1. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS PROGRAMME F11

2. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Exponential and logarithmic functions Odd and even functions

3. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Exponential and logarithmic functions Odd and even functions

4. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Rotation The tangent Period Amplitude Phase difference Inverse trigonometric functions Trigonometric equations Equations of the form acos x + bsin x = c

5. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Rotation For angles greater than zero and less than π/2 radians the trigonometric ratios are well defined and can be related to the rotation of the radius of a unit circle:

6. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Rotation By continuing to rotate the radius of a unit circle the trigonometric ratios can extended into the trigonometric functions, valid for any angle:

7. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Rotation The sine function:

8. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Rotation The cosine function:

9. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions The tangent The tangent is the ratio of the sine to the cosine:

10. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Period Any function whose output repeats itself over a regular interval is called a periodic function, the regular interval being called the period of the function. The sine and cosine functions are periodic with period 2π. The tangent function is periodic with period π.

11. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Amplitude Every periodic function possesses an amplitude that is given as the difference between the maximum value and the average value of the output taken over a single period.

12. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Phase difference The phase difference of a periodic function is the interval of the input by which the output leads or lags behind the reference function.

13. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Inverse trigonometric functions If the graph of y = sin x is reflected in the line y = x a function does not result.

14. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Inverse trigonometric functions Cutting off the upper and lower parts of the graph produces a single-valued function that is the inverse sine function.

15. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Inverse trigonometric functions Similarly for the inverse cosine function and the inverse tangent function.

16. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Trigonometric equations A simple trigonometric equation is one that involves just a single trigonometric expression: For example:

17. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Trigonometric equations As another example:

18. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Equations of the form acos x + bsin x = c The equation acos x + bsin x = c can be rewritten as: Remembering that multiple solutions can be found by using the graph.

19. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Exponential and logarithmic functions Odd and even functions

20. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Exponential and logarithmic functions Odd and even functions

21. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Exponential and logarithmic functions Exponential functions Indicial equations

22. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Exponential and logarithmic functions Exponential functions The exponential function is expressed by the equation: Where e is the exponential number 2.7182818... The value of e x can be found to any level of precision desired from the series expansion:

23. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Exponential and logarithmic functions Exponential functions The graphs of e x and e –x are:

24. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Exponential and logarithmic functions Exponential functions The general exponential function is given by y = a x where a > 0. Since a = e lna the general exponential function can be written as:

25. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Exponential and logarithmic functions Exponential functions The inverse exponential function is the logarithmic function expressed by the equation:

26. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Exponential and logarithmic functions Indicial equations An indicial equation is an equation where the variable appears as an index and the solution of such equations requires application of logarithms.

27. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Exponential and logarithmic functions Odd and even functions

28. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Trigonometric functions Exponential and logarithmic functions Odd and even functions

29. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Odd and even functions Odd and even parts Odd and even parts of the exponential function Limits of functions The rules of limits

30. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Odd and even functions Given a function f with output f (x) then, assuming f (−x) is defined: If f (−x) = f (x) the function f is called an even function If f (−x) = −f (x) the function f is called an odd function

31. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Odd and even functions Odd and even parts If, given f (x) where f (−x) is defined then:

32. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Odd and even functions Odd and even parts of the exponential function The even part of the exponential function is: The odd part of the exponential function is: Notice:

33. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Limits of functions There are times when a function has no defined output for a particular value of x, say x 0, but that it does have a defined value for values of x arbitrarily close to x 0. For example: However, so when x is close to 1 f (x) is close to 2. it is said that: the limit of f (x) as x approaches the value x = 1 is 2

34. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions The limit of f (x) as x approaches the value x = 1 is 2. Symbolically this is written as: Limits of functions

35. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions The rules of limits Limits of functions

36. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Limits of functions The rules of limits

37. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Learning outcomes Identify a function as a rule and recognize rules that are not functions Determine the domain and range of a function Construct the inverse of a function and draw its graph Construct compositions of functions and de-construct them into their components Develop the trigonometric functions from the trigonometric ratios Find the period, amplitude and phase of a periodic function More...

38. STROUD Worked examples and exercises are in the text Programme F11: Trigonometric and exponential functions Learning outcomes Distinguish between the inverse of a trigonometric function and the inverse trigonometric function Solve trigonometric equations using the inverse trigonometric functions Recognize that the exponential and natural logarithmic functions are mutual inverses and solve indicial and logarithmic equations Construct the hyperbolic functions from the odd and even parts of the exponential function Evaluate limits of simple functions