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Have you ever wondered how engineers design huge buildings, how sailors and pilots navigate through their journey, or how shadows change their length throughout the day? All of these answers can be found in Trigonometry, a fascinating branch of mathematics. As per the latest syllabus, this chapter contains the basic concepts of trigonometry, like trigonometric ratios, trigonometric identities, trigonometric ratios of some specific angles, and trigonometric ratios of complementary angles. NCERT Solutions for Class 10 can help the students understand these concepts and will make them more efficient in solving problems involving height, distance, and angles.
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These NCERT Solutions for class 10 Maths are designed by our experienced subject experts at Careers360 to offer clear and step-by-step solutions for the exercise problems and help students to prepare well for exams and to gain knowledge about all the natural processes happening around them through a series of solved questions. It covers questions from all the topics and will help you improve your speed and accuracy. Many teachers recommend NCERT Solutions because they closely match the exam pattern. For full syllabus coverage and solved exercises as well as a downloadable PDF, please visit the NCERT article.
Students who wish to access the NCERT solutions for class 10 chapter 8 can click on the link below to download the entire solution in PDF.
NCERT Introduction to Trigonometry Class 10 Questions and Answers Exercise: 8.1 Page number: 121 Total questions: 11 |
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Q1. In $\Delta \: ABC$ , right-angled at $B, AB = 24 \: cm$ , $BC = 7 \: cm$.
Determine : $(i)\; \sin A, \cos A$ $(ii)\; \sin C, \cos C$
Answer:
We have,
In $\Delta \: ABC$ , $\angle$B = 90, and the length of the base (AB) = 24 cm and length of perpendicular (BC) = 7 cm
So, by using Pythagoras' theorem,
$AC^2 = AB^2 + BC^2$
⇒ $AC = \sqrt{AB^2+BC^2}$
⇒ $AC = \sqrt{576+49}$
⇒ $AC = \sqrt{625}$
⇒ $AC = 25$ cm
Now,
(i) $\sin A = \frac{P}{H} = \frac{BC}{AC} = \frac{7}{25}$
$\cos A = \frac{B}{H} = \frac{BA}{AC} = \frac{24}{25}$
(ii) For angle C, AB is perpendicular to the base (BC).
Here, B indicates Base and P means perpendicular wrt angle $\angle$C
So, $\sin C = \frac{P}{H} = \frac{BA}{AC} = \frac{24}{25}$
and $\cos C = \frac{B}{H} = \frac{BC}{AC} = \frac{7}{25}$
Q2. In Fig. 8.13, find $\tan P - \cot R$ .
Answer:
We have $\Delta$PQR
According to Pythagoras, in a right-angled triangle, the lengths of PQ and PR are 12 cm and 13 cm, respectively.
So, by using Pythagoras' theorem,
$QR = \sqrt{13^2-12^2}$
$QR = \sqrt{169-144}$
$QR = \sqrt{25} = 5\ cm$
Now, according to the question,
$\tan P -\cot R$ = $\frac{RQ}{QP}-\frac{QR}{PQ} = \frac{5}{12} - \frac{5}{12} = 0$
Q3. If $\sin A=\frac{3}{4},$ calculate $\cos A$ and $\tan A$ .
Answer:
Suppose $\Delta$ ABC is a right-angled triangle in which $\angle B = 90°$ and we have $\sin A=\frac{3}{4},$
So,
Let the length of AB be 4 units and the length of BC = 3 units. So, by using Pythagoras' theorem,
$AB = \sqrt{16-9} = \sqrt{7}$ units
Therefore,
$\cos A = \frac{AB}{AC} = \frac{\sqrt{7}}{4}$ and $\tan A = \frac{BC}{AB} = \frac{3}{\sqrt{7}}$
Q4. Given $15 \cot A=8,$ find $\sin A$ and $\sec A$ .
Answer:
We have,
$15 \cot A=8$ $\Rightarrow \cot A =\frac{8}{15}$
It implies that in the triangle ABC, in which $\angle B =90°$. The length of AB is 8 units, and the length of BC = 15 units
Now, by using Pythagoras' theorem,
$AC = \sqrt{64 +225} = \sqrt{289}$
$\Rightarrow AC =17$ units
So, $\sin A = \frac{BC}{AC} = \frac{15}{17}$
and $\sec A = \frac{AC}{AB} = \frac{17}{8}$
Q5. Given $\sec \theta =\frac{13}{12},$ calculate all other trigonometric ratios.
Answer:
We have,
$\sec \theta =\frac{13}{12},$
This means that the hypotenuse of the triangle is 13 units, and the base is 12 units.
Let ABC be a right-angled triangle in which $\angle$ B is 90 and AB is the base, BC is the perpendicular height, and AC is the hypotenuse.
By using Pythagoras' theorem,
$BC = \sqrt{169-144}=\sqrt{25}$
$\Rightarrow BC = 5$ units
Therefore,
$\sin \theta = \frac{BC}{AC}=\frac{5}{13}$
$\cos \theta = \frac{BA}{AC}=\frac{12}{13}$
$\tan \theta = \frac{BC}{AB}=\frac{5}{12}$
$\cot \theta = \frac{BA}{BC}=\frac{12}{5}$
$\sec \theta = \frac{AC}{AB}=\frac{13}{12}$
$\csc \theta = \frac{AC}{BC}=\frac{13}{5}$
Answer:
We have, A and B are two acute angles of triangle ABC and $\cos A =\cos B$.
