This document provides steps to solve a trigonometric identity involving logarithms. It breaks the problem down into solving the logarithm and proving the resulting identity. To solve the logarithm, it simplifies the base, base of the argument, and sets the logarithm equal to the exponent of the argument. This reduces the identity. The final steps prove the identity by separately solving each side of the reduced identity using trigonometric identities and properties.
- QTPIE has become faster and researchers now understand why previously calculated dipole moments and polarizabilities were not translationally invariant or size extensive.
- Researchers have optimized Gaussian type orbitals to closely match Slater type orbitals, resulting in very little error (<0.00001e) in calculations using QTPIE.
- Using sparse matrix data structures and conjugate gradients optimization allows QTPIE to solve problems more efficiently in both memory usage and speed compared to conventional methods.
This document compares JPEG and JPEG2000 image compression techniques using objective and perceptual quality measures. JPEG2000 provides higher PSNR values at all bitrates but JPEG has better picture quality scale (PQS) scores, a perceptual measure, at moderate and high bitrates. At very low bitrates below 0.5 bpp, JPEG2000 produces higher quality images according to PQS due to its wavelet-based compression method. The study uses four test images with different spatial and frequency characteristics to evaluate the compression methods.
The document outlines an approach to summarize stability margins for multivariable feedback systems. It begins by introducing the problem of defining meaningful stability margins for multivariable systems. Next, it proposes using a PID controller of the form K(s) = K1 + K2/s + K3s with scalar values for each term. The problem is then defined as finding the ranges of these scalar values that ensure closed-loop stability. Finally, it proposes definitions for common and individual loop gain margins based on the stabilizing ranges of the scalar values. The approach aims to generalize stability margin concepts from single-input single-output systems to multivariable systems.
The document discusses a mountainous highway in Bolivia called Stremnaya Road that is nicknamed "the road of death". It is a dangerous road situated in Bolivia that winds through mountainous terrain. Driving on this road is risky due to its steep cliffs and lack of safety features.
D = f(x), the original parent function
B = f'(x), the first derivative of D
A = f''(x), the second derivative of D
This determination is made based on the properties that the derivative is positive when the slope of the parent function is positive, the second derivative is positive when the parent function is concave up, and the second derivative is the derivative of the first derivative. Graphical analysis of the relationships between the graphs of A, B, C and D support this determination.
- QTPIE has become faster and researchers now understand why previously calculated dipole moments and polarizabilities were not translationally invariant or size extensive.
- Researchers have optimized Gaussian type orbitals to closely match Slater type orbitals, resulting in very little error (<0.00001e) in calculations using QTPIE.
- Using sparse matrix data structures and conjugate gradients optimization allows QTPIE to solve problems more efficiently in both memory usage and speed compared to conventional methods.
This document compares JPEG and JPEG2000 image compression techniques using objective and perceptual quality measures. JPEG2000 provides higher PSNR values at all bitrates but JPEG has better picture quality scale (PQS) scores, a perceptual measure, at moderate and high bitrates. At very low bitrates below 0.5 bpp, JPEG2000 produces higher quality images according to PQS due to its wavelet-based compression method. The study uses four test images with different spatial and frequency characteristics to evaluate the compression methods.
The document outlines an approach to summarize stability margins for multivariable feedback systems. It begins by introducing the problem of defining meaningful stability margins for multivariable systems. Next, it proposes using a PID controller of the form K(s) = K1 + K2/s + K3s with scalar values for each term. The problem is then defined as finding the ranges of these scalar values that ensure closed-loop stability. Finally, it proposes definitions for common and individual loop gain margins based on the stabilizing ranges of the scalar values. The approach aims to generalize stability margin concepts from single-input single-output systems to multivariable systems.
The document discusses a mountainous highway in Bolivia called Stremnaya Road that is nicknamed "the road of death". It is a dangerous road situated in Bolivia that winds through mountainous terrain. Driving on this road is risky due to its steep cliffs and lack of safety features.
D = f(x), the original parent function
B = f'(x), the first derivative of D
A = f''(x), the second derivative of D
This determination is made based on the properties that the derivative is positive when the slope of the parent function is positive, the second derivative is positive when the parent function is concave up, and the second derivative is the derivative of the first derivative. Graphical analysis of the relationships between the graphs of A, B, C and D support this determination.
Este documento presenta una serie de operaciones matemáticas básicas como suma, resta, división y multiplicación para enseñar álgebra, pero al final contiene un engaño con un mensaje ofensivo que insta al receptor a enviarlo a otros para vengarse.
