Prove the following :
Question: Prove the following : (i) $\sin \theta \sin \left(90^{\circ}-\theta\right)-\cos \theta \cos \left(90^{\circ}-\theta\right)=0$ (ii) $\frac{\cos \left(90^{\circ}-\theta\right) \sec \left(90^{\circ}-\theta\right) \tan \theta}{\operatorname{cosec}\left(90^{\circ}-\theta\right) \sin \left(90^{\circ}-\theta\right) \cot \left(90^{\circ}-\theta\right)}+\frac{\tan \left(90^{\circ}-\theta\right)}{\cot \theta}=2$ (iii) $\frac{\tan \left(90^{\circ}-A\right) \cot A}{\operatorname{cosec}^{2} A}-\cos...
Read More →Write the value of cos
Question: Write the value of $\cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7}$. Solution: We have, $\cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7}=\frac{2 \sin \frac{\pi}{7} \cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7}}{2 \sin \frac{\pi}{7}}$ $\left[\right.$ On dividing and multiplying by $\left.2 \sin \frac{\pi}{7}\right]$ $=\frac{2 \times \sin \frac{2 \pi}{7} \cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7}}{2 \times 2 \sin \frac{\pi}{7}}$ Proceeding in the ...
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Question: If $\frac{\pi}{4}x\frac{\pi}{2}$, then write the value of $\sqrt{1-\sin 2 x}$. Solution: We have, $\sqrt{1-\sin 2 x}$ $=\sqrt{\sin ^{2} x+\cos ^{2} x-2 \sin x \cos x}$ $=\sqrt{(\sin x-\cos x)^{2}}$ $=|\sin x-\cos x|$ $=\sin x-\cos x$ $\left[\because \sin x\cos x\right.$ for $\left.\frac{\pi}{4}x\frac{\pi}{2}\right]$ $\therefore \sqrt{1-\sin 2 x}=\sin x-\cos x$...
Read More →Write the value of cos
Question: Write the value of $\cos ^{2} 76^{\circ}+\cos ^{2} 16^{\circ}-\cos 76^{\circ} \cos 16^{\circ} .$ Solution: We have, $\cos ^{2} 76^{\circ}+\cos ^{2} 16^{\circ}-\cos 76^{\circ} \cos 16^{\circ}$ $=\frac{1}{2}\left[1+\cos 2(76)^{\circ}+1+\cos 2(16)^{\circ}-\cos (76+16)^{\circ}-\cos (76-16)^{\circ}\right]$ $\left[\because 2 \cos ^{2} \theta=1+\cos 2 \theta\right.$ and $\left.2 \cos A \cos B=\cos (A+B)+\cos (A-B)\right]$ $=\frac{1}{2}\left[2+\cos 152^{\circ}+\cos 32^{\circ}-\cos 92^{\circ}-\...
Read More →State whether the following statements are true or false. Give reasons for your answer.
Question: State whether the following statements are true or false. Give reasons for your answer.(i) Every natural number is a whole number.(ii) Every whole number is a natural number.(iii) Every integer is a whole number.(iv) Every integer is a rational number.(v) Every rational number is an integer.(vi) Every rational number is a whole number. Solution: (i) Every natural number is a whole number.True, since natural numbers are counting numbers i.e N = 1, 2,...Whole numbers are natural numbers ...
Read More →In a right angled triangle ABC,
Question: In a right angled triangleABC, write the value of sin2A+ Sin2B+ Sin2C. Solution: Let, $\angle B=90^{\circ}$ $\therefore A+C=90^{\circ}=\frac{\pi}{2}$ $\Rightarrow C=\frac{\pi}{2}-A$ $\Rightarrow \sin C=\sin \left(\frac{\pi}{2}-A\right)$ $\Rightarrow \sin C=\cos A \ldots(\mathrm{i})$ Now, $\sin ^{2} A+\sin ^{2} B+\sin ^{2} C=\sin ^{2} A+1+\sin ^{2} C \quad\left(\because \sin B=\sin 90^{\circ}=1\right)$ $=\sin ^{2} A+\cos ^{2} A+1 \quad[$ Using $(\mathrm{i})]$ $=1+1$ $=2$...
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Question: If $\pix\frac{3 \pi}{2}$, then write the value of $\sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}$. Solution: We have, $\sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}=\sqrt{\frac{2 \sin ^{2} x}{2 \cos ^{2} x}}$ $=\frac{|\sin x|}{|\cos x|}$ $=\frac{|\sin x|}{|\cos x|}$ $=\frac{-\sin x}{-\cos x} \quad\left(\because \pix\frac{3 \pi}{2}\right)$ $\therefore \sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}=\tan x$...
