Unit 3 - 1 Trigonometric function and equations
What are the four steps to use substitution to solve these equations?
- Make a substitution for ax. For example, let A=ax. - Change the interval for x to the interval for A by multiplying through by a. - Solve the equation in the usual way. - Divide the solutions through by a to find all possible values of x in the required range.
How do we solve an equation in the form sin(x) = a for multiple values of a?
- Use your calculator to find the first solution x1=sin^−1 a The second solution is given by x2 = 180°−x1 All other solutions are found by adding or subtracting multiples of 360° to x1 and x2
What does sin(-x) equal?
-sinx
What range are the principle values in given from the calculator?
0°≤x≤180° for cos^−1 and in the range −90°≤x≤90° for sin^−1 and tan^−1
What is the cos graphs amplitude?
1
What does tan^2x + 1 equal?
1/cos^2x
As the angles in a right angled triangle add up to 180 degrees, if one of the two acute angles is theta, what does the other one equal to?
90 degrees - theta
What does Cosine theta equal?
Adj/hyp
Where does tanx cross the y axis and x axis?
At the origin and crosses the x axis every 180 degrees
How is it easier to answer these questions?
By sketching a graph
How are positive and negative angles generally measured in mathematics?
From the positive x-axis moving anticlokcwise and negative clockwise
Where does the cos graph cross the y axis and the x axis?
It crosses the y− axis at (0,1) and the x− axis at (90°,0) every 180° thereafter
What is the period of the cos graph?
It has a period of 360°.
How would you solve an equation that looks like this: 6cos^2x−5cos x+1=0?
It is is a disguised quadratic, so cosx = c. Remember that cos2x means (cos x)2. If we substitute c for cos x in the trigonometric equation 6cos2x−5cos x+1=0 we can write it as 6c2−5c+1=0. Factorising this quadratic gives us (3c−1)(2c−1)=0 so c=13 or 12. Substituting the cos x back in gives us two trigonometric equations to solve: cos x=13 and cos x=12 Using the inverse cos button on the calculator gives us that x=70.5° (to 1 d.p.) or 60°. Other solutions in any given range can then be found by sketching the graph of y=cos x.
Where is the line of symmetry in the cos graph?
It is symmetrical about the y− axis
What does Tangent theta equal?
Opp/Adj
What does sine theta equal?
Opp/Hyp
What are the three basic trignometric ratios?
SOH CAH TOA
What is the equation of tan theta in a unit circle?
Sin theta / cos theta
where does the graph of sin start and what are its ranges/amplitude?
Starts at 0 and ranges from 1 to -1
How is the graph of sin(2x) different from sin(x)?
The x values are squashed by a factor of 2, so there are twice the amount of x intercepts in the given range
What are the problems with these definitions?
They are limited to define angles under 90 degrees
How do you solve sinx(3cosx-2)=0 for a range 0 to 360 degrees?
Treat it as two separate equations, sinx=0 and cosx=2/3, then find the points both intersect the x axis within the range.
How can we solve sin(0.5) for the principle value?
Type sin^−1 (0.5) into calculator and it will say 30
When you are given an equation that contains a tan x term as well as terms in sin x or cos x, what can it be useful to do to solve the equation?
Use the identity tan x=sinx/cos x to eliminate the tan x term, leaving you with an equation that is just in terms of sin x or cos x.
How do we solve an equation in the form cos(x) = a for multiple values of a?
Use your calculator to find the first solution x1 = cos^−1 a The second solution is given by x2 = −x1 All other solutions are found by adding or subtracting multiples of 360° to x1 and x2
How do we solve an equation in the form tan(x) = a for multiple values of a?
Use your calculator to find the first solution x1 = tan^−1 a All other solutions are found by adding or subtracting multiples of 180° to x1
What is the easiest method of solving equations in the form: in ax=k , cos ax=k and tan ax=k (where a and k are fixed numbers)?
Using substitution
What must you remember, when answering a question like 3sin^2(x)=2?
When squarerooting 2/3, there are positive and negative answers
If the solution to a trignometric equation = 0, are there solutions?
Yes there are , for example for sinx = 0: 0 degrees, 180, 360...
What is a unit circle?
a circle with a radius of 1
How do you use substitution to solve equations in the form sin, cos or tan(ax+b)=k (where a, b, and k are fixed numbers).?
a substitution for ax+b is made to change the interval before solving the equation in the usual way. Remember when altering the solutions to fit the actual range at the end, to add or subtract b first then divide by a.
What does cos x equal?
cos (-x)
What values is the graph of tan x undefined?
cos x=0 when x=±90°,x=±270°,x=±450°
what values on the graph does cos x equal as it is periodic and has a line of symmetry on the y axis?
cos x=cos(−x) cos x=cos(360+x) And in general, cos x=cos(x±360n°)=cos(360n°−x) where n is an integer.
With reference to the above, what does Cos Theta equal?
cos θ = sin(90°−θ)
As cosx has rotational symmetry around 180 degrees what does cos(180°+x) equal?
cos(180°−x)=cos(180°+x)=−cos x
What are the undefined lines (asymptotes identified by on the graph)?
dotted lines
When does sin cross the x axis?
every 180 degrees
How often does sin repeat itself and what do we call this?
every 360 degrees, periodic
How would you find all the solution to tan x=k?
if the solution given by the calculator is α then x=α±180n° where n is an integer.
What is the difference between the graph of y=sinx and y=cosx?
it's the same as the graph of y=sin x but translated 90° to the left so cos x=sin(x+90°).
What are the periods of the tan x graph?
period of 180°.
How are sin and cosine related?
sin x = cos(90°−x) = cos(x−90°) cos x = sin(90°−x) = sin(x+90°)
For which values is tanx=0?
sin x=0 when x=0°,x=±180°,x=±360°... and so these are the values for which tan x is 0.
What values on the graph does sin x equal as it is periodic?
sin x=sin(180°−x) sin x=sin(360°+x) sin x=sin x±360n°=sin(180°−x)±360n° where n is an integer.
What does sin θ equal?
sin θ=cos(90°−θ)
As sinx has rotational symmetry around 90 degrees what does sin(180°+x) equal?
sin(180°+x)=sin(−x)=−sin x
What is pythagoras' theorem with regards to trig of a unit circle?
sin^2x + cos^2x ≡ 1
How can you rearrange that equation to be useful?
sin^2x ≡ 1−cos^2x and cos^2x ≡ 1−sin^2x
therefore what does tanx equal?
tan x = tan(θ+180°) = tan(θ+360°) =...= tan(x+180n°) where n is an integer.
When is tanx undefined?
tan x is undefined at x=±90°,x=±270°,x=±450°... so there are asymptotes on the graph for these values.
As the graph of tan x has rotational symettry aroudn the origin what does tan(-x) equal?
tan(−x) = −tan x .
Where are the graph of sines line of symmetry?
x= 90° + 180n° In other words, at x=90° and every 180° before and after