This document summarizes a calculus lesson on implicit differentiation. The lesson began with examples of implied relationships in pictures and led into a discussion of implicit functions hidden within explicit functions. The class then learned how to take the derivative of an implicit function using the example of finding the derivative of any point on a circle. The process involves rewriting the equation in terms of y, applying the chain rule, and solving for the derivative of y. The lesson concluded by emphasizing that the derivative is zero when the numerator is zero, indicating a horizontal tangent line, and when the denominator is zero, indicating a vertical tangent line.
The document outlines several ministries of Called 2 Ministry.org, including ministries focused on: women aiming to exemplify Proverbs 31 (For Women Only); training godly men to lead their households (For Da Fellaz); guiding young couples to build godly relationships (Couples Ministry); serving young families (Youth Family Life Ministry); spreading the gospel through music (Ps. 150 - Music Ministry); volunteer outreach opportunities (Each One, Reach One!); providing recipes (Kareem's Kuisine); and health and exercise (Fitness Corner). The ministries aim to train and guide people in different stages of life and relationships to serve God.
The document contains graphs that are divided into four sections of a full period. The graphs show lines that increase at quarter period intervals and contain minimum values at those intervals, with the last section showing a minimum value of 151.
The document discusses the transition to a global digital network era driven by advances in information technology. Key developments include the rise of social media platforms with hundreds of millions of users, the transformation of industries like automobiles which now have computers and connectivity, and the evolution of the music business from physical to digital formats. It argues that businesses must shift from vertical integration to virtual integration based on relationships across networks to succeed in this new environment.
This document summarizes a presentation given by Professor N. Venkatraman on business innovation and transformation opportunities in the media sector. It discusses how emerging technologies like increased bandwidth, connectivity through platforms like Facebook, and mobile access are shifting products, processes and services. It introduces concepts like Moore's Law, Metcalfe's Law and the Bandwidth Law to explain the technology trends driving this change and outlines the challenges of aligning business and technology strategies during times of innovation and transformation.
This document discusses how digital technologies are transforming business through five interconnected webs: the mobile web, social web, media web, real-time web, and machine web. It provides examples of how each web is impacting industries and consumer behavior. The presentation raises questions for business leaders about how these shifts are affecting their companies and whether their CIOs are involved in the strategic conversations around digital transformation.
The document shows the step-by-step work to solve a multi-step math problem about riders on a ferris wheel. It first finds that Francis and Kieran will get off the ride after 14 minutes, when Wilfred and John are 117 meters high. It then determines it will take another 25 - 14 = 11 more minutes for Wilfred and John to reach the same height of 117 meters again.
- Video recording of this lecture in English language: https://youtu.be/kqbnxVAZs-0
- Video recording of this lecture in Arabic language: https://youtu.be/SINlygW1Mpc
- Link to download the book free: https://nephrotube.blogspot.com/p/nephrotube-nephrology-books.html
- Link to NephroTube website: www.NephroTube.com
- Link to NephroTube social media accounts: https://nephrotube.blogspot.com/p/join-nephrotube-on-social-media.html
Integrating Ayurveda into Parkinson’s Management: A Holistic ApproachAyurveda ForAll
Explore the benefits of combining Ayurveda with conventional Parkinson's treatments. Learn how a holistic approach can manage symptoms, enhance well-being, and balance body energies. Discover the steps to safely integrate Ayurvedic practices into your Parkinson’s care plan, including expert guidance on diet, herbal remedies, and lifestyle modifications.
Local Advanced Lung Cancer: Artificial Intelligence, Synergetics, Complex Sys...Oleg Kshivets
Overall life span (LS) was 1671.7±1721.6 days and cumulative 5YS reached 62.4%, 10 years – 50.4%, 20 years – 44.6%. 94 LCP lived more than 5 years without cancer (LS=2958.6±1723.6 days), 22 – more than 10 years (LS=5571±1841.8 days). 67 LCP died because of LC (LS=471.9±344 days). AT significantly improved 5YS (68% vs. 53.7%) (P=0.028 by log-rank test). Cox modeling displayed that 5YS of LCP significantly depended on: N0-N12, T3-4, blood cell circuit, cell ratio factors (ratio between cancer cells-CC and blood cells subpopulations), LC cell dynamics, recalcification time, heparin tolerance, prothrombin index, protein, AT, procedure type (P=0.000-0.031). Neural networks, genetic algorithm selection and bootstrap simulation revealed relationships between 5YS and N0-12 (rank=1), thrombocytes/CC (rank=2), segmented neutrophils/CC (3), eosinophils/CC (4), erythrocytes/CC (5), healthy cells/CC (6), lymphocytes/CC (7), stick neutrophils/CC (8), leucocytes/CC (9), monocytes/CC (10). Correct prediction of 5YS was 100% by neural networks computing (error=0.000; area under ROC curve=1.0).
