Question 13 Marks
Express the following in terms of trigonometric ratios of angles between 0 and 45.
$\sin 59+\cos 56$
$\sin 59+\cos 56$
Answer
View full question & answer→$\cos 31+\sin 34$
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Question 23 Marks
Express he following in terms of trigonometric ratios of angles between 0 and 45.
$\sec 63+\operatorname{cosec} 49$
$\sec 63+\operatorname{cosec} 49$
Answer
View full question & answer→$\operatorname{cosec} 27+\sec 41$
Question 33 Marks
Express the following in terms of trigonometric ratios of angles between 0 and 45.
$\cos 76+\sec 76$
$\cos 76+\sec 76$
Answer
View full question & answer→$\sin 14+\operatorname{cosec} 14$
Question 43 Marks
Express the following in terms of trigonometric ratios of angles between 0 and 45.
$\cos 81+\cot 81$
$\cos 81+\cot 81$
Answer
View full question & answer→$\sin 9+\tan 9$
Question 53 Marks
Without using trigonometric table, prove that :
$\tan 10 \tan 15 \tan 75 \tan 80=1$
$\tan 10 \tan 15 \tan 75 \tan 80=1$
Answer
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Question 63 Marks
Without using trigonometric table, prove that :
$\operatorname{cosec}^2 56-\tan ^2 34=1$
$\operatorname{cosec}^2 56-\tan ^2 34=1$
Answer
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Question 73 Marks
Without using trigonometric table, prove that :
$\cos ^2 23+\cos ^2 67=1$
$\cos ^2 23+\cos ^2 67=1$
Answer
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Question 83 Marks
Without using trigonometric table, prove that :
$\sec ^2 75-\cot ^2 15=1$
$\sec ^2 75-\cot ^2 15=1$
Answer
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Question 93 Marks
Without using trigonometric table, prove that :
$\sin 40 \sec 50+\cos 40 \operatorname{cosec} 50=2$
$\sin 40 \sec 50+\cos 40 \operatorname{cosec} 50=2$
Answer
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Question 103 Marks
Without using trigonometric table, prove that :
$\sin 73 \cos 17+\cos 73 \sin 17=1$
$\sin 73 \cos 17+\cos 73 \sin 17=1$
Answer
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Question 113 Marks
If $A =60$ and $B =30$, show that :
$(\sin A \cos B+\cos A \sin B)^2+(\cos A \cos B-\sin A \sin B)^2=1$.
$(\sin A \cos B+\cos A \sin B)^2+(\cos A \cos B-\sin A \sin B)^2=1$.
Answer
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Question 123 Marks
If $A=B=45$, show that :
$\cos ( A + B )=\cos A \cos B -\sin A \sin B$.
$\cos ( A + B )=\cos A \cos B -\sin A \sin B$.
Answer
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Question 133 Marks
If $A=B=45$, show that :
$\sin ( A - B )=\sin A \cos B -\cos A \sin B$.
$\sin ( A - B )=\sin A \cos B -\cos A \sin B$.
Answer
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Question 143 Marks
If $A=30$, prove that :
$\cos 2 A=\left(\frac{1-\tan ^2 A}{1+\tan ^2 A}\right)$
$\cos 2 A=\left(\frac{1-\tan ^2 A}{1+\tan ^2 A}\right)$
Answer
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Question 153 Marks
If $A=30$, prove that :
$\sin 2 A=\frac{2 \tan A}{\left(1+\tan ^2 A\right)}$
$\sin 2 A=\frac{2 \tan A}{\left(1+\tan ^2 A\right)}$
Answer
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Question 163 Marks
Evaluate : $\frac{3 \sin 3 A+2 \cos \left(2 A+5^{\circ}\right)}{2 \cos 3 A-\sin \left(2 A-10^{\circ}\right)}$, when $A =20$.
Answer
View full question & answer→$(3 \sqrt{3}+2 \sqrt{2})$
Question 173 Marks
Evaluate : $\frac{\cos 3 A+2 \cos 4 A}{\sin 3 A+2 \sin 4 A}$, when $A=15$.
Answer
View full question & answer→$\left(\frac{1+\sqrt{2}}{1+\sqrt{6}}\right)$
Question 183 Marks
If $A=60$ and $B=30$, prove that:
$\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B}$.
$\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B}$.
Answer
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Question 193 Marks
If $A=60$ and $B=30$, prove that:
$\cos ( A - B )=\cos A \cos B +\sin A \sin B$.
$\cos ( A - B )=\cos A \cos B +\sin A \sin B$.
Answer
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Question 203 Marks
If $A=60$ and $B=30$, prove that:
$\cos (A+B)=\cos A \cos B-\sin A \sin B$.
