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Topic 2: Trigonometry
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Achievement Standard 91575: Apply trigonometric methods in solving problems.
This standard is an Internal worth 4 credits.
Week 8: Trigonometry definitions and transformations of trig graphs.
This needs to be completed by Wednesday 30 March.
Learning Objectives
1. To understand the formal definition of the 3 trig functions from the unit circle.
2. To know the basic features of the graphs of the 3 trig functions.
3. To understand transformations of trig graphs and how these transformations are linked to the
equation.
4. To be able to find the amplitude and period from a trig equation.
Topic
Notes
Definitions from
the unit circle.
The unit circle is a circle radius 1 centre (0,0).
As a point rotates
around the
circumference of the
circle, the horizontal
component is defined
as cosθ (the pink line),
the vertical
component is defined
as sinθ ( the blue line)
and if the horizontal
distance is fixed at 1
the distance to the
intersection with the
tangent to the circle is
defined as tanθ (the
red line).
The basic graph of any function can be transformed
using reflections, changes of scale and translations.
This work was covered in the Level 2 graphs
achievement standard. We are now looking at the
effect of these transformations on the trig graphs.
Graphs of y=sinθ,
y=cosθ and y=tanθ
and their basic
features.
Transformations of
trig graphs
If we put all the possible transformations together in
one equation we get equations like
𝑦 = 𝐴𝑠𝑖𝑛𝐵(𝑥 + 𝐶) + 𝐷
A changes the amplitude
B changes the period
C translates horizontally
D translates vertically
Finding period and
amplitude from
transformed
equations.
Before we start using these trig equations to model
real life situations we need to be able to find the
period and amplitude from the equation.
From the equation
𝑦 = 𝐴𝑠𝑖𝑛𝐵(𝑥 + 𝐶) + 𝐷
The amplitude is equal to A and
2𝜋
The period=
𝐵
Delta Mathematics 2nd
Edition Old Book
The basic graphs and
their features are
summarised on Page
304-305.
Also go to the websites
listed on the next page
to watch the
construction of the 3
graphs from the unit
circle.
Delta Maths NCEA Level 3
New Book
The basic graphs and their
features are summarised
on Page 72.
Read through pages 305
to 308 and the summary
of these transformations
on page 308.
Read through pages 73 to
76 and the summary of
these transformations on
page 77.
Use the link on the next
page to an online
graphing package to
have a play with
transforming the trig
graphs. The graphs are
easier to see than when
using your graphical
calculators.
Use the link on the next
page to an online graphing
package to have a play
with transforming the trig
graphs. The graphs are
easier to see than when
using your graphical
calculators.
Page 309-310
Ex 33.2
Q 2, 3, 9, 10, 11 and 12
Page 79-80
Ex 5.02
Q 2, 3, 10, 11, 12 and 13
Also go to the websites
listed on the next page to
watch the construction of
the 3 graphs from the unit
circle.
Useful web links for trigonometry definitions and graphs.
Description
Interactive graphing of
sine and cosine from the
unit circle
Web link
https://tube.geogebra.org/student/m1525
Click on either the sine or cosine curve and click the
trace on. Then use the mouse to rotate the point around
the unit circle. A trace of dots is left to show the curve
being drawn. The sine curve is red and the cosine is blue.
You can erase the trace or have both curves drawn on
the same axes.
Graphing of sine and
cosine from the unit
circle
https://tube.geogebra.org/student/m220433
Select sine, cosine or both and click start animation to
see the curves drawn as the point rotates around the
unit circle.
Video explaining the 3
trig graphs.
https://www.youtube.com/watch?v=OjHgoZOdRKM
Video explaining special
triangles
https://www.youtube.com/watch?v=ZT65zTqMjXQ
The special triangles are given on the formula sheet and
this video explains how to use them to find exact values
for the 3 trig fucnctions and multiples of 30, 45 and 60
degrees.
Online graphing program
https://www.desmos.com/calculator
Use this graphing package to enter equations of the
following form and play with changing the values of A, B,
C and D. Make sure you understand how each value
changes the standard trig graph.
𝑦 = 𝐴𝑠𝑖𝑛𝐵(𝑥 + 𝐶) + 𝐷
A changes the amplitude
B changes the period
C translates horizontally
D translates vertically
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