These lecture slides, by Dr Sidra Arshad, offer a simplified look into the mechanisms involved in the regulation of respiration:
Learning objectives:
1. Describe the organisation of respiratory center
2. Describe the nervous control of inspiration and respiratory rhythm
3. Describe the functions of the dorsal and respiratory groups of neurons
4. Describe the influences of the Pneumotaxic and Apneustic centers
5. Explain the role of Hering-Breur inflation reflex in regulation of inspiration
6. Explain the role of central chemoreceptors in regulation of respiration
7. Explain the role of peripheral chemoreceptors in regulation of respiration
8. Explain the regulation of respiration during exercise
9. Integrate the respiratory regulatory mechanisms
10. Describe the Cheyne-Stokes breathing
Study Resources:
1. Chapter 42, Guyton and Hall Textbook of Medical Physiology, 14th edition
2. Chapter 36, Ganong’s Review of Medical Physiology, 26th edition
3. Chapter 13, Human Physiology by Lauralee Sherwood, 9th edition
Adhd Medication Shortage Uk - trinexpharmacy.comreignlana06
The UK is currently facing a Adhd Medication Shortage Uk, which has left many patients and their families grappling with uncertainty and frustration. ADHD, or Attention Deficit Hyperactivity Disorder, is a chronic condition that requires consistent medication to manage effectively. This shortage has highlighted the critical role these medications play in the daily lives of those affected by ADHD. Contact : +1 (747) 209 – 3649 E-mail : sales@trinexpharmacy.com
TEST BANK For Community Health Nursing A Canadian Perspective, 5th Edition by...Donc Test
TEST BANK For Community Health Nursing A Canadian Perspective, 5th Edition by Stamler, Verified Chapters 1 - 33, Complete Newest Version Community Health Nursing A Canadian Perspective, 5th Edition by Stamler, Verified Chapters 1 - 33, Complete Newest Version Community Health Nursing A Canadian Perspective, 5th Edition by Stamler Community Health Nursing A Canadian Perspective, 5th Edition TEST BANK by Stamler Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Pdf Chapters Download Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Pdf Download Stuvia Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Study Guide Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Ebook Download Stuvia Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Questions and Answers Quizlet Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Studocu Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Quizlet Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Stuvia Community Health Nursing A Canadian Perspective, 5th Edition Pdf Chapters Download Community Health Nursing A Canadian Perspective, 5th Edition Pdf Download Course Hero Community Health Nursing A Canadian Perspective, 5th Edition Answers Quizlet Community Health Nursing A Canadian Perspective, 5th Edition Ebook Download Course hero Community Health Nursing A Canadian Perspective, 5th Edition Questions and Answers Community Health Nursing A Canadian Perspective, 5th Edition Studocu Community Health Nursing A Canadian Perspective, 5th Edition Quizlet Community Health Nursing A Canadian Perspective, 5th Edition Stuvia Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Pdf Chapters Download Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Pdf Download Stuvia Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Study Guide Questions and Answers Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Ebook Download Stuvia Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Questions Quizlet Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Studocu Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Quizlet Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Stuvia
Este documento presenta una serie de operaciones matemáticas básicas como suma, resta, división y multiplicación para enseñar álgebra, pero al final contiene un engaño con un mensaje ofensivo que insta al receptor a enviarlo a otros para vengarse.
This document summarizes a calculus lesson on implicit differentiation. The lesson began with examples of implied relationships in pictures and led into a discussion of implicit functions hidden within explicit functions. The class then learned how to take the derivative of an implicit function using the example of finding the derivative of any point on a circle. The process involves rewriting the equation in terms of y, applying the chain rule, and solving for the derivative of y. The lesson concluded by emphasizing that the derivative is zero when the numerator is zero, indicating a horizontal tangent line, and when the denominator is zero, indicating a vertical tangent line.
The document outlines several ministries of Called 2 Ministry.org, including ministries focused on: women aiming to exemplify Proverbs 31 (For Women Only); training godly men to lead their households (For Da Fellaz); guiding young couples to build godly relationships (Couples Ministry); serving young families (Youth Family Life Ministry); spreading the gospel through music (Ps. 150 - Music Ministry); volunteer outreach opportunities (Each One, Reach One!); providing recipes (Kareem's Kuisine); and health and exercise (Fitness Corner). The ministries aim to train and guide people in different stages of life and relationships to serve God.
The document contains graphs that are divided into four sections of a full period. The graphs show lines that increase at quarter period intervals and contain minimum values at those intervals, with the last section showing a minimum value of 151.