$\cos (A+B)=\cos A \cos B-\sin A \sin B$.
Answer
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Question 213 Marks
If $A=60$ and $B=30$, prove that:
$\sin ( A + B )=\sin A \cos B +\cos A \sin B$.
$\sin ( A + B )=\sin A \cos B +\cos A \sin B$.
Answer
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Question 223 Marks
If $\operatorname{cosec} \theta=\sqrt{10}$, find the values of other trigonometrical ratios for $\theta$.
Answer
View full question & answer→$\sin \theta=\frac{1}{\sqrt{10}}, \cos \theta=\frac{3}{\sqrt{10}}, \tan \theta=\frac{1}{3}, \sec \theta=\frac{\sqrt{10}}{3}, \cot \theta=3$.
Question 233 Marks
If $\tan \theta=\frac{8}{15}$, find the values of other trigonometrical ratios for $\theta$.
Answer
View full question & answer→$\sin \theta=\frac{8}{17}, \cos \theta=\frac{15}{17}, \operatorname{cosec} \theta=\frac{17}{8}, \sec \theta=\frac{17}{15}, \cot \theta=\frac{15}{8}$.
Question 243 Marks
If $\sin \theta=\frac{1}{\sqrt{2}}$, find the values of other trigonometrical ratios for $\theta$.
Answer
View full question & answer→$\cos \theta=\frac{1}{\sqrt{2}}, \tan \theta=1, \sec \theta=\sqrt{2}, \operatorname{cosec} \theta=\sqrt{2}, \cot \theta=1$.
Question 253 Marks
In the given figure, $A D \perp B C$. If $A B=13 cm, B D=5 cm$ and $D C=16 cm$,
find the values of : (i) $\sin B$ (ii) $\sec B$ (iii) $\cot B$ (iv) $\cos C$ (v) $\operatorname{cosec} C$ (vi) $\tan C$
find the values of : (i) $\sin B$ (ii) $\sec B$ (iii) $\cot B$ (iv) $\cos C$ (v) $\operatorname{cosec} C$ (vi) $\tan C$
Answer
View full question & answer→(i) $\frac{12}{13}$ (ii) $\frac{13}{5}$ (iii) $\frac{5}{12}$ (iv) $\frac{4}{5}$ (v) $\frac{5}{3}$ (vi) $\frac{3}{4}$
Question 263 Marks
From the given figure, write down the values of :
(i) $\sin B$ (ii) $\tan B$ (iii) $\cos C$ (iv) $\cot C$ (v) $(\sin B \cos C +\cos B \sin C )$ (vi) $\left(\sec ^2 C -\tan ^2 C \right)$
(i) $\sin B$ (ii) $\tan B$ (iii) $\cos C$ (iv) $\cot C$ (v) $(\sin B \cos C +\cos B \sin C )$ (vi) $\left(\sec ^2 C -\tan ^2 C \right)$
Answer
View full question & answer→(i) $\frac{8}{17}$ (ii) $\frac{8}{15}$ (iii) $\frac{8}{17}$ (iv) $\frac{8}{15}$ (v) 1 (vi) 1
Question 273 Marks
If $\cos \theta=\frac{2 x}{1+x^2}$, find the values of $\sin \theta$ and $\tan \theta$ in terms of $x$.
Answer
View full question & answer→$\sin \theta=\frac{x^2-1}{x^2+1}, \tan \theta=\frac{\left(x^2-1\right)}{2 x}$.
[Hint.
$\cos \theta=\frac{\text { Base }}{\text { Hyp. }}=\frac{A B}{A C}=\frac{2 x}{\left(1+x^2\right)}$
Perp. $B C=\sqrt{A C^2-A B^2}=\sqrt{\left(1+x^2\right)^2-4 x^2}=\sqrt{x^4-2 x^2+1}=\left(x^2-1\right)$
$\left.\therefore \sin \theta=\frac{\text { Perp. }}{\text { Hyp. }}=\frac{\left(x^2-1\right)}{\left(1+x^2\right)} ; \tan \theta=\frac{\text { Perp. }}{\text { Base }}=\frac{\left(x^2-1\right)}{2 x}\right]$
[Hint.
$\cos \theta=\frac{\text { Base }}{\text { Hyp. }}=\frac{A B}{A C}=\frac{2 x}{\left(1+x^2\right)}$
Perp. $B C=\sqrt{A C^2-A B^2}=\sqrt{\left(1+x^2\right)^2-4 x^2}=\sqrt{x^4-2 x^2+1}=\left(x^2-1\right)$
$\left.\therefore \sin \theta=\frac{\text { Perp. }}{\text { Hyp. }}=\frac{\left(x^2-1\right)}{\left(1+x^2\right)} ; \tan \theta=\frac{\text { Perp. }}{\text { Base }}=\frac{\left(x^2-1\right)}{2 x}\right]$
Question 283 Marks
In the given figure, $\triangle A B C$ is right angled at $B$.
