Functions & Graphs: Trigonometric Functions


Hello and welcome back to another Sqa Past Papers Website. we’re going to be continuing our functions and graphs topic and we’re going to be looking at the final type of functions of higher math’s and those are trigonometric functions so so far and higher 

 

we have looked at several types of functions we have looked at composite functions we’ve looked at inverse functions we’ve looked at exponential functions and we’ve even looked at logarithmic functions today we’re going to be doing i think you’ll be happy 

 

when i say this probably the easiest type of functions and those are trigonometric functions and the reason they are the easiest is well there’s not really much to them and we’ve done them quite a lot at national five now you can see from this diagram
(SQA Higher) Functions & Graphs: Trigonometric Functions


that i’ve drawn on screen we have what looks to be a graph that is going to keep going up and down and up and down and it keeps on doing this this graph will keep going up and down between these max values and these minimum values we call this graph a periodic graph because it repeats its pattern


(SQA Higher) Functions & Graphs: Trigonometric Functions

so that’s an important definition a graph that repeats is called periodic and this is because we call the period of the graph one period of it and it will keep repeating that period so this graph goes up down and back up and it will repeat that period it will keep doing that and i think you can already tell from this graph that it is the sine graph so this graph in this case is y equals sine x

(SQA Higher) Functions & Graphs: Trigonometric Functions
now if the repeating pattern has a minimum and a maximum value then half of the distance between these is called the amplitude so as you can see this graph will never get higher than this value here which we call the maximum value and it will never go lower on the y-axis than this value down here so we call this the minimum value and halfway between there the distance from the x-axis to the max value or the distance from the x to the min value is called the amplitude now we have looked at three trigonometric functions at national five





and we’re only going to look at the same three at higher advanced higher you will look at more complex trigonometric functions but higher these are the three main ones we’re talking about and you might recognize them from when we talked about them at national five and we call these our trig functions 

(SQA Higher) Functions & Graphs: Trigonometric Functions

we’ll start with this one on the left our function in this case is y equals sine x this is the sine wave the sine curve which we can see starts at zero reaches a max value at one goes back down to zero 

 

then reaches a min value at minus one before finishing back up at the end of its period back at zero and as we can see after this one wave after this period it will go back up and do the exact same thing again so in this case we say that the amplitude

 

 which we talked about is the distance from uh it’s just half of the distance from the max value to the min value so the distance from the x to the max we say that the amplitude is one now the period in this case is the distance it takes along the x-axis to complete one wave or one motion of the function and in our case it’s 360 degrees so the period is going to be 360. it looks a bit like a 6. 360 degrees 

(SQA Higher) Functions & Graphs: Trigonometric Functions

now we’ll move on to the second graph which we can see is going to be our cause graph cos x now some people put degrees it’s completely up to you you don’t need to it’s implied that it’s the degrees and we can see for our cause graph it starts at -1 which is actually its max value goes down to 0 then goes down to -1 back up to 0 before finishing its period at -1 so in this case we say that the amplitude it’s going to be the exact same it’s simply just going to be one it’s half of the distance from the max to the min so it’s one and in this case we also have the period 360 degrees just like the period of a sine wave 

(SQA Higher) Functions & Graphs: Trigonometric Functions

now if we look over at our final graph it’s the weird one in this case the graph is tan x which if you remember we said tan x is equal to sine x over cos x it’s not really necessary to write this i’m just reminding us so things make sense now the graph of tan x is a bit weird it starts at zero 

 

and goes up kind of like an exponential and then after it goes up it shoots back down and curls back up a kind of like a log and then it has this little split here which is actually the end of its period and then it starts back up again and repeats in between 

 

these two dotted lines so we can see in this case that our period is not going to be the same our period in this case is going to be 180 degrees because if we sort of shield out these two dotted lines we can say that this is the same as this so it’s period simply 180 degrees

 

but what about its amplitude well its amplitude is undefined this bit here will keep going up and up and up like an exponential and this bit here will keep going down and down and down and down and down like a log so the amplitude of a trigonometric function is simply undefined it does not have an amplitude and that’s all we’ve got on trigonometric functions.

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