The document discusses the transition to a global digital network era driven by advances in information technology. Key developments include the rise of social media platforms with hundreds of millions of users, the transformation of industries like automobiles which now have computers and connectivity, and the evolution of the music business from physical to digital formats. It argues that businesses must shift from vertical integration to virtual integration based on relationships across networks to succeed in this new environment.
This document summarizes a presentation given by Professor N. Venkatraman on business innovation and transformation opportunities in the media sector. It discusses how emerging technologies like increased bandwidth, connectivity through platforms like Facebook, and mobile access are shifting products, processes and services. It introduces concepts like Moore's Law, Metcalfe's Law and the Bandwidth Law to explain the technology trends driving this change and outlines the challenges of aligning business and technology strategies during times of innovation and transformation.
This document discusses how digital technologies are transforming business through five interconnected webs: the mobile web, social web, media web, real-time web, and machine web. It provides examples of how each web is impacting industries and consumer behavior. The presentation raises questions for business leaders about how these shifts are affecting their companies and whether their CIOs are involved in the strategic conversations around digital transformation.
The document shows the step-by-step work to solve a multi-step math problem about riders on a ferris wheel. It first finds that Francis and Kieran will get off the ride after 14 minutes, when Wilfred and John are 117 meters high. It then determines it will take another 25 - 14 = 11 more minutes for Wilfred and John to reach the same height of 117 meters again.
- Video recording of this lecture in English language: https://youtu.be/kqbnxVAZs-0
- Video recording of this lecture in Arabic language: https://youtu.be/SINlygW1Mpc
- Link to download the book free: https://nephrotube.blogspot.com/p/nephrotube-nephrology-books.html
- Link to NephroTube website: www.NephroTube.com
- Link to NephroTube social media accounts: https://nephrotube.blogspot.com/p/join-nephrotube-on-social-media.html
Integrating Ayurveda into Parkinson’s Management: A Holistic ApproachAyurveda ForAll
Explore the benefits of combining Ayurveda with conventional Parkinson's treatments. Learn how a holistic approach can manage symptoms, enhance well-being, and balance body energies. Discover the steps to safely integrate Ayurvedic practices into your Parkinson’s care plan, including expert guidance on diet, herbal remedies, and lifestyle modifications.
Local Advanced Lung Cancer: Artificial Intelligence, Synergetics, Complex Sys...Oleg Kshivets
Overall life span (LS) was 1671.7±1721.6 days and cumulative 5YS reached 62.4%, 10 years – 50.4%, 20 years – 44.6%. 94 LCP lived more than 5 years without cancer (LS=2958.6±1723.6 days), 22 – more than 10 years (LS=5571±1841.8 days). 67 LCP died because of LC (LS=471.9±344 days). AT significantly improved 5YS (68% vs. 53.7%) (P=0.028 by log-rank test). Cox modeling displayed that 5YS of LCP significantly depended on: N0-N12, T3-4, blood cell circuit, cell ratio factors (ratio between cancer cells-CC and blood cells subpopulations), LC cell dynamics, recalcification time, heparin tolerance, prothrombin index, protein, AT, procedure type (P=0.000-0.031). Neural networks, genetic algorithm selection and bootstrap simulation revealed relationships between 5YS and N0-12 (rank=1), thrombocytes/CC (rank=2), segmented neutrophils/CC (3), eosinophils/CC (4), erythrocytes/CC (5), healthy cells/CC (6), lymphocytes/CC (7), stick neutrophils/CC (8), leucocytes/CC (9), monocytes/CC (10). Correct prediction of 5YS was 100% by neural networks computing (error=0.000; area under ROC curve=1.0).