According to the question, in triangle ABC,
$\cos A =\cos B$
$\frac{AC}{AB}=\frac{BC}{AB}$
$\Rightarrow AC = AB$
Therefore, $\angle$ A = $\angle$ B [angles opposite to equal sides are equal]
Q7. If $\cot \theta =\frac{7}{8},$ evaluate: $(i)\; \frac{(1+\sin \theta)(1-\sin \theta)}{(1+\cos \theta)(1-\cos \theta)}$ $(ii)\; \cot ^{2}\theta$
Answer:
Given that,
$\cot \theta =\frac{7}{8}$
$\therefore$ perpendicular (AB) = 8 units and Base (BC) = 7 units
Draw a right-angled triangle ABC in which $\angle B =90°$
Now, by using Pythagoras' theorem,
$AC^2 = AB^2+BC^2$
$AC = \sqrt{64 +49} =\sqrt{113}$
So, $\sin \theta = \frac{AB}{AC} = \frac{8}{\sqrt{113}}$
and $\cos \theta = \frac{BC}{AC} = \frac{7}{\sqrt{113}}$
$\Rightarrow \cot \theta =\frac{\cos \theta}{\sin \theta} = \frac{7}{8}$
$(i)\; \frac{(1+\sin \theta)(1-\sin \theta)}{(1+\cos \theta)(1-\cos \theta)}$
$\Rightarrow \frac{(1-\sin^2\theta)}{(1-\cos^2\theta)} = \frac{\cos^2\theta}{\sin^2\theta} = \cot ^2\theta=(\frac{7}{8})^2 = \frac{49}{64}$
$(ii)\; \cot ^{2}\theta$
$=(\frac{7}{8})^2 = \frac{49}{64}$
Q8. If $3\cot A=4,$ check wether $\frac{1-\tan ^{2}A}{1+\tan ^{2}A}=\cos ^{2}A-\sin ^{2}A$ or not.
Answer:
Given that,
$3\cot A=4,$
$\Rightarrow \cot = \frac{4}{3} = \frac{base}{perp.}$
ABC is a right-angled triangle in which $\angle B =90°$ and the length of the base AB is 4 units and the length of the perpendicular is 3 units
By using Pythagoras' theorem, in triangle ABC,
$\\AC^2=AB^2+BC^2\Rightarrow AC = \sqrt{16+9}\Rightarrow AC = \sqrt{25}$
$\Rightarrow AC = 5$ units
So,
$\tan A = \frac{BC}{AB} = \frac{3}{4}$
$\cos A = \frac{AB}{AC} = \frac{4}{5}$
$\sin A = \frac{BC}{AC} = \frac{3}{5}$
$\frac{1-\tan ^{2}A}{1+\tan ^{2}A}=\cos ^{2}A-\sin ^{2}A$
Putting the values of the above trigonometric ratios, we get;
$\Rightarrow \frac{1-\frac{9}{16}}{1+\frac{9}{16}} = \frac{16}{25}-\frac{9}{25}$
$\Rightarrow \frac{7}{25} = \frac{7}{25}$
LHS $=$ RHS
Q9. In triangle $ABC$ , right-angled at $B$ , if $\tan A =\frac{1}{\sqrt{3}},$ find the value of:
$(i) \sin A\: \cos C + \cos A\: \sin C$
$(ii) \cos A\: \cos C + \sin A\: \sin C$
Answer:
Given a triangle ABC, right-angled at B and $\tan A =\frac{1}{\sqrt{3}}$ $\Rightarrow A=30°$
According to question, $\tan A =\frac{1}{\sqrt{3}} = \frac{BC}{AB}$
By using Pythagoras' theorem,
$\\AC^2 = AB^2+BC^2\Rightarrow AC = \sqrt{1+3} =\sqrt{4}$
$\Rightarrow AC = 2$
Now,
$\\\sin A = \frac{BC}{AC} = \frac{1}{2}, \sin C =\frac{AB}{AC} = \frac{\sqrt{3}}{2}, \cos A = \frac{AB}{AC} = \frac{\sqrt{3}}{2}, \cos C = \frac{BC}{AC} = \frac{1}{2}$
Therefore,
$(i) \sin A\: \cos C + \cos A\: \sin C$
$= \frac{1}{2}\times\frac{1}{2}+\frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{2}=\frac{1}{4} +\frac{3}{4} = 1$
$(ii) \cos A\: \cos C + \sin A\: \sin C$
$= \frac{\sqrt{3}}{2}\times \frac{1}{2}+\frac{1}{2}\times\frac{\sqrt{3}}{2}= \frac{\sqrt{3}}{4}+\frac{\sqrt{3}}{4}= \frac{\sqrt{3}}{2}$
Answer:
We have, PR + QR = 25 cm.............(i)
PQ = 5 cm and $\angle Q =90°$
According to the question,
In triangle $\Delta$ PQR,
By using Pythagoras' theorem,
$PR^2 = PQ^2+QR^2$
$\Rightarrow PQ^2 =PR^2-QR^2$
$\Rightarrow 5^2= (PR-QR)(PR+QR)$
$\Rightarrow 25 = 25(PR-QR)$
$\Rightarrow PR - QR = 1$........(ii)
From equation(i) and equation(ii), we get;
PR = 13 cm and QR = 12 cm.
