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Syllabus

30. Using the measurements given in figure alongside,

(a) find the values of :

(1) sin $\varphi $ (ii) tan $\theta $.

(b) write an expression for AD in terms of $\theta $.

Hint: (b) CD = 5. Draw DE perpendicular to AB, BE = 5, EA = 9.

^{2}+ 4y^{2 }^{}?(a) 2 (b) 1 (c) 0 (d) None of these.

(xxii) If tan $\theta $ + cot $\theta $ = 2, then the value of tan $\theta $ - cot $\theta $ is

(a) 1 (b) 2 (c) - 1 (d) None of these.

(xxiii) If 3 cos $\theta $ - 4 sin $\theta $ = 5, then the value of 3 sin $\theta $ + 4 cos $\theta $ is

(a) 1 (b) 2 (c) 0 (d) None of these.

Q.37. In the following figure, AB = 4 cm and ED = 3 m.

If $\mathrm{sin}\alpha =\frac{3}{5}and\mathrm{cos}\beta =\frac{12}{13}$ find the length of BD.

i) cos angle CBD

ii) cot angle ABD

7. Find AB.

^{o}, AB = 8 cm and BC = 25 cm . Calculate i) BE ii) ACQ12. If $cot\theta =\frac{1}{\sqrt{3}},showthat\left(\frac{1-{\mathrm{cos}}^{2}\theta}{2-{\mathrm{sin}}^{2}\theta}\right)=\frac{3}{5}$

Q13. If $sec\theta =\frac{13}{5},showthat\left(\frac{2\mathrm{sin}\theta -3\mathrm{cos}\theta}{4\mathrm{sin}\theta -9\mathrm{cos}\theta}\right)=3.$

Q14. If $3\mathrm{tan}\theta =4,showthat\left(\frac{3\mathrm{sin}\theta +2\mathrm{cos}\theta}{3\mathrm{sin}\theta -2\mathrm{cos}\theta}\right)=3.$

Q12. Without using trigonometrical tables, evaluate :

(i) $2{\left(\frac{\mathrm{tan}35\xb0}{\mathrm{cot}55\xb0}\right)}^{2}+\left(\frac{\mathrm{cot}55\xb0}{\mathrm{tan}35\xb0}\right)-3\left(\frac{\mathrm{sec}40\xb0}{\mathrm{cosec}50\xb0}\right)$ (ii) $\frac{\mathrm{sin}35\xb0\mathrm{cos}55\xb0+\mathrm{cos}35\xb0\mathrm{sin}55\xb0}{{\mathrm{cosec}}^{2}10\xb0-{\mathrm{tan}}^{2}80\xb0}$

(iii) sin

^{2}34° + sin^{2}56° + 2 tan 18° tan 72° – cot^{2}30°.Q13. Prove the following :

(i) $\frac{\mathrm{cos}\mathrm{\theta}}{\mathrm{sin}\left(90\xb0-\mathrm{\theta}\right)}+\frac{\mathrm{sin}{\displaystyle}{\displaystyle \mathrm{\theta}}}{\mathrm{cos}{\displaystyle}{\displaystyle \left(90\xb0-\mathrm{\theta}\right)}}=2$

(ii) cos $\mathrm{\theta}\mathrm{sin}\left(90\xb0-\mathrm{\theta}\right)+\mathrm{sin}\mathrm{\theta}\mathrm{cos}\left(90\xb0-\mathrm{\theta}\right)=1$

(iii) $\frac{\mathrm{tan}\theta}{\mathrm{tan}\left(90\xb0-\theta \right)}+\frac{{\displaystyle \mathrm{sin}\left(90\xb0-\theta \right)}}{{\displaystyle \mathrm{cos}\theta}}={\mathrm{sec}}^{2}\mathrm{\theta}$.

Q14. Prove the following :

(i) $\frac{\mathrm{cos}\left(90\xb0-\mathrm{A}\right)\mathrm{sin}\left(90\xb0-\mathrm{A}\right)}{\mathrm{tan}\left(90\xb0-\mathrm{A}\right)}=1-{\mathrm{cos}}^{2}\mathrm{A}$

(ii) $\frac{\mathrm{sin}\left(90\xb0-\mathrm{A}\right)}{\mathrm{cosec}\left(90\xb0-\mathrm{A}\right)}+\frac{\mathrm{cos}{\displaystyle}{\displaystyle \left(90\xb0-\mathrm{A}\right)}}{\mathrm{sec}\left(90\xb0-\mathrm{A}\right)}=1$

Q15. Simplify the following :

(i) $\frac{\mathrm{cos}\mathrm{\theta}}{\mathrm{sin}\left(90\xb0-0\right)}+\frac{\mathrm{cos}{\displaystyle}{\displaystyle \left(90\xb0-\mathrm{\theta}\right)}}{\mathrm{sec}\left(90\xb0-\mathrm{\theta}\right)}-3{\mathrm{tan}}^{2}30\xb0$

1. (Cos 13 + sin 13)/ (cos 13 - sin 13) = tan A

What is A?

2. (sin 20 cos 70 + cos 20 sin 70)/ (sin 23 cosec 23 + cos 23 sec 23)= ?

3. If tan a = 5/6 and tan b = 1/11, then what is the value of a + b ?

2. If tan $\theta $ = cot $\theta $ and $0\le \theta \le 90$, State the value of $\theta $.

3. If sin x = cos y; write the relation between x and y, if both the angles x and y are acute.

^{2}+ y^{2}= a^{2}+ b^{2}8 (c) In the figure (3) given below, AD is perpendicular to BC, BD=15 cm, sin B= $\frac{4}{5}$ and tan C=1

(i) Calculate the lengths of AD, AB, DC and AC.

(i) sin A (ii) cos A (iii) tan C

^{2}A=1/(1-cos A)Q.4. Prove that

(ii). $4\left({\mathrm{sin}}^{4}30\xb0+{\mathrm{cos}}^{4}60\xb0\right)-3\left({\mathrm{cos}}^{2}45\xb0-{\mathrm{sin}}^{2}90\xb0\right)=2$

Find

(a)tanx

(b)sinx

$iv.\mathrm{tan}\left(55\xb0-A\right)-cot\left(35\xb0+A\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}v.\mathrm{cos}ec\left(65\xb0+A\right)-sec\left(25\xb0-A\right)$

Tan

^{2}θ/ (sec^{2}θ-1)^{2}= 1+ cosθ/(1- cos θ)Q6. If 3$\mathrm{\theta}$ is an acute angle, solve the following equation for $\mathrm{\theta}$ :

(cosec 3$\mathrm{\theta}$ – 2) (cot 2$\mathrm{\theta}$ – 1) = 0.