These lecture slides, by Dr Sidra Arshad, offer a simplified look into the mechanisms involved in the regulation of respiration:
Learning objectives:
1. Describe the organisation of respiratory center
2. Describe the nervous control of inspiration and respiratory rhythm
3. Describe the functions of the dorsal and respiratory groups of neurons
4. Describe the influences of the Pneumotaxic and Apneustic centers
5. Explain the role of Hering-Breur inflation reflex in regulation of inspiration
6. Explain the role of central chemoreceptors in regulation of respiration
7. Explain the role of peripheral chemoreceptors in regulation of respiration
8. Explain the regulation of respiration during exercise
9. Integrate the respiratory regulatory mechanisms
10. Describe the Cheyne-Stokes breathing
Study Resources:
1. Chapter 42, Guyton and Hall Textbook of Medical Physiology, 14th edition
2. Chapter 36, Ganong’s Review of Medical Physiology, 26th edition
3. Chapter 13, Human Physiology by Lauralee Sherwood, 9th edition
Adhd Medication Shortage Uk - trinexpharmacy.comreignlana06
The UK is currently facing a Adhd Medication Shortage Uk, which has left many patients and their families grappling with uncertainty and frustration. ADHD, or Attention Deficit Hyperactivity Disorder, is a chronic condition that requires consistent medication to manage effectively. This shortage has highlighted the critical role these medications play in the daily lives of those affected by ADHD. Contact : +1 (747) 209 – 3649 E-mail : sales@trinexpharmacy.com
TEST BANK For Community Health Nursing A Canadian Perspective, 5th Edition by...Donc Test
TEST BANK For Community Health Nursing A Canadian Perspective, 5th Edition by Stamler, Verified Chapters 1 - 33, Complete Newest Version Community Health Nursing A Canadian Perspective, 5th Edition by Stamler, Verified Chapters 1 - 33, Complete Newest Version Community Health Nursing A Canadian Perspective, 5th Edition by Stamler Community Health Nursing A Canadian Perspective, 5th Edition TEST BANK by Stamler Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Pdf Chapters Download Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Pdf Download Stuvia Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Study Guide Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Ebook Download Stuvia Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Questions and Answers Quizlet Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Studocu Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Quizlet Test Bank For Community Health Nursing A Canadian Perspective, 5th Edition Stuvia Community Health Nursing A Canadian Perspective, 5th Edition Pdf Chapters Download Community Health Nursing A Canadian Perspective, 5th Edition Pdf Download Course Hero Community Health Nursing A Canadian Perspective, 5th Edition Answers Quizlet Community Health Nursing A Canadian Perspective, 5th Edition Ebook Download Course hero Community Health Nursing A Canadian Perspective, 5th Edition Questions and Answers Community Health Nursing A Canadian Perspective, 5th Edition Studocu Community Health Nursing A Canadian Perspective, 5th Edition Quizlet Community Health Nursing A Canadian Perspective, 5th Edition Stuvia Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Pdf Chapters Download Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Pdf Download Stuvia Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Study Guide Questions and Answers Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Ebook Download Stuvia Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Questions Quizlet Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Studocu Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Quizlet Community Health Nursing A Canadian Perspective, 5th Edition Test Bank Stuvia
Cell Therapy Expansion and Challenges in Autoimmune DiseaseHealth Advances
There is increasing confidence that cell therapies will soon play a role in the treatment of autoimmune disorders, but the extent of this impact remains to be seen. Early readouts on autologous CAR-Ts in lupus are encouraging, but manufacturing and cost limitations are likely to restrict access to highly refractory patients. Allogeneic CAR-Ts have the potential to broaden access to earlier lines of treatment due to their inherent cost benefits, however they will need to demonstrate comparable or improved efficacy to established modalities.
In addition to infrastructure and capacity constraints, CAR-Ts face a very different risk-benefit dynamic in autoimmune compared to oncology, highlighting the need for tolerable therapies with low adverse event risk. CAR-NK and Treg-based therapies are also being developed in certain autoimmune disorders and may demonstrate favorable safety profiles. Several novel non-cell therapies such as bispecific antibodies, nanobodies, and RNAi drugs, may also offer future alternative competitive solutions with variable value propositions.
Widespread adoption of cell therapies will not only require strong efficacy and safety data, but also adapted pricing and access strategies. At oncology-based price points, CAR-Ts are unlikely to achieve broad market access in autoimmune disorders, with eligible patient populations that are potentially orders of magnitude greater than the number of currently addressable cancer patients. Developers have made strides towards reducing cell therapy COGS while improving manufacturing efficiency, but payors will inevitably restrict access until more sustainable pricing is achieved.
Despite these headwinds, industry leaders and investors remain confident that cell therapies are poised to address significant unmet need in patients suffering from autoimmune disorders. However, the extent of this impact on the treatment landscape remains to be seen, as the industry rapidly approaches an inflection point.
Osteoporosis - Definition , Evaluation and Management .pdfJim Jacob Roy
Osteoporosis is an increasing cause of morbidity among the elderly.
In this document , a brief outline of osteoporosis is given , including the risk factors of osteoporosis fractures , the indications for testing bone mineral density and the management of osteoporosis
1. 2
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2 log 2 16
(1+ cot θ) 4 sin θ
At first glance, this problem can look a little bit complicated
and intimidating. A good way to make it easier to solve is to
break it down:
Step 1- Solve the logarithm.
Step 2- Prove the resulting identity.
4. 1
1
1 (sec 2 θ −tan 2 θ −cos2 θ )
2 2 2
(sec θ − tan θ − cos θ)
log[1−(1−sin 2 θθ)] 2
2
[1− (1− sin )]
(1+ cot θ)
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€
Step 1- Solve the logartithm.
