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Khuram Ali Khan
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<[email protected]>\\ \\ **Field of Research:** Difference and functional equations, Real functions, Mathema... - Khuram Ali Khan, Josip Pečarić and Ivan Perić, Differences of Weighted Mixed Symmetric Means and Related Re... Behavior of a competitive system of second-order difference equations, The Scientific World Journal, Vol. 201... en, Stability Analysis of a System of Exponential Difference Equations, Discrete Dynamics in Nature and Societ
FSc Part 1 (KPK Boards) @fsc
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ries and its sum of $n$ terms. * know method of differences and its uses. * use the partial fraction to fi... = Chapter 10: Trigonometric Identities of Sum and Difference of Angles ===== === Objectives === After reading ... xpress the product of sines and cosines as sum or differences of sines and cosines. * express the sums or differences of sines and cosines as product. ===Download=== <
Atiq ur Rehman, PhD
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rg>, <[email protected]> **Field of Research:** Difference and functional equations, Real functions, Inequal... Y. Mehboob, Mean value theorems associated to the differences of Opial–type inequalities and their fractional ... convex functions and related results, Advances in Difference Equations, 2020:163, (2020), 1-18. 46. L. N. Mis... iated results in fractional calculus, Advanced in Difference Equations, 2019:152 (2019), 1-13. 44. G. Farid,
Numerical Analysis II @notes
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| ====Summary & contents==== * Operator * Difference Equation * Homogeneous Linear Difference Equation * Non-Homogeneous Linear Difference Equation * Non-Homogeneous Difference Equation * Formation of Difference Equation * Numerical Integrati
Chapter 10: Trigonometric Identities of Sum and Difference of Angles @fsc:kpk_fsc_part_1
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= Chapter 10: Trigonometric Identities of Sum and Difference of Angles ====== Notes of Chapter 10: Trigonometric Identities of Sum and Difference of Angles of "A Textbook of Mathematics for Class... c_part_1:ch10-trigonometric-identities-of-sum-and-difference-of-angles-fsc1-kpk.pdf}} **{{:fsc:kpk_fsc_part_1:ch10-trigonometric-identities-of-sum-and-difference-of-angles-fsc1-kpk.pdf|Download PDF ~ ch10-trigon
Question 1, Exercise 10.3 @fsc-part1-kpk:sol:unit10
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3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathemat... =Question 1(i)===== Express the product as sum or difference $2\sin 6x\sin x$. ====Solution==== We have an id... Question 1(ii)===== Express the product as sum or difference $\sin {{55}^{\circ }}\cos {{123}^{\circ }}$. ====... uestion 1(iii)===== Express the product as sum or difference: $$\sin \dfrac{A+B}{2}\cos \dfrac{A-B}{2}.$$ ====
Question 2, Exercise 10.3 @fsc-part1-kpk:sol:unit10
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3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathemat... istan. =====Question 2(i)===== Convert the sum or difference as product: $$\sin {{37}^{\circ }}+\sin {{43}^{\c... $$ =====Question 2(ii)===== Convert the sum or difference as product $\cos {{36}^{\circ }}-\cos {{82}^{\cir... .$$ =====Question 2(iii)===== Convert the sum or difference as product: $$\sin \dfrac{P+Q}{2}-\sin \dfrac{P-Q
Question 1, Exercise 10.3 @math-11-kpk:sol:unit10
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3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathemat... =Question 1(i)===== Express the product as sum or difference $2\sin 6x\sin x$. ====Solution==== We have an id... Question 1(ii)===== Express the product as sum or difference $\sin {{55}^{\circ }}\cos {{123}^{\circ }}$. ====... uestion 1(iii)===== Express the product as sum or difference: $$\sin \dfrac{A+B}{2}\cos \dfrac{A-B}{2}.$$ ====
Question 2, Exercise 10.3 @math-11-kpk:sol:unit10
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3 of Unit 10: Trigonometric Identities of Sum and Difference of Angles. This is unit of A Textbook of Mathemat... istan. =====Question 2(i)===== Convert the sum or difference as product: $$\sin {{37}^{\circ }}+\sin {{43}^{\c... $$ =====Question 2(ii)===== Convert the sum or difference as product $\cos {{36}^{\circ }}-\cos {{82}^{\cir... .$$ =====Question 2(iii)===== Convert the sum or difference as product: $$\sin \dfrac{P+Q}{2}-\sin \dfrac{P-Q
Question 2(i, ii, iii, iv and v) Exercise 8.3 @math-11-nbf:sol:unit08
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tan. =====Question 2(i)===== Rewrite the sum or difference as a product of two function: $\sin 70^{\circ} + ... OOD =====Question 2(ii)===== Rewrite the sum or difference as a product of two function: $\sin 76^{\circ} - ... n*} =====Question 2(iii)===== Rewrite the sum or difference as a product of two function: $\cos 58^{\circ} + ... gn*} =====Question 2(iv)===== Rewrite the sum or difference as a product of two function: $\cos \frac{p-q}{2}
Question 1(i, ii, iii & iv) Exercise 8.3 @math-11-nbf:sol:unit08
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e the product-to-sum formula to change the sum or difference: $$4 \sin 16x \cos 10x $$ ** Solution. ** \beg... e the product-to-sum formula to change the sum or difference: $10 \cos 10y \cos 6y$. ** Solution. ** \begin{... e the product-to-sum formula to change the sum or difference: $2 \cos5t \sin 3t$. ** Solution. ** \begin{doc... e the product-to-sum formula to change the sum or difference: $6\cos 5x \sin 10x$. ** Solution. ** \begin{al
