Graphs of Trigonometric Functions

∴ Sine ratio is a function of q. Consider y = sinq. determines each value of q exactly one value of y ∴ Sine ratio is a function of q. Actually, each trigonometric ratio is a function of q. e.g. f( ) = sin  , f( ) = cos  , f( ) = tan  Sine function Cosine function Tangent function In general, they are called trigonometric functions.

What do the graphs of trigonometric functions look like? Let’s complete the following table and then plot the graph of y = sinq for 0 ≤ q ≤ 360.

Consider y = sinq for 0 ≤ q ≤ 360. 0.5 0.87 1 0.87 0.5 -0.5 -0.87 -1 -0.87 -0.5 (The values of sinq are correct to 2 decimal places if necessary.)

Graph of y = sin  y is maximum at  = 90 1  sin   1 min. value max. value 1  sin   1 for 0    360 sin  = 0 at  = 0, 180 and 360 y is minimum at  = 270  = 0 to 90:  = 90 to 270:  = 270 to 360: y increases from 0 to 1. y decreases from 1 to 1. y increases from 1 to 0.

Similarly, plot the graph of y = cosq for 0 ≤ q ≤ 360.

Consider y = cosq for 0 ≤ q ≤ 360. 1 0.87 0.5 -0.5 -0.87 -1 -0.87 -0.5 0.5 0.87 1 (The values of cosq are correct to 2 decimal places if necessary.)

Graph of y = cos  y is maximum 1  cos   1 at  = 0 and 360 min. value max. value y is maximum at  = 0 and 360 1  cos   1 for 0    360 cos  = 0 at  = 90 and 270 y is minimum at  = 180  = 0 to 180:  = 180 to 360: y decreases from 1 to 1. y increases from 1 to 1.

Similarly, we can obtain the graph of y = tanq for 0 ≤ q ≤ 360.

Graph of y = tan  The graph is discontinuous at q = 90 and 270.  tan  is undefined at  = 90 and 270. 90 270 tanq has neither a maximum nor a minimum. tan  = 0 at  = 0, 180 and 360  = 0 to 90:  = 90 to 270:  = 270 to 360: y increases from 0 to positive infinity. y increases from negative infinity to positive infinity. y increases from negative infinity to 0.

Periodicity of Trigonometric Functions This is the graph of y = sin q for –720 ≤ q ≤ 720. Can you see any pattern?

The graphs of y = sin q in the four intervals are the same. –720   < – 360 –360   < 0 0   < 360 360   < 720 The graphs of y = sin q in the four intervals are the same.

The graph repeats itself every 360. –720   < – 360 –360   < 0 0   < 360 360   < 720 The graph repeats itself every 360.

y = sin  is a periodic function with period 360. The graph of y = sin  repeats itself every 360. y = sin  is a periodic function with period 360.

I notice that the graph of y = sin  also repeats itself every 720, 1080, etc. We take the smallest positive value in which the graph repeats itself as the period of the function. So, the period of y = sin  is 360.

Besides, since sin  is defined for any angle , the domain of the sine function is all real numbers.

The following shows the graph of y = cos . Is y = cos  a periodic function? If yes, what is its period?

y = cos  is a periodic function with period 360. The following shows the graph of y = cos . The graph of y = cos  repeats itself every 360. y = cos  is a periodic function with period 360.

The following shows the graph of y = cos . Since cos  is defined for any angle , the domain of the cosine function is all real numbers.

The following shows the graph of y = tan . Is y = tan  a periodic function? If yes, what is its period?

y = tan  is a periodic function with period 180. The following shows the graph of y = tan . The graph of y = tan  repeats itself every 180. y = tan  is a periodic function with period 180.

The following shows the graph of y = tan . Since tan  is undefined only when  = 90, 270, 450, 630, etc., the domain of the tangent function is all real numbers except 90, 270, 450, 630, etc.

Follow-up question Refer to the graph of a periodic function y = f(x) for 0 ≤ x ≤ 1080. (a) Find the maximum and minimum values of y. (b) Find the period of the function from its graph. y = f(x) x y 180 360 540 720 900 1080 1 0.5 1 0.5 (a) From the graph, the maximum and the minimum values of y are 1 and –1 respectively.

Follow-up question Refer to the graph of a periodic function y = f(x) for 0 ≤ x ≤ 1080. (a) Find the maximum and minimum values of y. (b) Find the period of the function from its graph. y = f(x) x y 180 360 540 720 900 1080 1 0.5 1 0.5 (b) ∵ The graph repeats itself every 540. ∴ The period of y = f(x) is 540.

In fact, for any value of , 1  sin   1; the minimum value of sin  is 1; the maximum value of sin  is 1.

In fact, for any value of , 1  cos   1; the minimum value of cos  is 1; the maximum value of cos  is 1.

In fact, for any value of , tan  has neither a maximum nor a minimum; the graph of tan  is discontinuous at  = 90, 270, 450, 630, …

Making use of the above facts, we can find the maximum and minimum values of a trigonometric function. e.g. Find the maximum and minimum values of y = 2 sin x. Maximum value of y = 2 (1) = 2 Minimum value of y = 2 (1) = 2

Follow-up question Find the maximum and minimum values of y = 5 – 3 cos x. ∵ ∴ The maximum value of y ∵ –1 ≤ cos x ≤ 1 ∴ 3 ≥ –3cos x ≥ –3 ∴ 8 ≥ 5 – 3cos x ≥ 2 The minimum value of y