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There are 3 parts to this logarithm:
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Part A. Base
Part B. Base of Argument
Part C. Exponent of Argument
5. 2
[1−(1−sin θ )]
Step 1- Solve the logartithm.
Part A. Simplify the Base.
6. 2
[1−(1−sin θ )]
Method 2
Method 1
Step 1- Solve the logartithm. Part A. Simplify the Base
There are two methods that can be used to get the simplified
version of this expression:
Method 1
Method 2
7. 2
[1−(1−sin θ )]
Method 1
2
1− (1− sin θ)
Step 1- Solve the logartithm. Part A. Simplify the Base
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8. 2
[1−(1−sin θ )]
Method 1
2
1−1+ sin θ
Step 1- Solve the logartithm. Part A. Simplify the Base
Method 1: Expand the original expression.
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9. 2
[1−(1−sin θ )]
Method 1
2
sin θ
Step 1- Solve the logartithm. Part A. Simplify the Base
Method 1: Simplify the resulting expression
10. 2
[1−(1−sin θ )]
Method 2
2
1− (1− sin θ)
Step 1- Solve the logartithm. Part A. Simplify the Base
€
11. 2
[1−(1−sin θ )]
Method 2
2
1− (cos θ)
Step 1- Solve the logartithm. Part A. Simplify the Base
sin2 θ + cos2 θ =1 so that
Method 2: Rearrange the identity
cos 2 θ = 1− sin2 θ and use it to simplify the expression.
€ €
12. 2
[1−(1−sin θ )]
Method 2
2
sin θ
Step 1- Solve the logartithm. Part A. Simplify the Base
Method 2: Simplify the resulting expression
13. 2
[1−(1−sin θ )]
2
2
[1− (1− sin θ)] = sin θ
Step 1- Solve the logartithm. Part A. Simplify the Base
2
sin θ
Therefore, the Base is equal to
€ €
14. 1
2
(1+ cot θ)
Step 1- Solve the logartithm.
Part B. Simplify the Base of Argument.
15. 1
2
(1+ cot θ)
Method 2
Method 1
€
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
There are two methods that can be used to get the simplified
version of this expression:
Method 1
Method 2
16. 1
2
(1+ cot θ)
Method 1
1
2
(1+ cot θ)
€
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
17. 1
2
(1+ cot θ)
Method 1
1
2 2
sin θ cos θ
+
€ 2 2
sin θ sin θ
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
cos2 θ
2
Method 1: Recognize that cot θ is the same thing as and sin2 θ
expand it as such. Also, rewrite “1” so it has the same LCD.
€
€ €
18. 1
2
(1+ cot θ)
Method 1
1
2 2
sin θ + cos θ
€ 2
sin θ
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
Method 1: Add the two fractions together to get one fraction.
€
19. 1
2
(1+ cot θ)
Method 1
1
1
€ 2
sin θ
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
sin2 θ + cos2 θ =1)
Method 1: Refer to the Pythagorean identity (
and use it to simplify the numerator of the
fraction in the denominator.
€
€
20. 1
2
(1+ cot θ)
Method 1
2
sin θ
€
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
Method 1: Multiply “1” by the reciprocal of the fraction in the
2
denominator ( sin θ ).
€
21. 1
2
(1+ cot θ)
Method 2
1
2
(1+ cot θ)
€
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
22. 1
2
(1+ cot θ)
Method 2
1
2
csc θ
€
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
Method 2: Recognize that the Pythagorean Identity
csc2 θ − cot 2 θ = 1 applies to the denominator and
simplify it.
€
23. 1
2
(1+ cot θ)
Method 2
1
1
€ 2
sin θ
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
csc2 θ is eqivalent to the reciprocal
Method 2: Recognize that
2
of sin θ ( sin θ ) and rewrite it as such.
1
2
€
€
€€
24. 1
2
(1+ cot θ)
Method 2
2
sin θ
€
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
Method 2: Multiply “1” by the reciprocal of the fraction in the
2
denominator ( sin θ ).
€
25. 1
2
(1+ cot θ)
1 2
sin θ
=
2
€(1+ cot θ)
Step 1- Solve the logartithm. Part B. Simplify the Base of Arugement
Therefore, the Base of the Argument
2
sin θ .
is equal to
€
26. 1
1
1 (sec 2 θ −tan 2 θ −cos2 θ )
2 2 2
(sec θ − tan θ − cos θ)
log[1−(1−sin 2 θθ)] 2
2
[1− (1− sin )]
(1+ cot θ)
(1+
€
€
Step 1- Solve the logartithm.
