Find the best tutors and institutes for Class 10 Tuition
Q1:
Express the trigonometric ratios $\sin A$, $\sec A$ and $\tan A$ in terms of $\cot A$.
Solution :
Given: The trigonometric ratios $\sin A$, $\sec A$, and $\tan A$.
To Find: Express $\sin A$, $\sec A$, and $\tan A$ in terms of $\cot A$.
Step 1: Expressing $\tan A$ in terms of $\cot A$
By the definition of reciprocal trigonometric identities, we know that the tangent ratio is the reciprocal of the cotangent ratio.
Formula: $\tan A = \frac{1}{\cot A}$
Thus, $\tan A = \frac{1}{\cot A}$.
Step 2: Expressing $\sin A$ in terms of $\cot A$
We use the trigonometric identity relating $\csc A$ and $\cot A$: $1 + \cot^2 A = \csc^2 A$.
Taking the square root on both sides: $\csc A = \sqrt{1 + \cot^2 A}$.
Since $\sin A = \frac{1}{\csc A}$ [Reciprocal identity], we substitute the expression for $\csc A$:
$\sin A = \frac{1}{\sqrt{1 + \cot^2 A}}$
Step 3: Expressing $\sec A$ in terms of $\cot A$
We use the trigonometric identity relating $\sec A$ and $\tan A$: $1 + \tan^2 A = \sec^2 A$.
Substitute the expression for $\tan A$ found in Step 1 ($\tan A = \frac{1}{\cot A}$):
$\sec^2 A = 1 + \left(\frac{1}{\cot A}\right)^2$
$\sec^2 A = 1 + \frac{1}{\cot^2 A}$
Find a common denominator:
$\sec^2 A = \frac{\cot^2 A + 1}{\cot^2 A}$
Taking the square root on both sides:
$\sec A = \sqrt{\frac{\cot^2 A + 1}{\cot^2 A}}$
$\sec A = \frac{\sqrt{1 + \cot^2 A}}{\cot A}$
Final Answer:
$\sin A = \frac{1}{\sqrt{1 + \cot^2 A}}$
$\sec A = \frac{\sqrt{1 + \cot^2 A}}{\cot A}$
$\tan A = \frac{1}{\cot A}$
More Questions from Class 10 Mathematics Introduction to Trigonometry EXERCISE 8.3
- Q2: Write all the other trigonometric ratios of $\angle A$ in terms of $\sec A$.
- Q3(i): Choose the correct option. Justify your choice. (i) $9 \sec^2 A – 9 \tan^2 A =$
- Q3(ii): Choose the correct option. Justify your choice. (ii) $(1 + \tan \theta + \sec \theta) (1 + \cot \theta – \text{cosec } \theta) =$
- Q3(iii): Choose the correct option. Justify your choice. (iii) $(\sec A + \tan A) (1 – \sin A) =$
- Q3(iv): Choose the correct option. Justify your choice. (iv) $\frac{1 + \tan^2 A}{1 + \cot^2 A} =$
- Q4(i): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (i) $(\text{cosec } \theta – \cot \theta)^2 = \frac{1 - \cos \theta}{1 + \cos \theta}$
- Q4(ii): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (ii) $\frac{\cos A}{1 + \sin A} + \frac{1 + \sin A}{\cos A} = 2 \sec A$
- Q4(iii): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (iii) $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \sec \theta \text{cosec } \theta$ [Hint : Write the expression in terms of $\sin \theta$ and $\cos \theta$]
- Q4(iv): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (iv) $\frac{1 + \sec A}{\sec A} = \frac{\sin^2 A}{1 – \cos A}$ [Hint : Simplify LHS and RHS separately]
- Q4(ix): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (ix) $(\text{cosec } A – \sin A) (\sec A – \cos A) = \frac{1}{\tan A + \cot A}$ [Hint : Simplify LHS and RHS separately]
- Q4(v): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (v) $\frac{\cos A – \sin A + 1}{\cos A + \sin A – 1} = \text{cosec } A + \cot A$, using the identity $\text{cosec}^2 A = 1 + \cot^2 A$.
- Q4(vi): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (vi) $\sqrt{\frac{1 + \sin A}{1 – \sin A}} = \sec A + \tan A$
- Q4(vii): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (vii) $\frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} = \tan \theta$
- Q4(viii): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (viii) $(\sin A + \text{cosec } A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$
- Q4(x): Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (x) $(\frac{1 + \tan^2 A}{1 + \cot^2 A}) = (\frac{1 - \tan A}{1 - \cot A})^2 = \tan^2 A$
CBSE Solutions for Class 10 Mathematics Introduction to Trigonometry
Chapters in CBSE - Class 10 Mathematics
Top Tutors who teach Introduction to Trigonometry
Top Tutor
Bataganj, Patna Sadar
2 yrs of
Exp
100
per hour
Online Classes
I am an experienced Spoken English and Class VI-X subject tutor with two years of experience in teaching. Currently, I am pursuing my bachelor's degree in BA from Magadh University. I can teach students who are in State, International Baccalaureate, CBSE and ICSE Board. My students have improved a lot through my teaching, and I've received five reviews till now with 100% positive feedback. I always try to understand my students and help them overcome their fear of failure in specific subjects like math and science.
Top Tutor
Ashok Vihar, Delhi
10 yrs of
Exp
2 students
121 hrs of classes completed
380
per hour
Online Classes
Student's Home
Being a professional tutor , I have an experience of more than 9 years , my students are my strength , i always dedicated for my students.
A+ tutor for Maths and Science class 10. Highly recommended, very humble and polite. You should go for him without doubts.
Top Tutor
Kharodi West, Mumbai
10 yrs of
Exp
1 student
700
per hour
Student's Home
An accommodating and versatile individual with the talent to develop inspiring hands-on lessons that will capture a child's imagination and breed success.
She’s the sweetest and the most lovable teacher i have ever met. The subject never felt tough with her.
Top Tutor
Ashok Nagar D' Souza Layout, Bangalore
6 yrs of
Exp
13 students
818 hrs of classes completed
400
per hour
Online Classes
I am a teacher , a statistician and a data analyst. I am willing to give online tution to uh in subject maths and statistics. I am a first ranker in my university in MSc (Statistics) and a gold medalist. I am certified in O level. I have cleared written exam for the post statistical officer in 2022 conducted by UPSC. Also cleared exam of MES but could not make in the interview. My key skills are teaching, solving queries, understanding students, visualising and analysing data. I have 6 years experience in teaching and i was a junior consultant having 4 years of experience in Ministry of labour and employment,Shimla . Currently i am working as a Monitoring Assistant in Department of Fisheries, Ministry of Fisheries,Animal Husbandry and Dairying, New Delhi. I know languages like R, python etc.
Top Tutor
Mahipalpur, Delhi
14 yrs of
Exp
Student's Home
As an experienced teacher for nine years, I have been teaching to make students interested in the subjects of Maths. My attitude is to help children improve in studies in the best possible way.
He is an excellent tutor who is very knowledgeable and conveys her knowledge of complex material in a manner easily understandable.
Find more Tutor for Introduction to Trigonometry in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
Download free CBSE - Class 10 Mathematics Introduction to Trigonometry EXERCISE 8.3 worksheets
Download Now