Therefore,
$\sin P= \frac{QR}{PR}= \frac{12}{13}$
$\cos P = \frac{PQ}{RP} = \frac{5}{13}$
$\tan P = \frac{\sin P}{\cos P} = \frac{12}{5}$
Q11. State whether the following are true or false. Justify your answer.
(i) The value of $\tan A$ is always less than 1.
(ii) $\sec A=\frac{12}{5}$ for some value of angle A.
(iii) $\cos A$ is the abbreviation used for the cosecant of angle A.
(iv) $\cot A$ is the product of cot and A.
(v) $\sin \theta =\frac{4}{3}$ for some angle $\theta .$
Answer:
(i) False,
because $\tan 60 = \sqrt{3}$ , which is greater than 1
(ii) True,
because $\sec A \geq 1$
(iii) False,
Because the $\cos A$ abbreviation is used for cosine A.
(iv) False,
because the term $\cot A$ is a single term, not a product.
(v) False,
because $\sin \theta$ lies between (-1 to +1) [ $-1\leq \sin \theta\leq 1$]
NCERT Introduction to Trigonometry Class 10 Questions and Answers Exercise: 8.2 Page number: 127 Total questions: 4 |
$(i) \sin 60°\cos 30°+\sin30° \cos 60°$
Answer:
As we know,
the value of $\sin 60° = \frac{\sqrt{3}}{2} = \cos 30°$ , $\sin 30° = \frac{1}{2}=\cos 60°$
$\sin 60°\cos 30°+\sin30° \cos 60°$
$= \frac{\sqrt{3}}{2}.\frac{\sqrt{3}}{2}+ \frac{1}{2}.\frac{1}{2}\\$
$=\frac{3}{4}+\frac{1}{4}$
$=1$
$(ii)\: 2\tan ^{2}45°+ 2\cos ^{2}30°- 2\sin ^{2}60°$
Answer:
We know the value of
$\tan 45°= 1$ and
$\cos 30° = \sin 60° = \frac{\sqrt{3}}{2}$
According to the question,
$=2\tan ^{2}45°+ 2\cos ^{2}30°- 2\sin ^{2}60°$
$\\=2(1)^2+ 2(\frac{\sqrt{3}}{2})^2-2(\frac{\sqrt{3}}{2})^2\\=2$
$(iii)\: \frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$
Answer:
$\frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$
We know the value of
$\cos 45^{\circ} = \frac{1}{\sqrt{2}}$ , $\sec 30^{\circ}= \frac{2}{\sqrt {3}}$ and $\operatorname{cosec}30^{\circ} =2$ ,
After putting these values
$=\frac{\frac{1}{\sqrt{2}}}{\frac{2}{\sqrt{3}}+2}$
$=\frac{\frac{1}{\sqrt{2}}}{\frac{(2+2\sqrt{3})}{ \sqrt{3}}}$
$\\=\frac{\sqrt{3}}{2\sqrt{2}+2\sqrt{6}}\times\frac{2\sqrt{2}-2\sqrt{6}}{2\sqrt{2}-2\sqrt{6}}$
$\\=\frac{2\sqrt{6}-2\sqrt{18}}{-16}\\ =2(\frac{\sqrt{6}-3\sqrt{3}}{-16}) = \frac{3\sqrt{3}-\sqrt{6}}{8}$
$(iv)\: \frac{\sin 30°+\tan 45°-\operatorname{cosec} 60°}{\sec 30°+\cos 60°+\cot 45°}$
Answer:
$\frac{\sin 30°+\tan 45°-\operatorname{cosec} 60°}{\sec 30°+\cos 60°+\cot 45°}$ ..................(i)
It is known that the values of the given trigonometric functions,
$\\\sin 30° = \frac{1}{2}=cos 60°, \tan 45° = 1=\cot 45°, \sec 30° = \frac{2}{\sqrt{3}}=\operatorname{cosec} 60°$
Putting all these values in equation (i), we get;
$\\\Rightarrow \frac{\frac{1}{2}+1-\frac{2}{\sqrt{3}}}{\frac{2}{\sqrt{3}}+\frac{1}{2}+1}=\frac{\frac{3}{2}-\frac{2}{\sqrt{3}}}{\frac{3}{2}+\frac{2}{\sqrt{3}}}= \frac{3\sqrt{3}-4}{3\sqrt{3}+4}\times\frac{4-3\sqrt{3}}{4-3\sqrt{3}}= \frac{12\sqrt{3}-27-16+12\sqrt{3}}{-11}= \frac{43-24\sqrt{3}}{11}$
$(v)\frac{5\cos^{2}60°+ 4\sec^{2}30°-\tan^{2}45°}{\sin^{2}30°+\cos^{2}30°}$
Answer:
$\frac{5\cos^{2}60°+ 4\sec^{2}30°-\tan^{2}45°}{\sin^{2}30°+\cos^{2}30°}$ .....................(i)
We know the values of,
$\\\cos 60° = \frac{1}{2}= \sin 30°, \sec 30° = \frac{2}{\sqrt{3}}, \tan 45° = 1, \cos 30° = \frac{\sqrt{3}}{2}$
By substituting all these values in equation(i), we get;
$\\\Rightarrow \frac{5.(\frac{1}{2})^2+4.(\frac{2}{\sqrt{3}}) ^2-1}{(\frac{1}{2})^2+(\frac{\sqrt{3}}{2})^2}= \frac{\frac{5}{4}-1+\frac{16}{3}}{1}= \frac{\frac{1}{4}+\frac{16}{3}}{1}=\frac{67}{12}$
Q2. Choose the correct option and justify your choice :
$(i)\, \frac{2\: \tan 30°}{1+\tan ^{2}30°}=$
$(A)\: \sin 60°$ $(B)\: \cos 60°$ $(C)\: \tan 60°$ $(D)\: \sin 30°$
Answer:
Put the value of $\tan 30°$ in the given question-
$\frac{2\: \tan 30°}{1+\tan ^{2}30°}=\frac{2\times\frac{1}{\sqrt{3}}}{1+(\frac{1}{\sqrt{3}})^2}=\frac{2}{\sqrt{3}}\times \frac{3}{4}=\frac{\sqrt{3}}{2} = \sin 60°$
The correct option is (A).