€
So far the logarithm goes from this...
27. 1
1
12 (sec 2 θθ−tan 2 θθ−cos2 θθ))
2 2 2
(sec − tan − cos
log[1−(1−sinθ2 θ )] sin θ
sin2 2
(1+ cot θ)
€
€
Step 1- Solve the logartithm.
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...to this.
28. 1
1
12 (sec 2 θθ−tan 2 θθ−cos2 θθ))
2 2 2
(sec − tan − cos
log[1−(1−sinθ2 θ )] sin θ
sin2 2
(1+ cot θ)
€
€
Step 1- Solve the logartithm.
€
We can recognize this about the logarithm: The Base is the
same as the Base of the Argument. If this is the case, no
matter what the Exponent of the Argument is, the entire
logarithm is equal to the Exponent of the Argument.
30. 1
2
1 (sec θ −tan θ −cos θ ) [sin(θ + θ)]
2 2 2
log[1−(1−sin 2 θ )] = +1
2 log 2 16
(1+ cot θ) 4 sin θ
Step 1- Solve the logartithm.
Now, we can take a look at the original identity.
Since we have solved the logarithm the identity no
longer looks like this...
31. 2
1 [sin(θ + θ)]
+1
=
2 2 2 log 2 16
(sec θ − tan θ − cos θ) 4 sin θ
€ Step 1- Solve the logartithm.
...it looks like this
32. 2
1 [sin(θ + θ)]
= +1
2 2 2 log 2 16
(sec θ − tan θ − cos θ) 4sin θ
€
Step 2- Prove the identity.
33. 2
1 [sin(θ + θ)]
+1
=
2 2 2 log 2 16
(sec θ − tan θ − cos θ) 4 sin θ
€ Step 2- Prove the identity.
To prove the identity, solve the two sides separately:
Side 1
Side 2
34. 1
2 2 2
(sec θ − tan θ − cos θ)
Step 2- Prove the identity.
Side 1.
35. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 2
Method 1
€
Step 2- Prove the identity. Side 1
There are two methods that can be used to get the simplified
version of this expression:
Method 1
Method 2
36. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 1
1
€ 2 2 2
(sec θ − tan θ − cos θ)
Step 2- Prove the identity. Side 1
€
37. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 1
1
2
1 sin θ
€ 2
[( 2 ) − ( 2 ) − cos θ]
cos θ cos θ
Step 2- Prove the identity. Side 1
Method 1: Simplify the first two terms so that they are
expressed in terms of sine and/or cosine.
€
38. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 1
1
2
€ 1− sin θ 2
[( ) − cos θ]
2
cos θ
Step 2- Prove the identity. Side 1
Method 1: Subtract the two fractions in the denominator to
get one fraction.
€
39. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 1
1
2
€ cos θ 2
[( 2 ) − cos θ]
cos θ
Step 2- Prove the identity. Side 1
2 2
Method 1: Due to the Pythagorean identity ( sin θ + cos θ =1),
2
cos θ
the fraction in the denominator is simplified to cos θ 2
€
.
€ €
40. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 1
1
€ 2
(1− cos θ)
Step 2- Prove the identity. Side 1
Method 1: Simplify the fraction in the denominator to “1”.
€
41. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 1
1
€
2
sin θ
Step 2- Prove the identity. Side 1
2 2
Method 1: Due to the Pythagorean identity ( sin θ + cos θ =1),
2
the denominator is simplified to sin θ .
€ €
€
42. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 1
2
csc θ
€
Step 2- Prove the identity. Side 1
csc 2 θ
1
Method 1: can also be rewritten as
sin2 θ
€
€
43. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 2
1
€ 2 2 2
(sec θ − tan θ − cos θ)
Step 2- Prove the identity. Side 1
€
44. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 2
1
€
2
(1− cos θ)
Step 2- Prove the identity. Side 1
Method 2: Recognize that the the Pythagorean Identity
sec 2 θ −tan 2 θ =1 applies to the denominator of the
expression and simplify it as such.
€
45. 1
2 2 2
(sec θ − tan θ − cos θ)
Method 2
1
€
2
sin θ
Step 2- Prove the identity. Side 1
2
θ + cos2 θ =1),
Method 2: Using the Pythagorean identity ( sin
2
simplify the denominator to sin θ .
€
€
46. 1
2 2 2
(sec θ − tan θ − cos θ)
2
csc θ
€
Step 2- Prove the identity. Side 1
csc 2 θ
1
Method 2: can also be rewritten as
sin2 θ
€
€
47. 2
1 [sin(θ + θ)]
+1
=
2 2 2 log 2 16
(sec θ − tan θ − cos θ) 4 sin θ
€ Step 2- Prove the identity.
