2 Unit Bridging Course Day 10

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1 1 / 31 Unit Bridging Course Day 10 Circular Functions III The cosine function, identities and derivatives Clinton Boys

2 / 31 The cosine function The cosine function, abbreviated to cos, is very similar to the sine function. In fact, the cos function is exactly the same, except shifted π/ units to the left.

3 3 / 31 The cosine function The cosine function, abbreviated to cos, is very similar to the sine function. In fact, the cos function is exactly the same, except shifted π/ units to the left.

4 4 / 31 Graph of y = cos(x) Below is the graph of y = cos(x) between x = 4π and x = 4π. 1 4π 7π 3π 5π π 3π π π 1 π π 3π π 5π 3π 7π 4π The graph continues forever in both directions. Notice the similarities between cos and sin, as well as the differences.

5 5 / 31 Graph of y = cos(x) Below is the graph of y = cos(x) between x = 4π and x = 4π. 1 4π 7π 3π 5π π 3π π π 1 π π 3π π 5π 3π 7π 4π The graph continues forever in both directions. Notice the similarities between cos and sin, as well as the differences.

6 6 / 31 Graph of y = cos(x) Below is the graph of y = cos(x) between x = 4π and x = 4π. 1 4π 7π 3π 5π π 3π π π 1 π π 3π π 5π 3π 7π 4π The graph continues forever in both directions. Notice the similarities between cos and sin, as well as the differences.

7 7 / 31 Graph of y = cos(x) Below is the graph of y = cos(x) between x = 4π and x = 4π. 1 4π 7π 3π 5π π 3π π π 1 π π 3π π 5π 3π 7π 4π The graph continues forever in both directions. Notice the similarities between cos and sin, as well as the differences.

8 8 / 31 Properties of cosine cos shares the following properties with sin: (i) 1 cos x 1 for all x. (ii) cos(x + π) = cos x for all x, i.e. cos x is periodic with period π, just like sin x.

9 9 / 31 Properties of cosine Unlike sin, however, cos is not odd: (iii) cos( x) = cos(x). 1 π π 1 y = cos x is symmetric about the y-axis we say it is an even function.

10 10 / 31 Sketching cosine curves Practice questions See if you can sketch the following cosine curves, using the same ideas we used to sketch sine curves. (i) y = cos x (ii) y = cos(x) (iii) y = 3 cos(x).

11 11 / 31 Sketching cosine curves Answers (i) y = cos x π π

12 1 / 31 Sketching cosine curves Answers (ii) y = cos(x) 1 1 π 4 π

13 13 / 31 Sketching cosine curves Answers (iii) y = 3 cos(x) 3 π 4 π 3

14 14 / 31 Identities involving circular functions Together, sin and cos are called the circular functions. There are many important identities involving circular functions which you should remember. (i) sin x + cos x = 1 (where sin x = (sin x) ) (ii) sin(x + y) = sin x cos y + cos x sin y (iii) cos(x + y) = cos x cos y sin x sin y (ii) and (iii) are known as double angle formulas. You can find plenty more such identities, for example on Wikipedia.

15 15 / 31 Identities involving circular functions Together, sin and cos are called the circular functions. There are many important identities involving circular functions which you should remember. (i) sin x + cos x = 1 (where sin x = (sin x) ) (ii) sin(x + y) = sin x cos y + cos x sin y (iii) cos(x + y) = cos x cos y sin x sin y (ii) and (iii) are known as double angle formulas. You can find plenty more such identities, for example on Wikipedia.

16 16 / 31 Identities involving circular functions Together, sin and cos are called the circular functions. There are many important identities involving circular functions which you should remember. (i) sin x + cos x = 1 (where sin x = (sin x) ) (ii) sin(x + y) = sin x cos y + cos x sin y (iii) cos(x + y) = cos x cos y sin x sin y (ii) and (iii) are known as double angle formulas. You can find plenty more such identities, for example on Wikipedia.

17 17 / 31 Derivatives of circular functions The circular functions, sin and cos, have particularly simple derivatives. Derivatives of the circular functions d d (sin x) = cos x Notice the derivative of cos is negative sin. (cos x) = sin x.

18 18 / 31 Derivatives of circular functions The circular functions, sin and cos, have particularly simple derivatives. Derivatives of the circular functions d d (sin x) = cos x Notice the derivative of cos is negative sin. (cos x) = sin x.

19 19 / 31 Derivatives of circular functions The circular functions, sin and cos, have particularly simple derivatives. Derivatives of the circular functions d d (sin x) = cos x Notice the derivative of cos is negative sin. (cos x) = sin x.

20 0 / 31 Derivatives of circular functions Example Find the derivative of the function f (x) = 3 sin(x). We need to use the chain rule.

21 1 / 31 Derivatives of circular functions Example Find the derivative of the function f (x) = 3 sin(x). We need to use the chain rule.

22 / 31 Derivatives of circular functions Example Find the derivative of the function f (x) = 3 sin(x). We need to use the chain rule. df = 3 cos(x) d (x)

23 3 / 31 Derivatives of circular functions Example Find the derivative of the function f (x) = 3 sin(x). We need to use the chain rule. df = 3 cos(x) d (x) = 3 cos(x)

24 4 / 31 Derivatives of circular functions Example Find the derivative of the function f (x) = 3 sin(x). We need to use the chain rule. df = 3 cos(x) d (x) = 3 cos(x) = 6 cos(x).

25 5 / 31 Derivatives of circular functions Example Find dy if y = sin x cos x. We need to use the product rule. Let u = sin x and v = cos x. Then

26 6 / 31 Derivatives of circular functions Example Find dy if y = sin x cos x. We need to use the product rule. Let u = sin x and v = cos x. Then

27 7 / 31 Derivatives of circular functions Example Find dy if y = sin x cos x. We need to use the product rule. Let u = sin x and v = cos x. Then dy = u dv + v du

28 8 / 31 Derivatives of circular functions Example Find dy if y = sin x cos x. We need to use the product rule. Let u = sin x and v = cos x. Then dy = u dv + v du = sin x ( sin x) + cos x (cos x)

29 9 / 31 Derivatives of circular functions Example Find dy if y = sin x cos x. We need to use the product rule. Let u = sin x and v = cos x. Then dy = u dv + v du = sin x ( sin x) + cos x (cos x) = sin x + cos x.

30 30 / 31 Derivatives of circular functions Practice questions Find the derivatives of the following functions: (i) f (x) = sin x (ii) f (x) = x cos x (iii) f (x) = sin(x ) (iv) f (x) = sin x (usually written tan x). cos x

31 31 / 31 Derivatives of circular functions Answers to practice questions df (i) df (ii) = sin x cos x = x sin x + cos x (iii) df = x cos(x ) (iv) df = cos x+sin x cos x = 1 cos x.

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