Download Trigonometry - NW 14-19

Trigonometry
Unit 4:Mathematics
Aims
•
•
Introduce
Pythagoras
therom .
Look at
Trigonometry
Objectives
• Investigate the
pythagoras therom.
• Calculate trigonometric
functions.
2
Mae'r onglau triongl yn adio i 180
180
– Using our knowledge of the
internal angles of a triangle
we can find an unknown
angle in a triangle given the
other two angles.
– Using Pythagoras’ theorem
we can find the length of a
side given the lengths of the
other two sides in a right
angled triangle.
– Where we have a
combination of lengths of
sides and angles in a right
angled triangle, we can use
trigonometry to calculate the
lengths of another side or an
unknown angle.
About 2,500 years
ago, a Greek
mathematician
named Pythagorus
discovered a
special relationship
between the sides
of right triangles.
Pythagorus realized that if you
have a right triangle,
5
3
4
and you square the lengths of the two
sides that make up the right angle,
5
3
4
2
3
4
2
and add them together,
5
3
4
3 4
2
2
you get the same number you would
get by squaring the other side.
5
3
3 4 5
2
4
2
2
And it is true for any right triangle.
6  8  10
2
2
2
10
8
36  64  100
6
Find the length of a diagonal of the
rectangle:
15"
?
8"
Find the length of a diagonal of the
rectangle:
15"
b=8
c
?
a = 15
8"
a b c
2
2
2
15  8  c
2
2
2
225  64  c
b=8
c  289
c  17
2
2
c
a = 15
Practice using
The Pythagorean Theorem
to solve these right
triangles:
12
c
5
b
15
12
b
10
26
Prawf pye 1 & 2
Test pye 1 & 2
Right angled triangle
– In a right angled triangle the
side opposite the right
angle (the longest side) is
known as the hypotenuse.
– The side opposite a given
angle (other than the right
angle) is known as the
opposite.
– The side next to the
given angle is
known as the
adjacent.
opposite
hypotenuse
adjacent
– Label each side on the diagrams below.
– 1.
hypotenuse
–
2.
adjacent
hypotenuse
opposite
opposite
– 3.
–
opposite
hypotenuse
adjacent
adjacent
4.
adjacent
Prawf trig 1
Test trig 1
hypotenuse
opposite
Sine (sin) ratio
opposite
sin x 
hypotenuse
hypotenuse
– In a right-angled triangle
opposite
x
b
– Calculate the length b in the diagram
opp
b
sin 52 

52
hyp
23
 b  23 sin 52
23
–
–
23  sin 52 =
 b = 18.1 (to 3 significant figures)
Prawf 3
Test 3
Calculate the angle x in the diagram
7
– x
opp
sin  
hyp
3
adj
cos  
hyp
opp
tan  
adj
Sin X = 3 = 0.428571
7
Sin -1 (0.428571) =25.377°
– Calculate the length p in the diagram
9.3
17.3
p
Cosine (cos) ratio
adjacent
cos x 
hypotenuse
– In a right-angled triangle
hypotenuse
x
adjacent
– Calculate the length c in the diagram
opp
sin  
hyp
15
–
c
–
–
–
27
adj
cos  
hyp
opp
tan  
adj
– Calculate the angle y in the diagram
opp
sin  
hyp
9.7
adj
cos  
hyp
–
6.1
y
opp
tan  
adj
Cos y = adj (6.1) = 0.628866
hyp (9.7)
Cos -1 (0.628866) = 51° 2”
Calculate the length q in the diagram
9.3
q
34
Tangent (tan) ratio
opposite
tan x 
adjacent
– In a right-angled triangle
opposite
x
adjacent
• Calculate the length d in the diagram
opp
sin  
hyp
–
37
adj
cos  
hyp
–
40
opp
tan  
adj
d
Tan 37 = opp (d) =
adj (40)
Opp (d) =tan 37 x 40 = 30.142
– Calculate the angle z in the diagram
opp
sin  
hyp
2.6
–
z
4.1
adj
cos  
hyp
opp
tan  
adj
Tan z = 2.6 = 0.6341
4.1
Z = tan-1 (0.6341) = 32.381
32.381 – 32 = .381 x60 =22.841
32° 22min
– Calculate the length marked r in the diagram
opp
sin  
hyp
79.2
adj
cos  
hyp
62
r
opp
tan  
adj
Prawf trig 2
Test trig 2