB be the height of the biggest tree and CD be the highest of the smaller tree and x is the point on the ground.
In the right triangle XCD, cos40° =\(\frac { CX }{ XD } \)
Therefore the distance between X and top of the smaller tree = XD =10.44m.
4.
\(\frac { sinA-sinB }{ cosA+cosB } +\frac { cosA-cosB }{ sinA+sinB } =0\)
\(\mathrm{LHS}=\frac{\sin A-\sin B}{\cos A+\cos B}+\frac{\cos A-\cos D}{\sin A+\sin B}
\)
\(=\frac{(\sin A-\sin B)(\sin A+\sin B)+(\cos A-\cos B)(\cos A+\cos B)}{(\cos A+\cos B)(\sin A+\sin B)}
\)
\(=\frac{\left(\sin ^{2} A-\sin ^{2} B\right)+\left(\cos ^{2} A-\cos ^{2} B\right)}{(\cos A+\cos B)(\sin A+\sin B)}
\)
\(=\frac{\sin ^{2} A-\sin ^{2} B+\cos ^{2} A-\cos ^{2} B}{(\cos A+\cos B)(\sin A+\sin B)}
\)
\(=\frac{\left(\sin ^{2} A+\cos ^{2} A\right)-\left(\sin ^{2} B+\cos ^{2} B\right)}{(\cos A+\cos B)(\sin A+\sin B)}
\)
\(=\frac{1-1}{(\cos A+\cos B)(\sin A+\sin B)}
\)
\(=\frac{0}{(\cos A+\cos B)(\sin A+\sin B)}=0=\mathrm{RHS}\)
5.
We have \(\frac { co{ s }^{ 2 }\theta }{ sin\theta } \) =p ...(1) and \(\frac { sin^{ 2 }\theta }{ cos\theta } \) =q. ..(2)
p2q2(p2 + q2 + 3)= \({ \left( \frac { co{ s }^{ 2 }\theta }{ sin\theta } \right) }^{ 2 }{ \left( \frac { sin^{ 2 }\theta }{ cos\theta } \right) }^{ 2 }\times \left[ { \left( \frac { co{ s }^{ 2 }\theta }{ sin\theta } \right) }^{ 2 }+{ \left( \frac { si{ n }^{ 2 }\theta }{ cos\theta } \right) }^{ 2 }+3 \right] \) [from (1) and (2)]
=\(\left( \frac { co{ s }^{ 4 }\theta }{ si{ n }^{ 2 }\theta } \right) \left( \frac { si{ n }^{ 2 }\theta }{ co{ s }^{ 2 }\theta } \right) \times \left[ \frac { co{ s }^{ 4 }\theta }{ si{ n }^{ 2 }\theta } +\frac { si{ n }^{ 4 }\theta }{ co{ s }^{ 2 }\theta } +3 \right] \)
= (cos2\(\theta \)\(\times \)sin2\(\theta \))\(\times \) \(\left[ \left( \frac { co{ s }^{ 6 }\theta +si{ n }^{ 6 }\theta +3si{ n }^{ 2 }\theta co{ s }^{ 2 }\theta }{ si{ n }^{ 2 }\theta co{ s }^{ 2 }\theta } \right) \right] \)
= cos6\(\theta \) + sin6\(\theta \) + 3sin2\(\theta \)cos2\(\theta \)
= (cos2\(\theta \))3 + (sin2\(\theta \))3 + 3sin2\(\theta \)cos2\(\theta \)
= [(cos2\(\theta \) + sin2\(\theta \))3 - 3cos2\(\theta \) sin2\(\theta \)(cos2\(\theta \) + sin2\(\theta \))] + 3sin2\(\theta \)cos2\(\theta \)
= 1 - 3cos2\(\theta \)sin2\(\theta \)(1) + 3cos2\(\theta \)sin2\(\theta \) = 1
6.
\(\left( \frac { co{ s }^{ 3 }A-si{ n }^{ 3 }A }{ cosA-sinA } \right) -\left( \frac { co{ s }^{ 3 }A+si{ n }^{ 3 }A }{ cosA+sinA } \right) \)
\(\left( \frac { (cosA-sinA)(co{ s }^{ 2 }A+si{ n }^{ 2 }A+cosAsinA) }{ cosA-sinA } \right) \)
\(\left[ since\quad { a }^{ 3 }-{ b }^{ 3 }=(a-b)({ a }^{ 2 }+{ b }^{ 2 }+ab \ { a }^{ 3 }+{ b }^{ 3 }=(a+b)({ a }^{ 2 }+{ b }^{ 2 }-ab) \right] \)
\(\left( \frac { (cosA+sinA)(co{ s }^{ 2 }A+si{ n }^{ 2 }A-cosAsinA) }{ cosA+sinA } \right) \)
= (1 + cos A sin A) - (1 - cos A sin A)
= 2cos A sin A
7.
(b)
\(\frac { x }{ 2\sqrt { 2 } } \)
8.
(b)
43.92 m
9.
(b)
\(\frac { b }{ 3 } \)
10.
(c)
2
11.
(a)
2a
12.
(b)
1
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Tamilnadu Stateboard 10th Standard Subjects
Tamilnadu Stateboard Standards
