Find the best tutors and institutes for Class 10 Tuition
Q1(iv):
Evaluate the following : (iv) $\frac{\sin 30^\circ + \tan 45^\circ – \text{cosec } 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ}$
Solution :
Given: The trigonometric expression $\frac{\sin 30^\circ + \tan 45^\circ - \text{cosec } 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ}$.
To Find: The numerical value of the given expression.
Step 1: Identify the values of the trigonometric ratios.
Using the standard trigonometric table for specific angles, we have:
$\sin 30^\circ = \frac{1}{2}$
$\tan 45^\circ = 1$
$\text{cosec } 60^\circ = \frac{1}{\sin 60^\circ} = \frac{1}{\sqrt{3}/2} = \frac{2}{\sqrt{3}}$
$\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\sqrt{3}/2} = \frac{2}{\sqrt{3}}$
$\cos 60^\circ = \frac{1}{2}$
$\cot 45^\circ = \frac{1}{\tan 45^\circ} = \frac{1}{1} = 1$
Step 2: Substitute the values into the expression.
Substituting the values identified in Step 1 into the given expression:
$\text{Expression} = \frac{\frac{1}{2} + 1 - \frac{2}{\sqrt{3}}}{\frac{2}{\sqrt{3}} + \frac{1}{2} + 1}$
Step 3: Simplify the numerator and the denominator.
For the numerator: $\frac{1}{2} + 1 - \frac{2}{\sqrt{3}} = \frac{3}{2} - \frac{2}{\sqrt{3}}$
Finding a common denominator ($2\sqrt{3}$):
$\frac{3\sqrt{3} - 4}{2\sqrt{3}}$
For the denominator: $\frac{2}{\sqrt{3}} + \frac{1}{2} + 1 = \frac{2}{\sqrt{3}} + \frac{3}{2}$
Finding a common denominator ($2\sqrt{3}$):
$\frac{4 + 3\sqrt{3}}{2\sqrt{3}}$
Step 4: Perform the division of the fractions.
$\text{Expression} = \frac{\frac{3\sqrt{3} - 4}{2\sqrt{3}}}{\frac{3\sqrt{3} + 4}{2\sqrt{3}}}$
Since the denominators are identical, they cancel out:
$\text{Expression} = \frac{3\sqrt{3} - 4}{3\sqrt{3} + 4}$
Step 5: Rationalize the denominator.
To rationalize, multiply the numerator and denominator by the conjugate of the denominator, which is $(3\sqrt{3} - 4)$:
$\frac{3\sqrt{3} - 4}{3\sqrt{3} + 4} \times \frac{3\sqrt{3} - 4}{3\sqrt{3} - 4} = \frac{(3\sqrt{3} - 4)^2}{(3\sqrt{3})^2 - (4)^2}$
Applying the algebraic identity $(a - b)^2 = a^2 - 2ab + b^2$ in the numerator and $a^2 - b^2$ in the denominator:
Numerator: $(3\sqrt{3})^2 - 2(3\sqrt{3})(4) + (4)^2 = (9 \times 3) - 24\sqrt{3} + 16 = 27 - 24\sqrt{3} + 16 = 43 - 24\sqrt{3}$
Denominator: $(3\sqrt{3})^2 - (4)^2 = 27 - 16 = 11$
Thus, the expression simplifies to:
$\frac{43 - 24\sqrt{3}}{11}$
Final Answer: \frac{43 - 24\sqrt{3}}{11}
More Questions from Class 10 Mathematics Introduction to Trigonometry EXERCISE 8.2
- Q1(i): Evaluate the following : (i) $\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ$
- Q1(ii): Evaluate the following : (ii) $2 \tan^2 45^\circ + \cos^2 30^\circ – \sin^2 60^\circ$
- Q1(iii): Evaluate the following : (iii) $\frac{\cos 45^\circ}{\sec 30^\circ + \text{cosec } 30^\circ}$
- Q1(v): Evaluate the following : (v) $\frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$
- Q2(i): Choose the correct option and justify your choice : (i) $\frac{2 \tan 30^\circ}{1 + \tan^2 30^\circ} =$
- Q2(ii): Choose the correct option and justify your choice : (ii) $\frac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ} =$
- Q2(iii): Choose the correct option and justify your choice : (iii) $\sin 2A = 2 \sin A$ is true when $A =$
- Q2(iv): Choose the correct option and justify your choice : (iv) $\frac{2 \tan 30^\circ}{1 - \tan^2 30^\circ} =$
- Q3: If $\tan (A + B) = \sqrt{3}$ and $\tan (A – B) = \frac{1}{\sqrt{3}}$; $0^\circ < A + B \le 90^\circ$; $A > B$, find $A$ and $B$.
