Trigonometry-5 Examples and Problems. Trigonometry Working with Greek letters to show angles in...
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Trigonometry-5
Examples and Problems
Trigonometry
Working with Greek letters to show angles in right angled triangles.
Exercises
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
Well Done!