If $AC =20 cm$ and $\tan A =\frac{3}{4}$, find the lengths of AB and BC .
If $AC =20 cm$ and $\tan A =\frac{3}{4}$, find the lengths of AB and BC .
Answer
View full question & answer→$AB =16 cm, BC =12 cm$
[Hint. $\tan A=\frac{B C}{A B}=\frac{3}{4}=\frac{3 x}{4 x}$.
Then, $B C=3 x$ units, $A B=4 x$ units and $A C=20 cm$.
Now, $\left.(A B)^2+(B C)^2=(A C)^2.\right]$
[Hint. $\tan A=\frac{B C}{A B}=\frac{3}{4}=\frac{3 x}{4 x}$.
Then, $B C=3 x$ units, $A B=4 x$ units and $A C=20 cm$.
Now, $\left.(A B)^2+(B C)^2=(A C)^2.\right]$
Question 293 Marks
Evaluate x and y from the given figure.
Answer
View full question & answer→$x=5 m, y=45$
Question 303 Marks
In the given figure, $\angle B=90, A B=4$ units and $B C=3$ units.
Find :(i) $\sin A$ (ii) $\cos A$ (iii) $\cot A$ (iv) $\sin C$ (v) $\sec C$ (vi) $\tan C$
Find :(i) $\sin A$ (ii) $\cos A$ (iii) $\cot A$ (iv) $\sin C$ (v) $\sec C$ (vi) $\tan C$
Answer
View full question & answer→(i) $\frac{3}{5}$ (ii) $\frac{4}{5}$ (iii) $\frac{4}{3}$ (iv) $\frac{4}{5}$ (v) $\frac{5}{3}$ (vi) $\frac{4}{3}$
Question 313 Marks
Answer
View full question & answer→(i) $\frac{12}{13}$
(ii) $\frac{4}{3}$
(iii) $\frac{5}{4}$
[Hint. Draw $C E \perp A D$.]
(ii) $\frac{4}{3}$
(iii) $\frac{5}{4}$
[Hint. Draw $C E \perp A D$.]
Question 323 Marks
Use the adjoining figure and write the values of :
(i) $\sin x$ (ii) $\cos y$ (iii) $3 \tan x-2 \sin y^{\circ}+4 \cos y$
(i) $\sin x$ (ii) $\cos y$ (iii) $3 \tan x-2 \sin y^{\circ}+4 \cos y$
Answer
View full question & answer→(i) $\frac{8}{17}$
(ii) $\frac{3}{5}$
(iii) $2 \frac{2}{5}$
[Hint. $A B=\sqrt{A C^2-B C^2}$.]
(ii) $\frac{3}{5}$
(iii) $2 \frac{2}{5}$
[Hint. $A B=\sqrt{A C^2-B C^2}$.]
Question 333 Marks
If $4 \cot \theta=3$, show that $\left(\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta}\right)=\frac{1}{7}$.
Answer
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Question 343 Marks
If $\cot \theta=\frac{q}{p}$, show that $\left(\frac{p \sin \theta-q \cos \theta}{p \sin \theta+q \cos \theta}\right)=\left(\frac{p^2-q^2}{p^2+q^2}\right)$.
Answer
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Question 353 Marks
If $3 \tan \theta=4$, show that $\left(\frac{3 \sin \theta+2 \cos \theta}{3 \sin \theta-2 \cos \theta}\right)=3$.
Answer
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Question 363 Marks
If $\sec \theta=\frac{13}{5}$, show that $\left(\frac{2 \sin \theta-3 \cos \theta}{4 \sin \theta-9 \cos \theta}\right)=3$.
Answer
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Question 373 Marks
If $\cot \theta=\frac{1}{\sqrt{3}}$, show that $\left(\frac{1-\cos ^2 \theta}{2-\sin ^2 \theta}\right)=\frac{3}{5}$.