Read More →Insert 16 rational numbers between 2.1 and 2.2.
Question: Insert 16 rational numbers between 2.1 and 2.2. Solution: Let:x= 2.1,y= 2.2 andn= 16 We know: $d=\frac{y-x}{n+1}=\frac{2.2-2.1}{16+1}=\frac{0.1}{17}=\frac{1}{170}=0.005$ (approx.) So, 16 rational numbers between 2.1 and 2.2 are: $(x+d),(x+2 d), \ldots(x+16 d)$ $=[2.1+0.005],[2.1+2(0.005)], \ldots[2.1+16(0.005)]$ = 2.105, 2.11, 2.115, 2.12, 2.125, 2.13, 2.135, 2.14, 2.145, 2.15, 2.155, 2.16, 2.165, 2.17, 2.175 and 2.18...
Read More →Prove that :Prove that :
Question: Prove that : (i) $\tan 20^{\circ} \tan 35^{\circ} \tan 45^{\circ} \tan 55^{\circ} \tan 70^{\circ}=1$ (ii) $\sin 48^{\circ} \sec 42^{\circ}+\cos 48^{\circ} \operatorname{cosec} 42^{\circ}=2$ (iii) $\frac{\sin 70^{\circ}}{\cos 20^{\circ}}+\frac{\operatorname{cosec} 20^{\circ}}{\sec 70^{\circ}}-2 \cos 70^{\circ} \operatorname{cosec} 20^{\circ}=0$ (iv) $\frac{\cos 80^{\circ}}{\sin 10^{\circ}}+\cos 59^{\circ} \operatorname{cosec} 31^{\circ}=2$ Solution: We are asked to find the value of $\t...
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Question: If $\frac{\pi}{2}x\pi$, then write the value of $\frac{\sqrt{1-\cos 2 x}}{1+\cos 2 x}$. Solution: WE have, $\sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}=\sqrt{\frac{2 \sin ^{2} x}{2 \cos ^{2} x}}$ $=\frac{|\sin x|}{|\cos x|}$ $=\frac{\sin x}{-\cos x} \quad\left(\because \frac{\pi}{2}x\pi\right)$ $=-\tan x$...
Read More →A merchant plans to sell two types of personal computers − a desktop model and a portable model that will cost Rs 25000 and Rs 40000 respectively.
Question: A merchant plans to sell two types of personal computers a desktop model and a portable model that will cost Rs 25000 and Rs 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs 70 lakhs and if his profit on the desktop model is Rs 4500 and on portable model is Rs 5000. Solution: Let th...
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Question: If $\frac{\pi}{2}x\pi$, the write the value of $\sqrt{2+\sqrt{2+2 \cos 2 x}}$ in the simplest form. Solution: We have, $\sqrt{2+\sqrt{2+2 \cos 2 x}}=\sqrt{2+\sqrt{2(1+\cos 2 x)}}$ $=\sqrt{2+\sqrt{2.2 \cos ^{2} x}}$ $=\sqrt{2+2|\cos x|}$ $=\sqrt{2-2 \cos x} \quad\left(\because \frac{\pi}{2}x\pi\right)$ $=\sqrt{2(1-\cos x)}$ $=\sqrt{2.2 \sin ^{2} \frac{x}{2}}$ $=2\left|\sin \frac{x}{2}\right|$ $=2 \sin \frac{x}{2} \quad\left(\because \frac{\pi}{4}\frac{x}{2}\frac{\pi}{2}\right)$...
Read More →Find five rational numbers between
Question: Find five rational numbers between $\frac{3}{5}$ and $\frac{2}{3}$. Solution: n= 5n+ 1 = 6 $x=\frac{3}{5}, y=\frac{2}{3}$ $d=\frac{y-x}{n+1}=\frac{\frac{2}{3}-\frac{3}{5}}{6}=\frac{10-9}{90}=\frac{1}{90}$ Thus, rational numbers between $\frac{3}{5}$ and $\frac{2}{3}$ will be $(x+d),(x+2 d),(x+3 d),(x+4 d),(x+5 d)$ $=\left(\frac{3}{5}+\frac{1}{90}\right),\left(\frac{3}{5}+\frac{2}{90}\right),\left(\frac{3}{5}+\frac{3}{90}\right),\left(\frac{3}{5}+\frac{4}{90}\right),\left(\frac{3}{5}+\f...
