4 The Trigonometric Functions

Transcription

1 Mathematics Learning Centre, University of Sydney 8 The Trigonometric Functions The definitions in the previous section apply to between 0 and, since the angles in a right angle triangle can never be greater than. The definitions given below are useful in calculus, as they extend sin, cos and tan without restrictions on the value of..1 The cosine function Let s begin with a definition of cos. Consider a circle of radius 1, with centre at the origin of the (x, y) plane. Let be the point on the circumference of the circle with coordinates (1, 0). is a radius of the circle with length 1. Let be a point on the circumference of the circle with coordinates (a, b). We can represent the angle between and,, by the arc length along the unit circle from to. This is the radian representation of. Q (a,b) The cosine of is defined to be the x coordinate of. Let s, for the moment, consider values of between 0 and. The cosine of is written cos, so in the diagram above, cos = a. Notice that as increases from 0 to, cos decreases from 1 to 0. For values of between 0 and, this definition agrees with the definition of cos as the ratio adjacent hypotenuse of the sides of a right angled triangle. Draw Q perpendicular. In Q, the hypotenuse has length 1, while Q has length a. The ratio adjacent hypotenuse = a = cos. The definition of cos using the unit circle makes sense for all values of. For now, we will consider values of between 0 and. The x coordinate of gives the value of cos. When =, is on the y axis, and it s x coordinate is zero. s increases beyond, moves around the circle into the second quadrant and therefore it s x coordinate will be negative. When =, the x coordinate is 1. cos is positive cos = 0 cos negative cos = 1

2 Mathematics Learning Centre, University of Sydney 9 s increases further, moves around into the third quadrant and its x coordinate increases from 1 to 0. Finally as increases from to the x coordinate of increases from 0 to 1. cos is negative cos = 0 cos positive cos =1.1.1 Exercise 1. Use the cosine (cos) key on your calculator to complete this table. (Make sure your calculator is in radian mode.) cos cos. Using this table plot the graph of y = cos for values of ranging from 0 to.. The sine function The sine of is defined using the same unit circle diagram that we used to define the cosine. (a,b) The sine of is defined to be the y coordinate of. Q The sine of is written as sin, so in the diagram above, sin = b. For values of between 0 and, this definition agrees with the definition of sin as the ratio opposite hypotenuse of sides of a right angled triangle. In the right angled triangle Q, the hypotenuse has length 1 while Q has length b. The ratio opposite hypotenuse = b 1 = sin. This definition of sin using the unit circle extends to all values of. consider values of between 0 and. Here, we will

3 Mathematics Learning Centre, University of Sydney 10 s moves anticlockwise around the circle from to B, increases from 0 to. When isat,sin = 0, and when is at B, sin =1. Soas increases from 0 to, sin increases from 0 to 1. The largest value of sin is 1. s increases beyond, sin decreases and equals zero when =. s increases beyond, sin becomes negative. sin is positive sin = 1 sin positive sin =0 sin is negative sin = 1 sin negative sin =0..1 Exercise 1. Use the sin key on your calculator to complete this table. Make sure your calculator is in radian mode sin sin. lot the graph of the y = sin using the table in the previous exercise.. The tangent function We can define the tangent of, written tan, in terms of sin and cos. tan = sin cos. Using this definition we can work out tan for values of between 0 and. You will

4 Mathematics Learning Centre, University of Sydney 11 be asked to do this in Exercise.. In particular, we know from this definition that tan is not defined when cos = 0. This occurs when = or =. When 0 << this definition agrees with the definition of tan as the ratio opposite adjacent of the sides of a right angled triangle. s before, consider the unit circle with points, and as shown. Drop a perpendicular from the point to which intersects at Q. s before has coordinates (a, b) and Q coordinates (a, 0). (a,b) opposite adjacent = Q Q (in triangle Q) Q = b a = sin cos = tan. If you try to find tan using your calculator, you will get an error message. Look at the definition. The tangent of is not defined as cos = 0. For values of near, tan is very large. Try putting some values in your calculator. (eg Try tan(1.57), tan(1.5707), tan( ).)..1 Exercise 1. Use the tan key on your calculator to complete this table. Make sure your calculator is in radian mode tan tan. Use the table above to graph tan.

5 Mathematics Learning Centre, University of Sydney 1 Your graph should look like this for values of between 0 and. Notice that there is a vertical asymptote at =. This is because tan is not defined at =. You will find another vertical asymptote at =. When = 0or, tan =0. For greater than 0 and less than, tan is positive. For values of greater than and less than, tan is negative. /. Extending the domain The definitions of sine, cosine and tangent can be extended to all real values of in the following way. 5 = + corresponds to the arc length of 1 1 revolutions around the unit circle going anticlockwise from to B. Since B has coordinates (0, 1) we can use the previous definitions to get: sin 5 =1, cos 5 =0, B tan 5 is undefined. Similarly, = , sin( 16) sin( 0.56 ) 0.9, cos( 16) 0.96, tan( 16) 0.0. B

6 Mathematics Learning Centre, University of Sydney 1..1 Exercise Evaluate the following trig functions giving exact answers where you are able. 1. sin 15. tan 1. cos 15. tan 1 5. sin 6 6 Notice The values of sine and cosine functions repeat after every interval of length. Since the real numbers x, x +, x, x +, x etc differ by a multiple of, they correspond to the same point on the unit circle. So, sin x = sin(x +) = sin(x ) = sin(x +) = sin(x ) etc. We can see the effect of this in the functions below and will discuss it further in the next chapter. 1 sin cos 0 1 The tangent function repeats after every interval of length. tan 1 0 1

7 Mathematics Learning Centre, University of Sydney 6 8 Solutions to Exercises Exercise cos cos

8 Mathematics Learning Centre, University of Sydney cos Exercise sin sin sin Exercise tan tan tan

9 Mathematics Learning Centre, University of Sydney 8. tan Exercise sin 15 = sin = 1 since 15 and differ by 6 =.. tan 1 6 = tan 6 = 1.. cos 15 = tan 1 = tan = tan = tan =. 5. sin = sin 5 = sin =.