Trig ch 6 Flashcards
How to find the value of cosec, sec and cot?
1/Cosx,secx, cot.
Work out the value on the sin, cos and tan and do the reciprocal of that value.
What does the cosec curve look like?
inverted parabola which intersects sin graph at max and min points.
Asymptotes beside where the U’s level off (asymptote every 180*)
What does the sec curve look like?
asymptotes at 90* and 270*.
n shaped parabola at minimum of cos graph.
Half parabolas either side of the asymptotes at 90 and 270.
What are two differences between sec and cosec curve?
In cosec asymptotes are every 180* (180,360), whilst in sec asymptotes are every 180* (90,270).
In cosec, there is a U and an n.
In sec, there is an l, n and l shape.
How do you know where the asymptotes will be on reciprocal curves?
Where the sin/cos/tan x=0 as this will cause the reciprocal curve to be 1/0 which is undefined and needs asymptote.
Why do the reciprocal trig curves have the shapes they take?
Because as the decimal values of the original trig graph gets smaller the larger the reciprocal graph gets.
It then gets to 0 which causes it to be undefined and have an asymptote.
What does the cot graph look like?
Reflected tan graph, asymptotes stay in the same place.
What are the maxima/ minima and asymptotes for cot graph?
asymptotes at 180 and 360 then negative cubics in between these asymptotes.
How to solve equations involving sec, cosec and cot
Convert them to expressions in times of sin, cos and tan
What is the domain and range of sec graph?
Domain = x is all real numbers but x cannot equal 180n-90. Range = f(x) is greater than of equaled to 1 and f(x) is less than -1
What is the domain and range of cosec graph?
Domain = x is all real values and not equal to 180n Range = f(x) greater than or equal to 1 and f(x) is less than or equaled to 1.
What is the domain and range of cosec graph?
Domain = x is all real values not equal to 180n Range g(x) is all real values.
What is the domain and range of the cot graph?
Domain= x is all real values except from 180n Range= f(x) is all real values
why would cos^3x=8 have no solutions?
once you cube root cos^2x=+-2
As cos graph has max at 1 and min at -1 there can be no solutions.
What to do when solving when there is a 2x inside the angle?
- Change the range by multiplying by 2.
- Find all the solutions in this range.
- Then divide all the solutions by 2 and see which ones fit the initial range.
How to solve trig equations involving reciprocal trig?
- make it 1/sin/cos/tan
- then using denominator rule switch it with value
- do the inverse to find the first angle
- check if the value of y is positive or negative
- find all solutions for given interval
How to solve equations that are in a negative range?
Go in an anti clockwise direction and check whether you would continue to add negative or add positive.
How large is an obtuse angle?
Between 90. and 180
How large is a reflex angle?
Between 180* and 360*
How large is an acute angle?
Less than 90*
How to workout a question that is like given that tana is x/y what is seca?
What is important about these questions?
Draw out triangle to find the value of the reciprocal of the one you want to find out.
They will tell you the angle is obtuse, reflex or acute so you must draw out cast and find out what the angle will be in the given quadrant.
How do you derive the two other Pythagorean identities from Sin^2+Cos^2=1
What is the difference between them?
To get the tan^2+1=sec^2 divide through by cos^2.
To get cos^2+1=cosec^2 divide through by sin^2 as you want to have a cos/sin which is the same as cot.
you divide one by cos^2 and divide the other by sin^2
What are the inverse functions of sin, cos and tan?
Arcsin
Arcos
Arctan
How does the rule of functions link to the inverse trig functions
The inverse of the trig functions which would become one-many which is not a function.
What is the condition on finding the inverse of sin cos and tan
You must restrict the theta values to when the sin, cos, tan is not many to one.
In what line is the reflection of an inverse function done in?
Reflected in the line y=x
How do the domain and range of trig and arc trig relate?
The domain of trig becomes the range of arctrig
The range of trig becomes the domain of arctrig
How do you sketch the arctrig functions?
You swap the x and y coordinates of the original trig function.
What is the domain and range of sin and arcsin.
Sin :Domain (-Pi/2 to Pi/2) Range (-1 to 1).
Arc sin: Domain (-1 to 1) Range (-Pi/2 to Pi/2)
What is the domain and range of cos and arccos.
Cos: Domain (0 to Pi) Range (-1 to 1)
Arcos: Domain (-1 to 1) Range (0 to PI)
Why do the domains and ranges of arctrig functions have to be restricted?
Because the trig curves are not many to one in these values.
Therefore, the inverse can be a function.
What is the domain of range of tran and arc tan?
What is unique about tan in this instance?
Tan: Domain (-Pi/2 to Pi/2) Range : all real values
Arctan: Domain: all real values Range: (-pi//2 to pi/2).
The vertical asymptotes of tan at -pi/2 and pi/2 become horizontal asymptotes at this location.