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Question: If $\frac{\pi}{2}x\frac{3 \pi}{2}$, then write the value of $\sqrt{\frac{1+\cos 2 x}{2}}$. Solution: $\because \frac{\pi}{2}x\frac{3 \pi}{2}$ $\therefore \sqrt{\frac{1+\cos 2 x}{2}}=\sqrt{\frac{2 \cos ^{2} x}{2}}=|\cos x|$ In second quadrant $\cos x$ is negative. $\therefore \sqrt{\frac{1+\cos 2 x}{2}}=-\cos x$...
Read More →If A, B, C are the interior angles of a triangle ABC, prove that
Question: If A, B, C are the interior angles of a triangle ABC, prove that (i) $\tan \left(\frac{C+A}{2}\right)=\cot \frac{B}{2}$ (ii) $\sin \left(\frac{B+C}{2}\right)=\cos \frac{A}{2}$ Solution: (i) We have to prove: $\tan \left(\frac{C+A}{2}\right)=\cot \frac{B}{2}$ Since we know that in triangle $A B C$ $A+B+C=180$ $\Rightarrow C+A=180^{\circ}-B$ $\Rightarrow \frac{C+A}{2}=90^{\circ}-\frac{B}{2}$ $\Rightarrow \tan \frac{C+A}{2}=\tan \left(90^{\circ}-\frac{B}{2}\right)$ $\Rightarrow \tan \frac...
Read More →If tan
Question: If $\tan \frac{x}{2}=\frac{m}{n}$, then write the value of $m \sin x+n \cos x$. Solution: Given: $\tan \frac{x}{2}=\frac{m}{n}$ $\Rightarrow \frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}=\frac{m}{n}$ Let $\sin \frac{x}{2}$ be $m k$ and $\cos \frac{x}{2}$ be $n k$. Now, $m \sin x+n \cos x=2 m \sin \frac{x}{2} \cos \frac{x}{2}+n\left(\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}\right)$ $=2 m \times m k \times n k+n\left(n^{2} k^{2}-m^{2} k^{2}\right)$ $=2 m^{2} k^{2} n+n k^{2}\left(n^{2}-m^{2...
Read More →Find six rational numbers between 2 and 3.
Question: Find six rational numbers between 2 and 3. Solution: x= 2,y= 3 andn= 6 $d=\frac{y-x}{n+1}=\frac{3-2}{6+1}=\frac{1}{7}$ Thus, the required numbers are $(x+d),(x+2 d),(x+3 d), \ldots,(x+n d)$ $=\left(2+\frac{1}{7}\right),\left(2+2 \times \frac{1}{7}\right),\left(2+3 \times \frac{1}{7}\right),\left(2+4 \times \frac{1}{7}\right),\left(2+5 \times \frac{1}{7}\right),\left(2+6 \times \frac{1}{7}\right)$ $=\frac{15}{7}, \frac{16}{7}, \frac{17}{7}, \frac{18}{7}, \frac{19}{7}, \frac{20}{7}$...
Read More →If cos 4x=1+k sin
Question: If $\cos 4 x=1+k \sin ^{2} x \cos ^{2} x$, then write the value of $k$. Solution: We have, $\cos 4 x=1+k \sin ^{2} \mathrm{x} \cos ^{2} x$ $\Rightarrow \cos (2 \times 2 x)=1+k \sin ^{2} x \cos ^{2} x$ $\Rightarrow 1-2 \sin ^{2} 2 x=1+k \sin ^{2} x \cos ^{2} x$ $\Rightarrow 1-2(2 \sin x \cos x)^{2}=1+k \sin ^{2} x \cos ^{2} x$ $\Rightarrow 1-8 \sin ^{2} x \cos ^{2} x=1+k \sin ^{2} x \cos ^{2} x$ $\Rightarrow \sin ^{2} x \cos ^{2} x(k+8)=0$ $\Rightarrow k+8=0$ $\therefore k=-8$...
Read More →If sin 3A = cos (A − 26°), where 3A is an acute angles, find the value of A.
Question: If $\sin 3 A=\cos \left(A-26^{\circ}\right)$, where $3 A$ is an acute angles, find the value of $A$. Solution: We are given 3Ais an acute angle We have: $\sin 3 A=\cos \left(A-26^{\circ}\right)$ $\Rightarrow \sin 3 A=\sin \left(90^{\circ}-\left(A-26^{\circ}\right)\right)$ $\Rightarrow \sin 3 A=\sin \left(116^{\circ}-A\right)$ $\Rightarrow 3 A=116^{\circ}-A$ $\Rightarrow 4 A=116^{\circ}$ $\Rightarrow A=29^{\circ}$ Hence the correct answer is $29^{\circ}$...