Q2. Choose the correct option and justify your choice :
$(ii)\: \frac{1-\tan^{2}45°}{1+ \tan^{2}45°}=$
$(A)\: \tan \: 90°$ $(B)\: 1$ $(C) \: \sin 45°$ $(D) \: 0$
Answer:
$\frac{1-\tan^{2}45°}{1+ \tan^{2}45°}=$
We know that $\tan 45° = 1$
So, $\frac{1-1}{1+1}=0$
The correct option is (D).
Q2. Choose the correct option and justify your choice :
$(iii)\sin \: 2A=2\: \sin A$ is true when $A$ =
$(A)0°$ $(B)\: 30°$ $(C)\: 45°$ $(D)\: 60°$
Answer:
The correct option is (A).
$\sin \: 2A=2\: \sin A$
We know that $\sin 2A = 2\sin A \cos A$
So, $2\sin A \cos A = 2\sin A$
$\\\cos A = 1, A = 0°$
Q2. Choose the correct option and justify your choice :
$(iv)\frac{2\: \tan 30°}{1-tan^{2}\: 30°}=$
$(A)\: \cos 60°$ $(B)\: \sin 60°$ $(C)\: \tan 60°$ $(D)\: \sin 30°$
Answer:
Put the value of $tan 30°={\frac{1}{\sqrt{3}}}$
$\\=\frac{2\: \tan 30°}{1-tan^{2}\: 30°}=\frac{2\times \frac{1}{\sqrt{3}}}{1-(\frac{1}{\sqrt{3}})^2}\\ =\frac{\frac{2}{\sqrt{3}}}{1-\frac{1}{3}}\\ =\frac{2}{\sqrt{3}}\times \frac{3}{2}\\ =\sqrt{3}= \tan 60°$
The correct option is (C).
Answer:
Given that,
$\tan (A+B) = \sqrt{3} =\tan 60°$
So, $A + B = 60^o$ ..........(i)
$\tan (A-B) = \frac{1}{\sqrt{3}} =\tan 30°$
therefore, $A - B = 30^o$ .......(ii)
By solving equations (i) and (ii), we get;
$A = 45^o$ and $B = 15^o$
Q4. State whether the following are true or false. Justify your answer.
$(i) \sin (A + B) = \sin A + \sin B$
$(ii)$ The value of $\sin \theta$ increases as $\theta$ increases.
$(iii)$ The value of $\cos \theta$ increases as $\theta$ increases.
$(iv)\sin \theta =\cos \theta$ for all values of $\theta$ .
$(v) \cot A$ is not defined for $A=0°$
Answer:
(i) False,
Let A = B = $45°$
Then, $\\\sin(45°+45°) = \sin 45°+\sin 45°$
$\sin 90° = \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}} \neq 1$
(ii) True,
Take $\theta = 0°,\ 30°,\ 45°$
when
$\theta$ = 0 then zero(0),
$\theta$ = 30 then value of $\sin \theta$ is $\frac{1}{2}$ = 0.5
$\theta$ = 45 then value of $\sin \theta$ is $\frac{1}{\sqrt2}$ = 0.707
(iii) False,
$\cos 0° = 1,\ \cos 30° = \frac{\sqrt{3}}{2}= 0.87,\ \cos 45° = \frac{1}{\sqrt2}= 0.707$
(iv) False,
Let $\theta = 0°$
$\\\sin 0° = 0$ and $\cos 0°=1$
(v) True,
$\cot 0° = \frac{\cos 0°}{\sin 0°}=\frac{1}{0}$ (not defined).
NCERT Introduction to Trigonometry Class 10 Questions and Answers |
Q1. Express the trigonometric ratios $\sin A,\sec, and \tan A$ in terms of $\cot A$.
