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Advanced Precalculus: Geometry, Trigonometry and Exponentials

University of Padova via FutureLearn


Take your precalculus knowledge to the next level

This course is the second of two precalculus courses from the University of Padova, providing you with the advanced mathematical skills required to complete a first course in calculus.

You’ll get to grips with more challenging precalculus topics: plane and solid geometry; logarithms and exponentials; equalities and inequalities; and trigonometric functions and identities.

By the end of this course you’ll have gained the skills to support further mathematical studies.

This course is for anyone who wants to brush up on their basic mathematical skills, particularly students in the last year of high school, or at the beginning of undergraduate studies, who intend to take a first course in calculus.

Ideally, you’ll have already taken the first course - Precalculus: the Mathematics of Numbers, Functions and Equations or be familiar with the topics covered in it, and have a basic knowledge of high school mathematics.


  • Exponential and logarithmic functions
    • Course presentation
    • Exponential functions
    • The Euler number
    • Functions and their inverses
    • Logarithms
    • It's quiz time
    • Test yourself
  • Trigonometry
    • Trigonometry by triangles
    • Trigonometry by the unit circle
    • Graphs of trigonometric functions
    • Trigonometric identities
    • Harmonic motion
    • The function arctan
    • More inverse trig functions, and a few others
    • It's quiz time
    • Test yourself
  • Geometry
    • Geometry of the plane: points, segments, lines
    • Distance and circles
    • Ellipses
    • Parabolas and hyperbolas
    • Level curves of quadratic functions
    • Lengths, areas, and volumes
    • It's quiz time
    • Test yourself
  • From equations to spirals
    • Equations with exponentials and logarithms
    • Growth and decay problems
    • Trigonometric equations
    • Inequalities with logarithms and exponentials
    • Trigonometric inequalities
    • Polar coordinates
    • It's quiz time
    • Test yourself
    • Final step and revision test

Taught by

Francis Clarke


4.8 rating, based on 44 Class Central reviews

4.8 rating at FutureLearn based on 17 ratings

Start your review of Advanced Precalculus: Geometry, Trigonometry and Exponentials

  • The best maths course I have ever did in my whole life with best professors. The basic precalculus course was a bit easy for me and I quickly advanced to this one. I refreshed lots of math knowledge from before and learned new things at the same time, much better than all other students I know. Fascinating course, my compliments. I already miss it!
  • Anonymous
    The instructors accompanied by the visualisations, the exercises and the quizzes made it very engaging, fun, intuitive and interesting. The course also spoke about next steps and gave a bit of history that added to its quality. Thank you for making this course.
  • Graham Divall

    Graham Divall completed this course, spending 25 hours a week on it and found the course difficulty to be hard.

    A very good course which I found, personally, difficult and demanding. A lot of fellow students appeared to have strong mathematical backgrounds and were probably familiar with many of the topics covered. I found a tendency for the videos to speak...
  • Anonymous
    The best maths course I have ever did in my whole life with best professors. The basic precalculus course was a bit easy for me and I quickly advanced to this one. I refreshed lots of math knowledge from before and learned new things at the same time, much better than all other students I know. Fascinating course, my compliments. I already miss it!
  • Profile image for Miha Veber
    Miha Veber
    Very well explained, just the right amount of problems to solve (not too much, not too little) in my opinion since this was just a refresher for me and I had some prior knowledge about the subject.
  • Excellent and amazing course. I found new things to learn. Teachers gave an excellent and professional support. I would recommend this amazing course.

    Thank you very much.
  • Anonymous

    Anonymous completed this course.

    This and the earlier precalculus course from the University of Padova are two of the best courses on FutureLearn, both challenging but enjoyable. They have a regular structure, the theory of each topic is discussed followed by practical examples of problem...
  • Anonymous

    Anonymous completed this course.

