Download strand 2 review

STRAND 2 REVIEW
Working
through
these
questions
will
help
you
assess
your
understanding of the learning outcomes listed here:
Question type
Review
Level
All
Learning outcome
use of the theorem of Pythagoras to solve problems (2D
only)
use the sine and cosine rules to solve problems solve
problems that involve calculating the cosine, sine and
tangent of angles between 0˚ and 90˚
graph trigonometric functions of type
aSin n , aCos n
for
a, n N
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Q. The diagram represents a plan for a new playground slide
B
5m
D
1m
C
0.5m
2.5m
m
E
F
G
A
Calculate the height of the slide.
Health and safety will not allow a slide in the playground if the angle between the ground
and the slide is bigger than 27˚.
Does the planned slide comply with safety regulations?
Look at the sketch of the slide. Can you see where the steps
would be? Where the bit you slide down would be? Can you
identify the ground? Where is the angle that must not be
greater than 27 ? The angle between the slide and the ground
How many triangles can you see in the diagram?
Can any of these triangles help you to find the height of the
slide or the size of the angle that must not be bigger than 27 ?
Can you use Pythagoras Theorem? Can you use the
trigonometric ratios Sin , Cos or Tan ?
Will the slide comply with Health and Safety regulations? Make
a decision based on your evidence
2
Q. Can a triangle have more than 2 obtuse angles? Explain.
What is an obtuse angle?
If you add two obtuse angles together what is the smallest
number you can get? When you add all the angles in a
triangle together what must they add to ?
Q. John is painting the house. He needs to use a stick to stir the paint.
He doesn’t want to get paint on his fingers so wants to make sure the stick is long enough
A diagonal line drawn on the diagram of the paint tin represents to smallest stick that could
be used. Draw this line.
Using a scale of 1cm=4cm
Take measurements from the diagram and use
these to calculate the minimum length the
stick can be.
Draw the diagonal line. Can you see a triangle?
What is the height of this triangle? The base? Is it
right angled? Try using both Pythagoras Theorem
and the trigonometric ratios Sin , Cos or Tan to
help find the length of the stick
Do you get the same answer?
3
Q. The diagram below shows two submarines, B and C and their distances from a sonar
station marked A. Point D represents an obstacle 2km from B, and 5km from A. How far is
submarine C from the obstacle?
27 km
6 km
5 km
2 km
Look at the diagram mark the distance you are being asked to find as x
How many triangles can you see?
Use the flow chart from the presentation found at
https://emea67395290.adobeconnect.com/_a858841383/maths/ to help decide on a solution
strategy
Don’t forget to check if your synthetic geometry theorems or axioms can help
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MIND MAPPING
CAN YOU SEE A
NO
TRIANGLE?
WILL MY SYNTHETIC GEOMETRY
THEOREMS OR AXIOMS HELP?
YES
NO
IS IT RIGHT
CAN I USE THE
ANGLED?
SINE
RULE?
YES
NO
YES
CAN I USE
PYTHAGORAS’
NO
YES
WHAT WILL
THIS HELP ME
FIND?
WHAT WILL
CAN I USE THE
THIS HELP
COSINE RULE?
CAN I USE THE
YES
TRIGONOMETRIC
RATIOS – SINE,
COSINE, TAN?
WHAT WILL THIS
HELP ME FIND?
YES
WHAT WILL THIS
HELP ME FIND?
5
Q. Use the notes on the next page to helpyou with this.
(a) Show that sin (x + 2π ) = sin x
(b) Draw the graph of the function y = sin x where x
(c)
R in the domain -2π ≤ x ≤ 2π
On the same axis sketch the graph of the function y = Sin 3x.
What is the period of this function?
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FACT
When the values of a function regularly repeat themselves, we say that the function is
periodic. By this definition is the function sin x periodic?
Back up your decision with a
table
graph
DEFINITION
If, for all values of x, the value of a function at (x + p) is equal to the value at x. That is:
If f(x + p) = f(x) then we say that the function is periodic and has period p.
What is the period of sin (x+2 )?
Back up your decision with
a table
a graph
an equation [Try to write sin (x + 2π) in a different way? This will help you to show
that sin (x + 2π) = sin x. Look in the tables for help on how to write sin (x + 2π) in a
different way]
Think about the following:
sin1x, sin 2x, sin 3x…….sin ax. What does the a stand for?
Use Geogebra to draw these graphs. What is different about each one? Focus on the x axis
in the interval between 0 and 2π .How many times does the pattern repeat itself in this
interval?
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What does the a stand for?
Number of times pattern repeats itself in
the
0 -2π interval
Sin1x
Sin2x
Sin3x
Sin ax
Conclude
What is the period of the function Sin ax?
Think: Can you decide on the period of sinax depend on the period of sinx?
8
Question type
Review: Strand 2
Level
LC /HL
Learning outcome
This question provides you with an opportunity to display evidence
that you can
calculate the area of a triangle
solve problems involving the length of the perpendicular
from a point to a line
recognise that x2+y2 +2gx+2fy+c = 0 represents the
relationship between the x and y co-ordinates of points on
a circle with centre (-g,-f) and radius r where r = √ (g2+f2 –
c)
recall that a diagonal divides a parallelogram into two
congruent triangles.
On the grid provided plot the points A (0, 4) B (4, 0) and C (-2,-2).
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a) (i) Find the mid-points of AB, BC and AC label these points X, Y and Z respectively
Join X to Y, Y to Z and X to Z.
(ii) What can you say about the area of ∆XYZ in terms of the area of ∆ABC? Show that
this is true
b) Find the equation of the circumcircle of ∆ABC.
Note to Students:
There are several ways you may approach this question.
a) In order to find the midpoint you may use the co-ordinate geometry formula to find the
co-ordinates of the midpoints or you could simply measure the length and half it. Which
method do you think will give you the most accurate result? Why is this? You should note
that the area of ∆XYZ is ¼ that of ∆ABC. Why is this true? The truth of this statement
can be shown in various ways. You may calculate the area of each triangle, or you can
show that AXYZ is a parallelogram which means that ∆AZX and ∆XZY are congruent
triangles. Similarly ZXBY is a parallelogram making ∆XZY and ∆XYB congruent pairs. A
similar argument means that ∆CZX is congruent to ∆YXB. So ∆ ABC is made up of 4
congruent triangles meaning that ∆ XYZ is ¼ the area of ∆ABC
b) The equation of the circumcircle can be found by substituting the vertices into the
general equation of the circle, or you could construct the circumcentre and calculate the
length of the radius and substitute these values into the equation.
Try all of the methods mentioned in the note
Think about other ways the question could be asked.
Write some notes to help organise your thoughts about circles, triangles and circumcircles and how
they are related.
Follow the link http://www.mathopenref.com/trianglecircumcircle.html to investigate properties of
circumcircles of triangles. When is the centre of the circle inside the triangle? outside the triangle?
Visit http://www.mathopenref.com/constcircumcircle.html here you will get help on how to construct
the circumcircle of a triangle. Think about the mathematics of each step why will these steps result in
you being able to construct the circumcircle of a triangle?
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