1. SANGEETH.S XI.B. Roll.No:50 K.V.PATTOM
TRIGONOMETRY
SANGEETH.S
XI.B.
Roll.No:50
K.V.PATTOM

2. When you have a right triangle there are 5 things you can know about it..
the lengths of the sides (A, B, and C)
the measures of the acute angles (a and b)
(The third angle is always 90 degrees) b C A a B

3. If you know two of the sides, you can use the Pythagorean theorem to find the other sideb C
A = 3 a
B = 4

4. And if you know either angle, a or b, you can subtract it from 90 to get the other one: a + b = 90
This works because there are 180º in a triangle and we are already using up 90º
For example:
if a = 30º
b = 90º – 30º
b = 60º b C A a B

5. But what if you want to know the angles?
Well, here is the central insight of trigonometry:
If you multiply all the sides of a right triangle by the same number (k), you get a triangle that is a different size, but which has the same angles:
k(C) b C b A
k(A) a a B
k(B)

6. How does that help us?
Take a triangle where angle b is 60º and angle a is 30º
If side B is 1unit long, then side C must be 2 units long, so that we know that for a triangle of this shape the ratio of side B to C is 1:2
There are ratios for every
shape of triangle!
C = 2
60 º
A = 1
30º B

7. But there are three pairs of sides possible!
Yes, so there are three sets of ratios for any triangle
They are mysteriously named:
sin…short for sine
cos…short for cosine
tan…short or tangent
and the ratios are already calculated, you just need to use them

8. So what are the formulas?
Tan is Opposite over Adjacent
Cos is Adjacent over Hypotenuse
Sin is Opposite over Hypotenuse
SOH
CAH
TOA
You can use this word if you
need to memorize the formulas!

9. Some terminology:
Before we can use the ratios we need to get a few terms straight
The hypotenuse (hyp) is the longest side of the triangle – it never changes
The opposite (opp) is the side directly across from the angle you are considering
The adjacent (adj) is the side right beside the angle you are considering

10. A picture always helps…
looking at the triangle in terms of angle b b
A is the adjacent (near the angle) C A
B is the opposite (across from the angle) B b
Near
C is always the hypotenuse
hyp
Longest
adj
opp
Across

11. But if we switch angles…
looking at the triangle in terms of angle a
A is the opposite (across from the angle) C A a
B is the adjacent (near the angle) B
Across
C is always the hypotenuse
hyp
Longest
opp a
adj
Near

12. Lets try an example Suppose we want to find angle a what is side A?
the opposite
what is side B?
the adjacent
with opposite and adjacent we use the…
tan formula b C
A = 3 a
B = 4

13. Lets solve it b C
A = 3 a
B = 4

14. Where did the numbers for the ratio come from?
Each shape of triangle has three ratios
These ratios are stored your scientific calculator
In the last question, tanθ = 0.75
On your calculator try 2nd, Tan 0.75 = °

15. Another tangent example…
we want to find angle b
B is the opposite
A is the adjacent
so we use tan b C
A = 3 a
B = 4

16. Calculating a side if you know the angle
you know a side (adj) and an angle (25°)
we want to know the opposite side b C A
25°
B = 6

17. Another tangent example
If you know a side and an angle, you can find the other side. b C
A = 6
25° B

18. An application
You look up at an angle of 65° at the top of a tree that is 10m away
the distance to the tree is the adjacent side
the height of the tree is the opposite side
65°
10m

19. Why do we need the sin & cos?
We use sin and cos when we need to work with the hypotenuse
if you noticed, the tan formula does not have the hypotenuse in it.
so we need different formulas to do this work
sin and cos are the ones! b
C = 10 A
25° B

20. Lets do sin first we want to find angle a
since we have opp and hyp we use sin b
C = 10
A = 5 a B

21. And one more sin example
find the length of side A
We have the angle and the hyp, and we need the opp b
C = 20 A
25° B

22. And finally cos We use cos when we need to work with the hyp and adj
so lets find angle b b
C = 10
A = 4 a B

23. Here is an example Spike wants to ride down a steel beam
The beam is 5m long and is leaning against a tree at an angle of 65° to the ground
His friends want to find out how high up in the air he is when he starts so they can put add it to the doctors report at the hospital
How high up is he?

24. How do we know which formula to use???
Well, what are we working with?
We have an angle
We have hyp
We need opp
With these things we will use the sin formula
C = 5 B
65°

25. So lets calculate
C = 5 B
so Spike will have fallen 4.53m
65°

26. One last example…
Lucretia drops her walkman off the Leaning Tower of Pisa when she visits Italy
It falls to the ground 2 meters from the base of the tower
If the tower is at an angle of 88° to the ground, how far did it fall?

27. First draw a triangle What parts do we have? We have an angle
We have the Adjacent
We need the opposite
Since we are working with the adj and opp, we will use the tan formula B
88° 2m

28. So lets calculate B
Lucretia’s walkman fell 57.27m
88° 2m

29. What are the steps for doing one of these questions?
Make a diagram if needed
Determine which angle you are working with
Label the sides you are working with
Decide which formula fits the sides
Substitute the values into the formula
Solve the equation for the unknown value
Does the answer make sense?

30. Two Triangle Problems
Although there are two triangles, you only need to solve one at a time
The big thing is to analyze the system to understand what you are being given
Consider the following problem:
You are standing on the roof of one building looking at another building, and need to find the height of both buildings.

31. Draw a diagram
You can measure the angle 40° down to the base of other building and up 60° to the top as well. You know the distance between the two buildings is 45m
60°
40°
45m

32. Break the problem into two triangles.
The first triangle:
The second triangle
note that they share a side 45m long
a and b are heights! a
60°
45m
40° b

33. The First Triangle
We are dealing with an angle, the opposite and the adjacent
this gives us Tan a
60°
45m

34. The second triangle
We are dealing with an angle, the opposite and the adjacent
this gives us Tan
45m
40° b

35. What does it mean? Look at the diagram now:
the short building is 37.76m tall
the tall building is 77.94m plus 37.76m tall, which equals m tall
77.94m
60°
40°
37.76m
45m

36. THANK YOU