Question 1(v, vi, vii & viii) Exercise 8.3 @math-11-nbf:sol:unit08
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e the product-to-sum formula to change the sum or difference: $ \sin(-u) \sin 5u$. ** Solution. ** \begin{al... e the product-to-sum formula to change the sum or difference: $-2 \sin 100^{\circ}\sin (-20^{\circ}) $. ** So... e the product-to-sum formula to change the sum or difference: $\cos 23^{\circ} \sin 17^{\circ}$. ** Solution.... e the product-to-sum formula to change the sum or difference: $2 \cos56^{\circ} \sin48^{\circ}$. ** Solution.
Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions) @fsc-part1-kpk:sol
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==== Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions) ===== This is a tenth unit... ess the product (of sines and cosines) as sums or differences (of sines and cosines) * Express the sums or differences (of sines and cosines) as products (of sines and
Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions) @math-11-kpk:sol
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==== Unit 10: Trigonometric Identities of Sum and Difference of Angles (Solutions) ===== This is a tenth unit... ess the product (of sines and cosines) as sums or differences (of sines and cosines) * Express the sums or differences (of sines and cosines) as products (of sines and
Question 1(ix, x & xi) Exercise 8.3 @math-11-nbf:sol:unit08
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e the product-to-sum formula to change the sum or difference: $2 \sin 75{\circ} \sin 15{\circ}$. ** Solution.... e the product-to-sum formula to change the sum or difference: $4 \sin \frac{u+v}{2} \cos \frac{u-v}{2} $. ** ... e the product-to-sum formula to change the sum or difference: $2 \cos \frac{2u+2v}{2}\sin \frac{2u-2v}{2} $.
MATH-510: Topology @atiq
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FSc Part 2 (KPK Boards) @fsc
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Unit 05: Miscellaneous Series (Solutions) @math-11-kpk:sol
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Unit 08: Fundamental of Trigonometry @math-11-nbf:sol
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Question 1 Exercise 4.3 @math-11-kpk:sol:unit04
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Question 10 Exercise 4.4 @math-11-kpk:sol:unit04
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Question 9 Review Exercise @math-11-kpk:sol:unit05
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Question 20 and 21, Exercise 4.4 @math-11-nbf:sol:unit04
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Question 1 and 2, Exercise 4.8 @math-11-nbf:sol:unit04
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Question 3 and 4, Exercise 4.8 @math-11-nbf:sol:unit04
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Question 5 and 6, Exercise 4.8 @math-11-nbf:sol:unit04
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MTH321: Real Analysis 1 @atiq
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MTH321: Real Analysis I (Fall 2015) @atiq
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MTH322: Real Analysis II (Fall 2017) @atiq
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MTH321: Real Analysis I (Fall 2018) @atiq
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MTH322: Real Analysis II (Fall 2018) @atiq
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MTH321: Real Analysis I (Fall 2019) @atiq
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MTH322: Real Analysis II (Fall 2019) @atiq
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MTH322: Real Analysis II (Fall 2020) @atiq
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MTH321: Real Analysis I (Fall 2021) @atiq
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MTH322: Real Analysis II (Fall 2021) @atiq
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MTH321: Real Analysis I (Fall 2022) @atiq
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MATH 103: Number Theory @atiq
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MATH-731: Convex Analysis @atiq
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MTH321: Real Analysis 1 @atiq
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MTH321: Real Analysis 1 (Spring 2015) @atiq
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MTH322: Real Analysis II (Spring 2017) @atiq
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MTH322: Real Analysis II (Spring 2019) @atiq
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MTH321: Real Analysis I (Spring 2020) @atiq
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MTH322: Real Analysis II (Spring 2022) @atiq
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MTH321: Real Analysis I (Spring 2023) @atiq
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MTH322: Real Analysis II (Spring 2023) @atiq
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Mathematics 9 (Science Group) @matric
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Mathematics 10 (Science Group) @matric
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Advance Analysis by Ms. Iqra Liaqat @notes
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Notes for Numerical Methods by M Usman Hamid @notes
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Operation Research: Handwritten Notes @notes
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Quotes for the March @quote-of-the-day
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Review exercise @matric:9th_science
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Question 1, Exercise 10.1 @fsc-part1-kpk:sol:unit10
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Question 2, Exercise 10.1 @fsc-part1-kpk:sol:unit10
