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What you have Learned
In any right angled triangle we can find the ratio of sides.
The ratios are:
O
A
H
OA
Tan =
OH
Sin =
AH
Cos =
Some Old Hens
Cackle And Howl
Till Old Age
In triangle ABC angle A = 25 and AB = 7,8 units. What is the length of BC?
Let BC = x
Example-1
25Ask yourself – we are working with angle A:
A
B
C
x7,8
Do we know the opposite side to 25?Do we know the adjacent side to 25?Do we know the hypotenuse?
Yes - xNo
Yes – 7,8
Work only with what you know or want to find.
We don’t know the adjacent side, so leave it out.
Which trig ratio does not use the adjacent side?
Example-1
25A
B
C
x7,8
OA
Tan =
OH
Sin =
AH
Cos =
x
x
This ratio does not use the adjacent side.This is the ratio we must use.
Start with the sine ratio and put in the values.
Example-1
25A
B
C
x7,8
OH
Sin =
x7,8
Sin 251
= To get x on its own multiply both sides by 7,8.
7,81
7,81
Now simplify.xSin 25 =7,8Find sin25 on calculator.
x0,4226
=7,8x = 3,296
In triangle PQR angle P = 37 and PQ = 11,3 units. What is the length of PR?
Let PR = x
Example-2
37Ask yourself – we are working with angle P:
P
Q
Rx
11,3
Do we know the opposite side to 37?Do we know the adjacent side to 37?Do we know the hypotenuse?
NoYes - x
Yes – 11,3
Work only with what you know or want to find.
We don’t know the opposite side, so leave it out.
Which trig ratio does not use the opposite side?
Example-2
37P
Q
R
11,3
OA
Tan =
OH
Sin =
AH
Cos =
xx This ratio does not
use the opposite side.This is the ratio we must use.
x
Start with the cosine ratio and put in the values.
Example-2
37P
Q
R
11,3A
Hcos =
x11,3
cos 371
= To get x on its own multiply both sides by 11,3.
11,31
11,31
Now simplify.xcos 37
=11,3 Find cos37 on
calculator.x0,7986
=11,3 x = 9,024
x
In triangle PQR angle P = 52 and QR = 6,7 units. What is the length of PQ?
Let PQ = x
Example-3
52Ask yourself – we are working with angle P:
P
Q
R
x6,7
Do we know the opposite side to 52?Do we know the adjacent side to 52?Do we know the hypotenuse?
Yes–6,7No
Yes – x
Work only with what you know or want to find.
We don’t know the adjacent side, so leave it out.
Which trig ratio does not use the adjacent side?
Example-3
52P
Q
ROA
Tan =
OH
Sin =
AH
Cos =
x
x
This ratio does not use the adjacent side.This is the ratio we must use.
x6,7
Start with the sine ratio and put in the values.
Example-3
52P
Q
ROH
sin =
6,7x
sin 521
= To get x on its own on top multiply both sides by x.
x1
x1
Cancel the x’s.6,
7sin 52 =x
Divide both sides by sin52.
x = 8,502
x6,7
Simplify and find sin52.sin 52 sin 52x =
sin 526,7
=0,788
6,7
In triangle PQR angle P = 52 and QR = 6,7 units. What is the length of PR?
Let PR = x
Example-4
52Ask yourself – we are working with angle P:
P
Q
Rx
6,7
Do we know the opposite side to 52?Do we know the adjacent side to 52?Do we know the hypotenuse?
Yes–6,7Yes-x
No
Work only with what you know or want to find.
We don’t know the hypotenuse, so leave it out.
Which trig ratio does not use the hypotenuse?
Example-4
52P
Q
ROA
Tan =
OH
Sin =
AH
Cos =
x
x
This ratio does not use the hypotenuse.
This is the ratio we must use.
6,7
x
Start with the tan ratio and put in the values.
Example-4
52P
Q
ROA
tan =
6,7x
tan 521
= To get x on its own on top multiply both sides by x.
x1
x1
Cancel the x’s.6,
7tan 52 =x
Divide both sides by tan52.
x = 5,238
6,7
Simplify and find tan52.tan 52 tan 52x =
tan 526,7
=1,279
6,7
x
You have to find the height of a tree. From where you stand, 76 metres from
the tree, the angle to the top of the tree is 32.
A Typical Problem
Always draw a diagram.
32 Ground
TreeLine of sight
You don’t have to draw a work of art, just a simple sketch.
32
The angle is 32 and we are 76 metres from the tree. How high is the tree?
Let tree height = x
Problem-1
Ask yourself – we are working with angle 32:
x
76
Do we know the opposite side?Do we know the adjacent sideDo we know the hypotenuse?
Yes - xYes - 76No
Work only with what you know or want to find.
We don’t know the hypotenuse, so leave it out.
Which trig ratio does not use the hypotenuse?
Problem-1
OA
Tan 32 =
OH
Sin 32 =
AH
Cos 32 =
x
x
This ratio does not use the adjacent side.This is the ratio we must use.
x32
76
Start with the tan ratio and put in the values.
Problem-1
OA
Tan 32 =
x76
Tan 321
= To get x on its own multiply both sides by 76.
761
761
Now simplify.xTan 32
=76Find tan 32 on calculator.x0,624
8=76
x = 4,873
x32
76
A surveyor has to find the distance from point A to point B on the other side of a lake. The distance AC is measured to be 275 m, and angle A is 40.
A Typical Problem
If you are given a diagram, then don’t waste time drawing another one.
40
275
m
A
B C
In triangle ABC angle A is 40 and AC = 275 m. What is the length of AB?
Let AB = x
Example-4
Ask yourself – we are working with angle A:Do we know the opposite side to 40?Do we know the adjacent side to 40?Do we know the hypotenuse?
NoYes - x
Yes - 275
Work only with what you know or want to find.
We don’t know the opposite side, so leave it out.
40
275
m
A
B C
x
Which trig ratio does not use the opposite side?
Example-4
OA
Tan =
OH
Sin =
AH
Cos =
x
x This ratio does not use the opposite side.This is the ratio we must use.
40
275
m
A
B C
x
Start with the cos ratio and put in the values.
Example-4
AH
cos =
x275
cos 401
= To get x on its own multiply both sides by 275.
2751
2751
Cancel the 275’s.xcos
40=27
5Find cos40 and multiply.
x = 210,7m
40
275
m
A
B C
x
x0.766 =275