Functions of periodic functions

Videos

• 18:15 (watch on YouTube, opens in a new tab)

  Transformations Part 5: Functions of Periodic Functions | TMUA (watch on YouTube, opens in a new tab)

  12 October 2026 · Duration 18:15

  How do you sketch sin(cos x), or even cos(sin(cos x))? Functions of periodic functions, built up from simple examples to nested trigonometric expressions. The key is to work from the inside out: track the values of the inner function, then see how the outer function responds, using ranges, increasing and decreasing behaviour, and periodicity to sketch graphs and count intersections. We finish by finding how many solutions cos(sin(cos x)) = sin(cos(sin x)) has.

  • Transformations
  • Trigonometric functions
  • Graph sketching
  • Number of solutions

  Watch on YouTube →: Transformations Part 5: Functions of Periodic Functions | TMUA (opens in a new tab)

• 10:36 (watch on YouTube, opens in a new tab)

  Transformations Part 5: Functions of Periodic Functions - BONUS | TMUA (watch on YouTube, opens in a new tab)

  13 October 2026 · Duration 10:36

  A bonus follow-up to Part 5, with two more demanding examples: a quadratic function of sin x and a rational function of cos x, and how turning points, restricted ranges and vertical asymptotes shape the graphs. The same approach — work from the inside out, track the input, and see how the outer function responds. Enrichment rather than typical TMUA questions.

  • Transformations
  • Trigonometric functions
  • Graph sketching
  • Asymptotes

  Watch on YouTube →: Transformations Part 5: Functions of Periodic Functions - BONUS | TMUA (opens in a new tab)