year 9 trigonometry dr j frost ([email protected]) last modified: 2 nd november 2014
TRANSCRIPT
Frost Childhood Story
x
y
θ
(a,b)
r
I was trying to write a program that would draw an analogue clock.
I needed to work out between what two points to draw the hour hand given the current hour, and the length of the hand.
Starter
3
4
x
13
5
y
Question: What do we require for the theorem to work?
What you already know
30°4
x
y
What is x and what is y?
What you’re less likely to know...
30°
hypotenuse
adjacent
opposite
Names of sides relative to an angle
?
?
?
60°
x
y
z
Hypotenuse Opposite Adjacent
x y z
√2 1 1
c a b
45°
1 √2
1
20°
a
c
b
? ? ?
? ? ?
? ? ?
Names of sides relative to an angle
𝜽𝒐
𝒉
𝒂
sin (𝜃 )=𝑜h
cos (𝜃 )=𝑎h
tan (𝜃 )=𝑜𝑎
“soh cah toa”
! sin, cos and tan are functions which take an angle and give us the ratio between pairs of sides in a right angle triangle.
Sin/Cos/Tan
?
?
?
Example
45
opposite
adjacent
Looking at this triangle, how many times bigger is the ‘opposite’ than the ‘adjacent’ (i.e. the ratio)
Ratio is 1 (they’re the same length!)
Therefore:
tan(45) = 1
?
??
40 °
4
𝒙
Find (to 3sf) 20 °
7
𝒙
Step 1: Determine which sides are hyp/adj/opp.Step 2: Work out which trigonometric function we need.
More Examples
𝑥=3.06 𝑥=2.39? ?
𝐜𝐨𝐬 (𝟒𝟎)=𝒙𝟒
𝐬𝐢𝐧 (𝟐𝟎 )= 𝒙𝟕
60 °𝒙
12
30°
4
𝒙
More Examples
𝑥=13.86?
𝑥=6.93?
𝒔𝒊𝒏 (𝟔𝟎 )=𝟏𝟐𝒙
𝒕𝒂𝒏 (𝟑𝟎 )=𝟒𝒙
Exercise 1Find , giving your answers to . Please copy the diagrams first.
𝟕𝟎° 15
𝒙
1
𝟒𝟎°
22
𝒙
a b
𝟖𝟎°20
𝒙
𝟓𝟓°10
𝒙
𝟕𝟎°4
𝒙
𝟕𝟎° 𝒙𝟒
c
d e f
I put a ladder 1.5m away from a tree. The ladder is inclined at above the horizontal. What is the height of the tree?
Ship B is 100m east of Ship A, and the bearing of Ship B from Ship A is . How far North is the ship?
Find the exact value of .
2
3
N1 𝟑𝟎°𝒙
𝒙+𝟏
𝑥=14.1
𝑥=16.9 𝑥=20.3
𝑥=7.00𝑥=11.0
𝑥=11.7
?
? ?
?
??
?
?
[IMC] The semicircle and isosceles triangle have equal areas. Find .
N2
?𝒙=𝟏
√𝟑−𝟏?
Frost Childhood Story
x
y
θ
(𝑎 ,𝑏)
𝑟
So what is ?
30 °
4
𝒙
RECAP: Find x
𝑥=4
tan 30=4 √3𝑜𝑟 6.93?
𝒂3
5
But what if the angle is unknown?
?
?
We can do the ‘reverse’ of sin, cos or tan to find the missing angle.
cos−1( 45 )cos−1( 54 )cos−1( 45 )sin−1( 54 )
What is the missing angle?
𝟓
𝟒𝒂
cos−1( 12 )sin−1 (2 )tan−1 (2 )tan−1( 12 )
What is the missing angle?
𝟏
𝟐
𝒂
cos−1( 35 )sin−1( 35 ) tan−1( 35 )sin−1( 53 )
What is the missing angle?
𝟓𝟑
𝒂
cos−1( 23 )sin−1( 23 )sin−1( 32 )tan−1( 23 )
What is the missing angle?
𝟑
𝟐
𝒂
The Wall of Trig Destiny
2
3
θ
1
3
“To learn secret way of math ninja, find θ you must.”
1 1θ
6
θ
8
1
2 3
4
θ
𝜃=33.7 °
𝜃=70.53 °
𝜃=45 °
𝜃=48.59 °
?
?
?
?
Exercise 2
𝜃
7
4𝜃
3
5𝜃
21
𝜃5
Find , giving your answer to 3sf.
4𝜃
3
4
13
1
11
1
1
𝜃1𝜃2𝜃3
The angles form a sequence. Give the formula for the th term of the sequence.
𝜃=cos− 1( 513 )=67.4 °
𝜃=55.2 °
𝜃=31.0 °𝜃=30.0 °
𝜃=51.3 ° 7
60 ° 𝜃
10𝜃=37.3 °
?
?
??
?
??
1
N
2
3
4
5
6
Real-World Example
x40° 60°
3mFind x
3.19m
Trig Challenge
Stage 1 The kind of problems that you’re likely to find in a landmark exam.
Stage 2
Stage 3
Problems you might find as a harder landmark question or in a GCSE exam.
More difficult problems that will help you become adept mathematicians.
Level 2 – Q3
Level 3 – Q1