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Question 3, Exercise 10.1 @fsc-part1-kpk:sol:unit10
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Question, Exercise 10.1 @fsc-part1-kpk:sol:unit10
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Question 5, Exercise 10.1 @fsc-part1-kpk:sol:unit10
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Question 6, Exercise 10.1 @fsc-part1-kpk:sol:unit10
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Question 7, Exercise 10.1 @fsc-part1-kpk:sol:unit10
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Question 8, Exercise 10.1 @fsc-part1-kpk:sol:unit10
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Question 9 and 10, Exercise 10.1 @fsc-part1-kpk:sol:unit10
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Question11 and 12, Exercise 10.1 @fsc-part1-kpk:sol:unit10
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Question 13, Exercise 10.1 @fsc-part1-kpk:sol:unit10
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Question 1, Exercise 10.2 @fsc-part1-kpk:sol:unit10
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Question 2, Exercise 10.2 @fsc-part1-kpk:sol:unit10
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Question 3, Exercise 10.2 @fsc-part1-kpk:sol:unit10
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Question 4 and 5, Exercise 10.2 @fsc-part1-kpk:sol:unit10
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Question 6, Exercise 10.2 @fsc-part1-kpk:sol:unit10
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Question 7, Exercise 10.2 @fsc-part1-kpk:sol:unit10
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Question 8 and 9, Exercise 10.2 @fsc-part1-kpk:sol:unit10
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Question 3, Exercise 10.3 @fsc-part1-kpk:sol:unit10
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Question 5, Exercise 10.3 @fsc-part1-kpk:sol:unit10
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Question 5, Exercise 10.3 @fsc-part1-kpk:sol:unit10
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Question 1, Review Exercise 10 @fsc-part1-kpk:sol:unit10
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Question 5 and 6 Exercise 4.2 @math-11-kpk:sol:unit04
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Question 7 Exercise 4.2 @math-11-kpk:sol:unit04
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Question 7 & 8 Exercise 4.3 @math-11-kpk:sol:unit04
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Question 3 Exercise 5.3 @math-11-kpk:sol:unit05
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Question 1, Exercise 10.1 @math-11-kpk:sol:unit10
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Question 2, Exercise 10.1 @math-11-kpk:sol:unit10
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Question 3, Exercise 10.1 @math-11-kpk:sol:unit10
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Question, Exercise 10.1 @math-11-kpk:sol:unit10
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Question 5, Exercise 10.1 @math-11-kpk:sol:unit10
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Question 6, Exercise 10.1 @math-11-kpk:sol:unit10
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Question 7, Exercise 10.1 @math-11-kpk:sol:unit10
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Question 8, Exercise 10.1 @math-11-kpk:sol:unit10
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Question 9 and 10, Exercise 10.1 @math-11-kpk:sol:unit10
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Question11 and 12, Exercise 10.1 @math-11-kpk:sol:unit10
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Question 13, Exercise 10.1 @math-11-kpk:sol:unit10
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Question 1, Exercise 10.2 @math-11-kpk:sol:unit10
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Question 2, Exercise 10.2 @math-11-kpk:sol:unit10
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Question 3, Exercise 10.2 @math-11-kpk:sol:unit10
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Question 4 and 5, Exercise 10.2 @math-11-kpk:sol:unit10
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Question 6, Exercise 10.2 @math-11-kpk:sol:unit10
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Question 7, Exercise 10.2 @math-11-kpk:sol:unit10
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Question 8 and 9, Exercise 10.2 @math-11-kpk:sol:unit10
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Question 3, Exercise 10.3 @math-11-kpk:sol:unit10
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Question 5, Exercise 10.3 @math-11-kpk:sol:unit10
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Question 5, Exercise 10.3 @math-11-kpk:sol:unit10
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Question 1, Review Exercise 10 @math-11-kpk:sol:unit10
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Question 4 & 5, Review Exercise 10 @math-11-kpk:sol:unit10
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Question 6 & 7, Review Exercise 10 @math-11-kpk:sol:unit10
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Question 8 & 9, Review Exercise 10 @math-11-kpk:sol:unit10
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Question 2, Exercise 4.2 @math-11-nbf:sol:unit04
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Question 9 and 10, Exercise 4.2 @math-11-nbf:sol:unit04
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Question 1,Review Exercise @math-11-nbf:sol:unit09
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Exercise 2.6 (Solutions) @matric:9th_science:unit_02
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