Read More →Find four rational numbers between
Question: Find four rational numbers between $\frac{3}{7}$ and $\frac{5}{7}$. Solution: n= 4n+ 1 = 4 + 1 = 5 $\frac{3}{7}=\frac{3}{7} \times \frac{5}{5}=\frac{15}{35}$ $\frac{5}{7}=\frac{5}{7} \times \frac{5}{5}=\frac{25}{35}$ Thus, rational numbers between $\frac{3}{7}$ and $\frac{5}{7}$ are $\frac{16}{35}, \frac{17}{35}, \frac{18}{35}, \frac{19}{35}$....
Read More →The value of 108 sin
Question: The value of $108 \sin \frac{\pi}{9}-144 \sin ^{3} \frac{\pi}{9}$ is Solution: $108 \sin \frac{\pi}{9}-144 \sin ^{3} \frac{\pi}{9}$ $=36\left(3 \sin \frac{\pi}{9}-4 \sin ^{3} \frac{\pi}{9}\right)$ $=36\left(\sin 3\left(\frac{\pi}{9}\right)\right)$ $\left[\right.$ using identity $\left.\because 3 \sin x-4 \sin ^{3} x=\sin 3 x\right]$ $=36\left(\sin \frac{\pi}{3}\right)$ $=36\left(\frac{\sqrt{3}}{2}\right)$ $=18 \sqrt{3}$ $\therefore 108 \sin \frac{\pi}{9}-144 \sin ^{3} \frac{\pi}{9}$ is...
Read More →Express cos 75° + cot 75° in terms of angles between 0° and 30°.
Question: Express $\cos 75^{\circ}+\cot 75^{\circ}$ in terms of angles between $0^{\circ}$ and $30^{\circ} .$ Solution: Given that: $\cos 75^{\circ}+\cot 75^{\circ}$ $=\cos 75^{\circ}+\cot 75^{\circ}$ $=\cos \left(90^{\circ}-15^{\circ}\right)+\cot \left(90^{\circ}-15^{\circ}\right)$ $=\sin 15^{\circ}+\tan 15^{\circ}$ Hence the correct answer is $\sin 15^{\circ}+\tan 15^{\circ}$...
Read More →Express each one of the following in terms of trigonometric ratios of angles lying between 0° and 45°
Question: Express each one of the following in terms of trigonometric ratios of angles lying between $0^{\circ}$ and $45^{\circ}$ (i) $\sin 59^{\circ}+\cos 56^{\circ}$ (ii) $\tan 65^{\circ}+\cot 49^{\circ}$ (iii) $\sec 76^{\circ}+\operatorname{cosec} 52^{\circ}$ (iv) $\cos 78^{\circ}+\sec 78^{\circ}$ (v) $\operatorname{cosec} 54^{\circ}+\sin 72^{\circ}$ (vi) $\cot 85^{\circ}+\cos 75^{\circ}$ (vii) $\sin 67^{\circ}+\cos 75^{\circ}$ Solution: (i) We have $\sin \left(90^{\circ}-\theta\right)=\cos \...
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Question: If $\frac{\pi}{4}x\frac{\pi}{2}$, then $\sqrt{2+\sqrt{2+2 \cos 4 x}}=$ Solution: Given for $\frac{\pi}{4}x\frac{\pi}{2}$ $\sqrt{2+\sqrt{2+2 \cos 4 x}}$ $=\sqrt{2+\sqrt{2(1+\cos 4 x)}}$ $=\sqrt{2+\sqrt{2\left(2 \cos ^{2} 2 x\right)}}$ $=\sqrt{2+\sqrt{4 \cos ^{2} 2 x}}$ $=\sqrt{2+2|\cos 2 x|}$ $\left\{\begin{array}{l}\text { Since } \frac{\pi}{4}x\frac{\pi}{2} \\ \Rightarrow \frac{\pi}{2}2 x\pi \\ \text { and } \cos \text { is negative in II quadrant } \\ \Rightarrow|\cos 2 x|=-\cos 2 x\...
Read More →Find three rational numbers lying between
Question: Find three rational numbers lying between $\frac{3}{5}$ and $\frac{7}{8}$. How many rational numbers can be determined between these two numbers? Solution: $x=\frac{3}{5}$ and $y=\frac{7}{8}$ n= 3 $d=\frac{(y-x)}{n+1}=\frac{\frac{7}{8}-\frac{3}{5}}{3+1}=\frac{11}{40} \times \frac{1}{4}=\frac{11}{160}$ Rational numbers between $x=\frac{3}{5}$ and $y=\frac{7}{8}$ will be $(x+d),(x+2 d), \ldots,(x+n d)$ $\Rightarrow\left(\frac{3}{5}+\frac{11}{160}\right),\left(\frac{3}{5}+2 \times \frac{1...
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