Answer
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Question 383 Marks
Look at the figures given below :
From these figures, write down the values of :
(i) $\sin x$ (ii) $\tan x$ (iii) $\sec x$ (iv) $\cos y$ (v) $\cot y$ (vi) $\operatorname{cosec} y$ (vii) $\sin z$ (viii) $\cos z$ (ix) $\tan z$
From these figures, write down the values of :
(i) $\sin x$ (ii) $\tan x$ (iii) $\sec x$ (iv) $\cos y$ (v) $\cot y$ (vi) $\operatorname{cosec} y$ (vii) $\sin z$ (viii) $\cos z$ (ix) $\tan z$
Answer
View full question & answer→(i) $\frac{q}{r}$ (ii) $\frac{q}{p}$ (iii) $\frac{r}{p}$ (iv) $\frac{b}{n}$ (v) $\frac{b}{m}$ (vi) $\frac{n}{m}$ (vii) $\frac{u}{n}$ (viii) $\frac{k}{n}$ (ix) $\frac{u}{k}$
Question 393 Marks
Prove that:
$\sin \left(90^{\circ}- A \right) \cos \left(90^{\circ}- A \right)=\frac{\tan A }{1+\tan ^2 A}$.
$\sin \left(90^{\circ}- A \right) \cos \left(90^{\circ}- A \right)=\frac{\tan A }{1+\tan ^2 A}$.
Answer
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Question 403 Marks
Prove that:
$\frac{\sin A }{\sin \left(90^{\circ}- A \right)}+\frac{\cos A }{\cos \left(90^{\circ}- A \right)}=\sec \left(90^{\circ}- A \right) \operatorname{cosec}\left(90^{\circ}- A \right).$
$\frac{\sin A }{\sin \left(90^{\circ}- A \right)}+\frac{\cos A }{\cos \left(90^{\circ}- A \right)}=\sec \left(90^{\circ}- A \right) \operatorname{cosec}\left(90^{\circ}- A \right).$
Answer
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Question 413 Marks
Prove that:
$\frac{\cos A}{\sin \left(90^{\circ}-A\right)}+\frac{\sin A}{\cos \left(90^{\circ}-A\right)}=2$.
$\frac{\cos A}{\sin \left(90^{\circ}-A\right)}+\frac{\sin A}{\cos \left(90^{\circ}-A\right)}=2$.
Answer
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Question 423 Marks
Express the following in terms of trigonometric ratios of angles between 0° and 45°.
$\sin 59^{\circ}+\cos 56^{\circ}$
$\sin 59^{\circ}+\cos 56^{\circ}$
Answer
View full question & answer→$\cos 31^{\circ}+\sin 34^{\circ}$
Question 433 Marks
Express the following in terms of trigonometric ratios of angles between 0° and 45°.
$\sec 63^{\circ}+\operatorname{cosec} 49^{\circ}$
$\sec 63^{\circ}+\operatorname{cosec} 49^{\circ}$
Answer
View full question & answer→$\operatorname{cosec} 27^{\circ}+\sec 41^{\circ}$
Question 443 Marks
Express the following in terms of trigonometric ratios of angles between 0° and 45°.
$\cos 76^{\circ}+\sec 76^{\circ}$
$\cos 76^{\circ}+\sec 76^{\circ}$
Answer
View full question & answer→$\sin 14^{\circ}+\operatorname{cosec} 14^{\circ}$
Question 453 Marks
Express the following in terms of trigonometric ratios of angles between 0° and 45°.
$\cos 81^{\circ}+\cot 81^{\circ}$
$\cos 81^{\circ}+\cot 81^{\circ}$
Answer
View full question & answer→$\sin 9^{\circ}+\tan 9^{\circ}$
Question 463 Marks
Without using trigonometric tables, prove that:
$\tan 10^{\circ} \tan 15^{\circ} \tan 75^{\circ} \tan 80^{\circ}=1$
$\tan 10^{\circ} \tan 15^{\circ} \tan 75^{\circ} \tan 80^{\circ}=1$
Answer
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Question 473 Marks
Without using trigonometric tables, prove that:
$\operatorname{cosec}^2 56^{\circ}-\tan ^2 34^{\circ}=1$
$\operatorname{cosec}^2 56^{\circ}-\tan ^2 34^{\circ}=1$
Answer
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Question 483 Marks
Without using trigonometric tables, prove that:
$\cos ^2 23^{\circ}+\cos ^2 67^{\circ}=1$
$\cos ^2 23^{\circ}+\cos ^2 67^{\circ}=1$
Answer
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Question 493 Marks
Without using trigonometric tables, prove that:
$\sec ^2 75^{\circ}-\cot ^2 15^{\circ}=1$
$\sec ^2 75^{\circ}-\cot ^2 15^{\circ}=1$
Answer
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Question 503 Marks
Without using trigonometric tables, prove that:
$\sin 40^{\circ} \sec 50^{\circ}+\cos 40^{\circ} \operatorname{cosec} 50^{\circ}=2$
$\sin 40^{\circ} \sec 50^{\circ}+\cos 40^{\circ} \operatorname{cosec} 50^{\circ}=2$
Answer
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