Answer:
We know that $\csc^2A -\cot^2A = 1$
(i)
$\\\Rightarrow \frac{1}{\sin^2A}= 1+\cot^2A\\ \Rightarrow\sin^2A = \frac{1}{1+\cot^2A}\\ \Rightarrow \sin A = \frac{1}{\sqrt{1+\cot^2A}}$
(ii) We know the identity of
$\sec ^2 A-\tan ^2 A=1$
$\Rightarrow \sec^2 A=1+\frac{1}{\cot^2\theta}$
$\Rightarrow \sec A=\frac{\sqrt{\cot^2\theta+1}}{\cot\theta}$
(iii) $\tan A = \frac{1}{\cot A}$
Q2. Write all the other trigonometric ratios of $\angle A$ in terms of $\sec A$.
Answer:
We know that the identity $\sin^2 A + \cos^2 =1$
$\\\sin^2 A =1- \cos^2 A$
$\sin^2A = 1-\frac{1}{\sec^2A}=\frac{\sec^2A -1}{\sec^2A}$
$\sin A =\sqrt{\frac{\sec^2A -1}{\sec^2A}}=\frac{{}\sqrt{\sec^2A -1}}{\sec A}$
$cosec A =\frac{\sec A}{\sqrt{\sec^2A -1}}$
$\tan A = \frac{\sin A}{\cos A} = \sqrt{\sec^2A -1}$
$\cot A =\frac{1}{ \sqrt{\sec^2A -1}}$
$\cos A=\frac{1}{\sec A}$
Q3. Choose the correct option. Justify your choice.
$(i) 9\sec^{2}A-9\tan^{2}A=$
(A) 1 (B) 9 (C) 8 (D) 0
Answer:
The correct option is (B) = 9
$9\sec^2A-9 \tan ^2A = 9(\sec^2A- \tan ^2A)$ .............(i)
And it is known that sec2A - tan2A = 1
Therefore, equation (i) becomes, $9\times 1 = 9$
Q3. Choose the correct option. Justify your choice.
$(ii)(1+\tan \theta +\sec \theta )(1+\cot \theta -cosec\: \theta )=$
(A) 0 (B) 1 (C) 2 (D) –1
Answer:
The correct option is (C)
$(1+\tan \theta +\sec \theta )(1+\cot \theta -cosec\: \theta )$ .......................(i)
We can write his above equation as;
$=(1+\frac{\sin \theta}{\cos \theta} +\frac{1}{\cos \theta})(1+\frac{\cos\theta}{\sin \theta} -\frac{1}{sin\theta})$
$= \frac{(1+\sin\theta+\cos\theta)}{\cos\theta.\sin\theta}\times (\frac{(\sin\theta+\cos\theta-1}{\sin\theta.\cos\theta})$
$= \frac{(\sin\theta+\cos\theta)^2-1^2}{\sin\theta.\cos\theta}$
$= \frac{\sin^2\theta+\cos^2\theta+2\sin\theta.\\cos\theta-1}{\sin\theta.\cos\theta}$
$= 2\times\frac{\sin\theta.\cos\theta}{\sin\theta.\cos\theta}$
$= 2$
Q3. Choose the correct option. Justify your choice.
$(iii) (\sec A+\tan A)(1-\sin A)=$
$(A)\sec A$ $(B)\sin A$ $(C)cosec A$ $(D) \cos A$
Answer:
The correct option is (D)
$(\sec A+\tan A)(1-\sin A)$
$= ( \frac{1}{\cos A}+\frac{\sin A}{\cos A})(1-\sin A)= \frac{1+\sin A}{\cos A}(1-\sin A)= \frac{1-\sin^2 A}{\cos A}= \cos A$
Q3. Choose the correct option. Justify your choice.
$(iv) \frac{1+\tan ^{2}A}{1+\cot ^{2}A}=$
$(A) \sec ^{2}A$ $(B) -1$ $(C) \cot ^{2}A$ $(D) \tan ^{2}A$
Answer:
The correct option is (D)
$\frac{1+\tan ^{2}A}{1+\cot ^{2}A}$
We know that $\cot A = \frac{1}{\tan A}$
therefore,
$\\\Rightarrow \frac{1+\tan ^{2}A}{1+\frac{1}{\tan ^{2}A}}= \tan^2A\times(\frac{1+\tan^2 A}{1+\tan^2A})= \tan^2A$
$(i) (\csc \theta -\cot \theta )^{2}= \frac{1-\cos \theta }{1+\cos \theta }$
Answer:
We need to prove-
$(\csc \theta -\cot \theta )^{2}= \frac{1-\cos \theta }{1+\cos \theta }$
Now, taking LHS,
$(\csc\theta-\cot\theta)^2 = (\frac{1}{\sin\theta}-\frac{\cos\theta}{\sin\theta})^2$
$\\= (\frac{1-\cos\theta}{\sin\theta})^2\\ =\frac{(1-\cos\theta)(1-\cos\theta)}{\sin^2\theta}\\$
$\\=\frac{(1-\cos\theta)(1-\cos\theta)}{1-\cos^2\theta}\\\\ =\frac{(1-\cos\theta)(1-\cos\theta)}{(1-\cos\theta)(1+\cos\theta)}\\\\ =\frac{1-\cos\theta}{1+\cos\theta}$
LHS = RHS
Hence proved.