So far the we have solved Side 1 of the identity. Now, it no
longer looks like this...
48. 2
[sin(θ + θ)]
2
csc θ = +1
log 2 16
4 sin θ
€
Step 2- Prove the identity.
...it looks like this
49. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Step 2- Prove the identity.
Side 2.
50. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
Method 1
€
Step 2- Prove the identity. Side 2
There are two methods that can be used to get the simplified
version of this expression:
Method 1
Method 2
51. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
2
[sin(θ + θ)]
+1
€ log 2 16
4 sin θ
Step 2- Prove the identity. Side 2
52. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
2
[sin(θ + θ)]
+1
€ 4
4 sin θ
Step 2- Prove the identity. Side 2
Method 1: Simplify the logarithm that is an exponent for
one of the terms in the denominator. This can be
done by converting it into a power (ie. 2 x =16 ).
€
53. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
2
(2sinθ cosθ)
+1
€ 4
4 sin θ
Step 2- Prove the identity. Side 2
Method 1: The expression inside the brackets can be
recognized as one of the “Double Angle Identity”
and therefore can be simplified to 2sinθ cosθ .(see last slide)
54. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
(2)(sinθ)(cosθ)(2)(sinθ)(cosθ)
+1
€ (2)(2)(sinθ)(sinθ)(sinθ)(sinθ)
Step 2- Prove the identity. Side 2
Method 1: Now both the numerator and the denominator
can be expanded. (Expand as much as possible).
55. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
(2)(sinθ)(cosθ)(2)(sinθ)(cosθ)
+1
€ (2)(2)(sinθ)(sinθ)(sinθ)(sinθ)
Step 2- Prove the identity. Side 2
Method 1: Now we can reduce many parts of the expression
and simplify the remaining terms.
56. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
2
(cosθ)
+1
2
€ (sinθ)
Step 2- Prove the identity. Side 2
Method 1: This is the resulting expression.
€
57. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
2 2
cos θ sin θ
+2
2
sin θ sin θ
€
Step 2- Prove the identity. Side 2
Method 1: From there we can rewrite “1” as a fraction with
the same denominator as the resulting fraction.
€
58. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
2 2
cos θ + sin θ
2
sin θ
€
Step 2- Prove the identity. Side 2
Method 1: After that, it simply becomes a matter of adding
the two fractions together...
59. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
1
2
€
sin θ
Step 2- Prove the identity. Side 2
Method 1: ...and applying the Pythagorean Identity.
60. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 1
2
csc θ
€
Step 2- Prove the identity. Side 2
csc 2 θ
1
Method 1: Fianlly, can also be rewritten as
sin2 θ
€
€
61. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
2
[sin(θ + θ)]
+1
€ log 2 16
4 sin θ
Step 2- Prove the identity. Side 2
62. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
2
[sin(θ + θ)]
+1
€ 4
4 sin θ
Step 2- Prove the identity. Side 2
Method 2: The second method starts out the same as the
first, solving the logarithm in the exponent of the
denominator.
63. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
2
(2sinθ cosθ)
+1
€ 4
4sin θ
Step 2- Prove the identity. Side 2
Method 2: Also the same as the first method, we recognize
the expression inside the brackets as a “Double
Angle Identity”...
64. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
(2)(sinθ)(cosθ)(2)(sinθ)(cosθ)
+1
€ (2)(2)(sinθ)(sinθ)(sinθ)(sinθ)
Step 2- Prove the identity. Side 2
Method 2: The expression, once again, is then expanded;
like terms are reduced; and remaining terms are
simplified to get...
66. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
2
(cosθ)
+1
2
€ (sinθ)
Step 2- Prove the identity. Side 2
Method 2: However, instead of rewriting “1” as a fraction,
due to the fact that (cosθ) is an identity itself,
2
2
(sinθ)
rewrite the expression as...
€
€
67. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
2
cot θ +1
€
Step 2- Prove the identity. Side 2
68. 2
[sin(θ + θ)]
+1
log 2 16
4 sin θ
Method 2
2
csc θ
€
Step 2- Prove the identity. Side 2
2 2
Method 2: Now, apply the Pythagorean Identity csc θ − cot θ =1
csc 2 θ .
and rewrite the expression as
€
€
69. 2
2
[sin(θ + θ)]
csc θ
=
+1
log 2 16
4 sin θ
Step 2- Prove the identity.
Now that we have solved Side 2, we can safely say...
€
70. Step 2- Prove the identity.
...SINCE...
1 2
= csc θ
2 2 2
(sec θ − tan θ − cos θ)