    Initial Videos gave complete step by step guidance and clarity of the topic. Though challenging, but not impossible. The task questions were well set, kind of covering most of the concepts and the solving videos/pdf were helpful to mark our work or address my misconception or error. Thoroughly enjoyed working on them, reignited the passion in me for mathematics and also enabled me to replenish my subject knowledge.
  • Anonymous
    I like the sequence of the topics increasing the complexity and culminating with the amalgamation of the knowledge in week 4 which i would have like to have more time to spend upon . Many thanks i did get all the information i was looking for and looking forward to progress and further my knowledge in calculus .
  • Profile image for Kevin Griffiths
    Kevin Griffiths

    Kevin Griffiths completed this course.

    Very enjoyable. All the tutors are knowledgable, enthusiastic and engaging. The structure of the course, with a lecture followed by worked examples which you are encouraged to try yourself, works really well. The pace is quite fast and some of the material...
  • Minh Nguyen

    Minh Nguyen completed this course.

    Very demanding zciriculum and fun, and serious concentration is needed,. Was a little tool fast to take in 8-10 minutes per section of the course though, I had to play it some time more than once to capture all of the technical information and answers...
  • Anonymous
    this course will help me to understand better the calculus. the tutor who delivered the course explained the main points in detailed. The presentation of the course in graphical way helps me to deeply understand the trigonometric ratios, exponential and logarithmic functions well. Combination of the sines and cosines function that results in waves was very interesting part of the course. Introduction to the limits could be a better idea in pre-calculus.
  • Anonymous
    This Advanced Precalculus refreshed my memory of topics from long ago, and introduced some expanded aspects of the topics I had never before seen. At 72 years old, it is good to be challenged and to acquire new skills and understanding. Thank you for the wide variety of courses I have completed in maths, science, social studies, astronomy and medical topics.
  • Anonymous
    This course was really well explained and structured. The practical exercises were very helpful to consolidate learning and recorded sessions managed to cover exactly the right areas. Just a shame that some of the videos were out of sync and some text in the answers was hidden
  • Profile image for Or Ka King
    Or Ka King
    This is very good topic as higher education programs. This is very interested in so many learner. This is very much famous topic.

    This is as follows as soon as possible.

    This is as very good in development at in-depth view's.

  • Anonymous

    Anonymous completed this course.

    A great deal of hard work was put into this MOOC, one of the most thorough (and demanding, but that is not a criticism) that I have done. All involved showed a love for their subject and the wish to share that. Excellent!
  • Anonymous
    Maths is for active engagement and this course is both engaging and varied in approach.

    The format is well balanced with ludid expositions, practical problem-solving demonstrations and post quiz comments and clarifications. Rather than a formulaic approach there were some unexpected twists and turns. There were also references to a wider field of Maths as well as a wider world.

    The course is suitable for any advanced student - post as well as pre-university, whether learning the material for the first time or in need of practice.
  • Anonymous

    Anonymous completed this course.

    I thought the course was quite interesting. I thought the course would be about differentiation or integration but I was pleasantly surprised. I enjoyed most of it and thought it was a nice difficulty and I didn't find it so hard that I couldn't sleep at night. I quite liked that you were shown how to do all the questions in week 1 but a little disappointed that it didn't return for the future weeks. I especially liked it when Francis showed his humorous side. His circle joke was legendary.
  • Anonymous
    This is an excellent course which I thoroughly enjoyed. The lecture videos are clear to understand and delivered with a sprinkling of dry humor. The chance to put into practice the content of the lectures in the 'Its Your Turn' lessons is both useful and fun, but at the same time challenging. Getting these problems right gives a real sense of achievement. I'd recommend this course to anyone wanting to learn the principles of maths.
  • Anonymous

    Anonymous completed this course.

    Inspiring course, urges to think about numbers and how it applies to surrounding environments. Currently I'm also studying bioinformatics, which I know this course will come into use. In the duration of this I also came to understand the relationship between math and art. Many of the equations presented in this course invoked visuals related to art, which helped me to understand the questions with greater clarity.

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