$(ii) \frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}= 2\sec A$
Answer:
We need to prove-
$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}= 2\sec A$
Taking LHS;
$\\=\frac{\cos^2A+1+\sin^2A+2\sin A}{\cos A(1+\sin A)}\\\\ =\frac{2(1+\sin A)}{\cos A(1+\sin A)}\\\\ =\frac{2}{\cos A} = 2\sec A$
= RHS
Hence proved.
$(iii)\frac{\tan \theta }{1-\cot \theta }+\frac{\cot \theta }{1-\tan \theta }=1+\sec \theta \csc \theta$
[ Hint: Write the expression in terms of $\sin \theta$ and $\cos\theta$ ]
Answer:
We need to prove-
$\frac{\tan \theta }{1-\cot \theta }+\frac{\cot \theta }{1-\tan \theta }=1+\sec \theta \:cosec \theta$
Taking LHS;
$\\\Rightarrow \frac{\tan^2 \theta }{\tan \theta-1 }+\frac{1}{\tan\theta(1-\tan \theta)}=\frac{\tan^3\theta-\tan^4\theta+\tan\theta-1}{(\tan\theta-1).\tan\theta.(1-\tan\theta)}= \frac{(\tan^3\theta-1)(1-\tan\theta)}{\tan\theta.(\tan\theta-1)(1-\tan\theta)}\\$
By using the identity $a^3-b^3=(a-b)(a^2+ab+b^2)$
$\\\Rightarrow \frac{(\tan\theta -1)(\tan^2\theta+1+\tan\theta)}{\tan\theta(\tan\theta -1)}= \tan\theta+1+\frac{1}{\tan\theta}= 1+\frac{1+\tan^2\theta}{\tan\theta}= 1+\sec^2\theta \times \frac{1}{\tan\theta}= 1+\sec\theta.\csc\theta\\\\ =RHS$
Hence proved.
$(iv)\frac{1+\sec A}{\sec A}=\frac{\sin ^{2}A}{1-\cos A}$
[ Hint: Simplify LHS and RHS separately.]
Answer:
We need to prove-
$\frac{1+\sec A}{\sec A}=\frac{\sin ^{2}A}{1-\cos A}$
Taking LHS;
$\\\Rightarrow \frac{1+\sec A}{\sec A}= \frac{(1+\frac{1}{\cos A})}{\sec A}= 1+\cos A$
Taking RHS;
We know that identity $1-\cos^2\theta = \sin^2\theta$
$\\\Rightarrow \frac{\sin ^{2}A}{1-\cos A}= \frac{1-\cos^2 A}{1-\cos A}= \frac{(1-\cos A)(1+\cos A)}{(1-\cos A)}= 1+\cos A$
LHS = RHS
Hence proved.
Q4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. $(v) \frac{\cos A-\sin A+1}{\cos A+\sin A-1}= \csc A+\cot A$ , using the identity $\csc ^{2}A= 1+\cot ^{2}A$
Answer:
We need to prove -
$\frac{\cos A-\sin A+1}{\cos A+\sin A-1}= cosec A+\cot A$
Dividing the numerator and denominator by $\sin A$ , we get;
$\\=\frac{\cot A-1+\csc A}{\cot A +1-\csc A}\\\\= \frac{(\cot A+\csc A)-(\csc^2 A-\cot^2A)}{\cot A +1-\csc A}\\\\= \frac{(\csc A+\cot A)(1-\csc A+\cot A)}{\cot A +1-\csc A}\\\\= \csc A+\cot A\\\\ =RHS$
Hence Proved.
$(vi)\sqrt{\frac{1+\sin A}{1-\sin A}}= \sec A+\tan A$
Answer:
We need to prove -
$\sqrt{\frac{1+\sin A}{1-\sin A}}= \sec A+\tan A$
Taking LHS;
By rationalising the denominator, we get;
$\\= \sqrt{\frac{1+\sin A}{1-\sin A}\times \frac{1+\sin A}{1+\sin A}}\\\\ = \sqrt{\frac{(1+\sin A)^2}{1-\sin^2A}}\\\\ =\frac{1+\sin A}{\cos A}\\\\ = \sec A + \tan A\\\\ = RHS$
Hence proved.
Q4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. $(vii)\frac{\sin \theta -2\sin ^{3}\theta }{2\cos ^{3}\theta -\cos \theta }= \tan \theta$
Answer:
We need to prove -
$\frac{\sin \theta -2\sin ^{3}\theta }{2\cos ^{3}\theta -\cos \theta }= \tan \theta$
Taking LHS;
[we know the identity $\cos2\theta = 2\cos^2\theta-1=\cos^2\theta-\sin^2\theta$ ]
$\\\Rightarrow \frac{\sin \theta(1 -2\sin ^{2}\theta) }{\cos\theta(2\cos ^{2}\theta -1) }= \frac{\sin\theta(\sin^2\theta+\cos^2\theta-2\sin^2\theta)}{\cos\theta.\cos2\theta}= \frac{\sin\theta.\cos2\theta}{\cos\theta.\cos2\theta}= \tan\theta =RHS$
Hence proved.