- Q4(i): State whether the following are true or false. Justify your answer. (i) $\sin (A + B) = \sin A + \sin B$.
- Q4(ii): State whether the following are true or false. Justify your answer. (ii) The value of $\sin \theta$ increases as $\theta$ increases.
- Q4(iii): State whether the following are true or false. Justify your answer. (iii) The value of $\cos \theta$ increases as $\theta$ increases.
- Q4(iv): State whether the following are true or false. Justify your answer. (iv) $\sin \theta = \cos \theta$ for all values of $\theta$.
- Q4(v): State whether the following are true or false. Justify your answer. (v) $\cot A$ is not defined for $A = 0^\circ$.
CBSE Solutions for Class 10 Mathematics Introduction to Trigonometry
Chapters in CBSE - Class 10 Mathematics
Top Tutors who teach Introduction to Trigonometry
Top Tutor
Nanded, Pune
12 yrs of
Exp
1 student
1000
per hour
Online Classes
Student's Home
Tutor's home
Dedicated and passionate educator with over 10 years of experience teaching Mathematics and Science to students of Classes 8th to 12th across multiple boards — CBSE, ICSE, IGCSE, IB MYP, and IBDP. Founder of Ishaan Tuitions, Pune, specializing in personalized one-to-one coaching and small group sessions designed to strengthen conceptual understanding and academic performance. Expertise also includes preparing students for competitive examinations such as SAT and IIT-JEE Mains, with a strong record of improving analytical thinking and problem-solving abilities.
Shankar Vihar, Delhi
18 yrs of
Exp
Student's Home
We are teaching 8th, 9th,10th,11th,12th for the last 12 years and providing 95%+ in our subjects. We are giving best results in DELHI CANTT and South Delhi region.
Top Tutor
Chitragupta Nagar, Patna Sadar
5 yrs of
Exp
11 students
5094 hrs of classes completed
300
per hour
Online Classes
I am Utkarsh Kumar. As an experienced, qualified teacher and tutor with over 8 years of experience in teaching Maths, Physics, Chemistry and Biology across different boards, including CBSE and State Board. Passionate about solving mathematical problems and chemical equations. Over the years, I have helped thousands of students overcome their fear of Maths, Physics, Chemistry and Biology. If u wish to learn in an interactive and fun manner, feel free to make a contact.
Super Tutor
Shahdara Chitrakoot, Delhi
10 yrs of
Exp
74 students
4025 hrs of classes completed
500
per hour
Online Classes
With extensive experience in providing Class 10 tuition, I specialize in teaching mathematics, science, and computer science. My approach emphasizes a thorough understanding of key concepts and practical applications. In mathematics, I cover topics such as algebra, geometry, and trigonometry. My science classes include physics, chemistry, and biology, focusing on fundamental principles and real-world relevance. In computer science, I teach programming basics, algorithms, and data management. I use interactive tools, clear presentations, and regular assessments to ensure comprehensive learning. My goal is to build strong foundational knowledge, boost confidence, and prepare students for their board exams and future academic pursuits.
I’m really glad I found Rahul sir through UrbanPro. He has been extremely helpful with my Computer studies and has made understanding different concepts so much easier for me. He is very patient while explaining things and always makes sure that I actually understand the topic instead of just memorising it. I started taking his help after my Unit Tests, and it has genuinely made a big difference. With my half-yearlies coming up, I feel much more prepared and confident in Computer, and a lot of that is because of his guidance and support. He has helped me improve so much and has always been really supportive whenever I’ve had doubts. I’m really thankful for all his help and would definitely recommend him to anyone looking for a Computer tutor!
Top Tutor
TT Nagar, Bhopal
5 yrs of
Exp
11 students
608 hrs of classes completed
350
per hour
Online Classes
Teaching students is just not my profession, this is my passion to give student the best that I could and bring them on the pathway of excellence. Previously, before the pandemic I have been teaching from 3 years back and after the pandemic I am teaching online on this platform. In general , I taught maths,science and english subject. I do adopt the easy and interesting path to teach which makes students curious , attentive and hard working. They enjoyed teaching and th classes became more comfortable.
I am truly grateful for Bhavana tutoring services. Her patient guidance and ability to explain complex concepts clearly have significantly improved my daughter's academic performance in Maths and Science.Her dedication and flexibility have made a positive impact on my daughter's learning experience. I wholeheartedly recommend her to any student seeking a knowledgeable and supportive tutor.
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
- Ghaziabad Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
Download free CBSE - Class 10 Mathematics Introduction to Trigonometry EXERCISE 8.2 worksheets
Download Now