$(viii)(\sin A+\csc A)^{2}+(\cos A+\sec A)^{2}= 7+\tan ^{2}A+\cot ^{2}A$
Answer:
Given the equation,
$(\sin A+\csc A)^{2}+(\cos A+\sec A)^{2}= 7+\tan ^{2}A+\cot ^{2}A$ ..................(i)
Taking LHS;
$(\sin A+\csc A)^{2}+(\cos A+\sec A)^{2}$
$\\\Rightarrow \sin^2 A+\csc^2A +2+\cos^2A+\sec^2A+2= 1+2+2+(1+\cot^2A)+(1+\tan^2A)$
[since $\sin^2\theta +\cos^2\theta = 1, \csc^2\theta-\cot^2\theta =1, \sec^2\theta-\tan^2\theta=1$ ]
$=7+\csc^2A+\tan^2A\\ =RHS$
Hence proved
$(ix)\:(cosec A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$
[ Hint: Simplify LHS and RHS separately.]
Answer:
We need to prove-
$(coesc A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$
Taking LHS;
$\\\Rightarrow (cosec A-\frac{1}{\csc A})(\sec A-\frac{1}{\sec A})=\frac{(cosec^2-1)}{cosec A}\times\frac{\sec^2A-1}{\sec A}=\frac{\cot^2A}{cosec A}.\frac{\tan^2A}{\sec A}=\sin A .\cos A$
Taking RHS;
$\\\Rightarrow\frac{1}{\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}}=\frac{\sin A .\cos A}{\sin^2A+\cos^2A}=\sin A.\cos A$
LHS = RHS
Hence proved.
$(x) (\frac{1+\tan ^{2}A}{1+\cot ^{2}A})= (\frac{1-\tan A}{1-\cot A})^{2}= \tan ^{2}A$
Answer:
We need to prove,
$(\frac{1+\tan ^{2}A}{1+\cot ^{2}A})= (\frac{1-\tan A}{1-\cot A})^{2}= \tan ^{2}A$
Taking LHS;
$\\\Rightarrow \frac{1+\tan ^{2}A}{1+\cot ^{2}A} = \frac{\sec^2A}{\csc^2A}=\tan^2A$
Taking RHS;
$\\=(\frac{1-\tan A}{1-\cot A})^2\\\\= (\frac{1-\frac{\sin A}{\cos A} }{1-\frac{\cos A}{\sin A}})^2\\\\ = \frac{(\cos A -\sin A)^2(\sin^2A)}{(\sin A-\cos A)^2(\cos^2A)}\\\\ =\tan^2A$
LHS = RHS
Hence proved.
Exercise-wise NCERT Solutions of Introduction to Trigonometry Class 10 Maths Chapter 8 are provided in the links below.
Question:
$(\sin \theta+\operatorname{cosec} \theta)^2+(\cos \theta+\sec \theta)^2=$?
Answer:
$(\sin \theta+\operatorname{cosec} \theta)^2+(\cos \theta+\sec \theta)^2$
= $(\sin^2 \theta+\operatorname{cosec}^2 \theta+2\sin \theta\operatorname{cosec} \theta)+(\cos^2 \theta+\sec^2 \theta+2\cos \theta\sec \theta)$
= $\sin^2 \theta+\cos^2 \theta+\operatorname{cosec}^2 \theta+2\sin \theta\operatorname{cosec} \theta+\sec^2 \theta+2\cos \theta\sec \theta$
= $1+1+\cot^2 \theta+2+1+\tan^2 \theta+2$
= $7+\cot^2 \theta+\tan^2 \theta$
Hence, the correct answer is $7+\tan^2 \theta+\cot^2 \theta$.
Students will explore the following topics in NCERT Class 10 Maths Chapter 8 Introduction to Trigonometry:
If an arc of length $l$ in a circle with radius $r$ subtends an angle of $\theta$ radians.
$l = r\theta$ and $θ = \frac{l}{r}$
Conversion Between Radian and Degree Measures:
The trigonometric ratios of the angle A in right triangle ABC are defined as follows-
$\begin{aligned} & \text { sine of } \angle A=\frac{\text { side oppositeto angle } A}{\text { hypotenuse }}=\frac{B C}{A C} \\ & \text { cosine of } \angle A=\frac{\text { side adjacent to angle } A}{\text { hypotenuse }}=\frac{A B}{A C} \\ & \text { tangent of } \angle A=\frac{\text { side opposite to angle } A}{\text { side adjacentto angle } A}=\frac{B C}{A B} \\ & \text { cosecant of } \angle A=\frac{\text { hypotenuse }}{\text { side opposite to angle } A}=\frac{A C}{B C} \\ & \text { secant of } \angle A=\frac{\text { hypotenuse }}{\text { side adjacent to angle } A}=\frac{A C}{A B} \\ & \text { cotangent of } \angle A=\frac{\text { side adjacent to angle } A}{\text { side opposite to angle } A}=\frac{A B}{B C}\end{aligned}$
$\angle A$ | 0° | 30° | 45° | 60° | 90° |
Sin A |
0 | $\frac{1}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{\sqrt{3}}{2}$ |
1 |
Cos A |
1 | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{2}$ |
0 |
Tan A |
0 | $\frac{1}{\sqrt{3}}$ |
1 | $\sqrt{3}$ |
Not Defined |
Cosec A |
Not Defined |
2 | $\sqrt{2}$ | $\frac{2}{\sqrt{3}}$ |
1 |
Sec A |
1 | $\frac{2}{\sqrt{3}}$ | $\sqrt{2}$ |
2 |
Not Defined |
Cot A |
Not Defined | $\sqrt{3}$ |
1 | $\frac{1}{\sqrt{3}}$ |
0 |
Also, in a right triangle with an angle $\theta$
sin θ = Opposite/Hypotenuse
cos θ = Adjacent/Hypotenuse
tan θ = Opposite/Adjacent
cosec θ = Hypotenuse/Opposite
sec θ = Hypotenuse/Adjacent
cot θ = Adjacent/Opposite
sin θ = 1/(cosec θ)
cosec θ = 1/(sin θ)
cos θ = 1/(sec θ)
sec θ = 1/(cos θ)
tan θ = 1/(cot θ)
cot θ = 1/(tan θ)
For an angle 'θ', the trigonometric ratios of its complementary angle (90° – θ) are:
sin (90° – θ) = cos θ
cos (90° – θ) = sin θ
tan (90° – θ) = cot θ
cot (90° – θ) = tan θ
sec (90° – θ) = cosec θ
cosec (90° – θ) = sec θ
sin2 θ + cos2 θ = 1
sin2 θ = 1 – cos2 θ
cos2 θ = 1 – sin2 θ
cosec2 θ – cot2 θ = 1
cot2 θ = cosec2 θ – 1
sec2 θ – tan2 θ = 1
tan2 θ = sec2 θ – 1
We at Careers360 compiled all the NCERT class 10 Maths solutions in one place for easy student reference. The following links will allow you to access them.
Also read,
NCERT Notes Class 10 Maths Chapter 8 Introduction to Trigonometry
NCERT Exemplar Class 10 Maths Solutions Chapter 8 - Introduction To Trigonometry And Its Equations
Given below are the subject-wise exemplar solutions of class 10 NCERT:
Here are some useful links for NCERT books and the NCERT syllabus for class 10:
Frequently Asked Questions (FAQs)
These NCERT Solutions for Class 10 Maths Chapter 8 contain the basic concepts of trigonometry, like trigonometric ratios, trigonometric identities, and their applications in the real world.
The easiest way to understand trigonometric ratios is by taking a right-angle triangle, where sine, cosine, and tangent represent ratios of different sides as
sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
tan θ = Opposite / Adjacent
To solve trigonometric identities in Class 10 Maths Chapter 8, use the basic trigonometric identities as sin2 θ + cos2 θ = 1 and try to express terms in sine and cosine. Also, use simplification and factorisation to solve these types of problems.
To score full marks in Class 10 Trigonometry, memorise all the important trigonometric ratios, identities, and formulas. Practice NCERT examples, exercises, and questions from the previous years. Also, regularly revise and give mock tests to improve scores.
Trigonometry is used in various fields of real life. It is mainly used for calculating the height and distance of real-life objects, and for building bridges and roads. Also, it is used in navigation systems like GPS.
On Question asked by student community
Hello,
Yes, you can give the CBSE board exam in 2027.
If your date of birth is 25.05.2013, then in 2027 you will be around 14 years old, which is the right age for Class 10 as per CBSE rules. So, there is no problem.
Hope it helps !
Hello! If you selected “None” while creating your APAAR ID and forgot to mention CBSE as your institution, it may cause issues later when linking your academic records or applying for exams and scholarships that require school details. It’s important that your APAAR ID correctly reflects your institution to avoid verification problems. You should log in to the portal and update your profile to select CBSE as your school. If the system doesn’t allow editing, contact your school’s administration or the APAAR support team immediately so they can correct it for you.
Hello Aspirant,
Here's how you can find it:
School ID Card: Your registration number is often printed on your school ID card.
Admit Card (Hall Ticket): If you've received your board exam admit card, the registration number will be prominently displayed on it. This is the most reliable place to find it for board exams.
School Records/Office: The easiest and most reliable way is to contact your school office or your class teacher. They have access to all your official records and can provide you with your registration number.
Previous Mark Sheets/Certificates: If you have any previous official documents from your school or board (like a Class 9 report card that might have a student ID or registration number that carries over), you can check those.
Your school is the best place to get this information.
Hello,
It appears you are asking if you can fill out a form after passing your 10th grade examination in the 2024-2025 academic session.
The answer depends on what form you are referring to. Some forms might be for courses or examinations where passing 10th grade is a prerequisite or an eligibility criteria, such as applying for further education or specific entrance exams. Other forms might be related to other purposes, like applying for a job, which may also have age and educational requirements.
For example, if you are looking to apply for JEE Main 2025 (a competitive exam in India), having passed class 12 or appearing for it in 2025 are mentioned as eligibility criteria.
Let me know if you need imformation about any exam eligibility criteria.
good wishes for your future!!
Hello Aspirant,
"Real papers" for CBSE board exams are the previous year's question papers . You can find these, along with sample papers and their marking schemes , on the official CBSE Academic website (cbseacademic.nic.in).
For notes , refer to NCERT textbooks as they are the primary source for CBSE exams. Many educational websites also provide chapter-wise revision notes and study material that align with the NCERT syllabus. Focus on practicing previous papers and understanding concepts thoroughly.
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