trigonometric ratios a ratio is a comparison of two numbers. for example; boys to girls cats : dogs...
TRANSCRIPT
Trigonometric
Ratios
A RATIO is a comparison of two numbers. For
example; boys to girls cats : dogs
right : wrong.
In Trigonometry, the comparison is between
sides of a triangle.
We need to do some housekeeping before we
can proceed…
In trigonometry, the ratio we are talking about is the comparison of the sides of a
RIGHT TRIANGLE.
Two things MUST BE understood:1. This is the hypotenuse.. This
will ALWAYS be the hypotenuse2. This is 90°… this makes the
right triangle a right triangle…. Without it, we can not do this trig… we WILL NOT use it in our calculations because we COULD NOT do calculations without it.
Now that we agree about the hypotenuse and right angle, there are only 4 things left; the 2 other
angles and the 2 other sides.
A We will refer to the sides in terms of their proximity to the angle
If we look at angle A, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse.
opposite
adjacent
hypotenuse
B
If we look at angle B, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse.
opposite
adjacent
hypotenuse
Remember we won’t use the right angle
X
θ this is the symbol for an unknown angle measure.
It’s name is ‘Theta’.
Don’t let it scare you… it’s like ‘x’ except for angle measure… it’s a way for us to keep our variables understandable and organized.
One more thing…
Here we go!!!!
Trigonometric RatiosName“say”
Sine Cosine tangent
AbbreviationAbbrev.
Sin Cos Tan
Ratio of an angle measure
Sinθ = opposite side hypotenuse
cosθ = adjacent side hypotenuse
tanθ =opposite side adjacent side
One more time…Here are the ratios:
One more time…Here are the ratios:
sinθ = opposite side hypotenuse
cosθ = adjacent side hypotenuse
tanθ =opposite side adjacent side
S OH
AHOA
C
T
SOH CAH TOA
Let’s practice…
B
c
a
C b A
Write the ratio for sin A
Sin A = a c
Write the ratio for cos A
Cos A = b c
Write the ratio for tan A
Tan A = a b
Let’s switch angles: Find the sin, cos and tan for Angle B:
Sin B = b
cCos B = a
c
Tan B = b
a
Make sure you have a calculator…
Given Ratio of sides Angle, side
Looking for Angle measure Missing side
UseSIN-1
COS-1
TAN-1
SIN, COS, TAN
Set your calculator to ‘Degree’…..
MODE (next to 2nd button)
Degree (third line down… highlight it)
2nd
Quit
Practice some more…
Find tan A: 24.19 12
A 21
Tan A = opp/adj = 12/21
Tan A = .5714
8
4A
Tan A = 8/4 = 2 8
Find tan A:
Using trig ratios in equations
Remember back in 1st grade when you had to solve:
12 = x What did you do? 6
(6) (6)
72 = xRemember back in 3rd grade when x was in
the denominator? 12 = 6 What did you do? x
(x) (x)
12x = 6__ __12 12 x = 1/2
x cm
15 cm
34°
Ask yourself:In relation to the angle,
what pieces do I have?
Opposite and hypotenuse
Ask yourself:
What trig ratio uses Opposite and Hypotenuse?
SINE
Set up the equation and solve:
Sin 34 = x 15
(15) (15)
(15)Sin 34 = x8.39 cm = x
x cm
12 cm
53°
Ask yourself:In relation to the angle,
what pieces do I have?
Opposite and adjacent
Ask yourself:
What trig ratio uses Opposite and adjacent?
tangent
Set up the equation and solve:
Tan 53 = x 12
(12) (12)
(12)tan 53 = x15.92 cm = x
x cm
18 cm
68°
Ask yourself:In relation to the angle,
what pieces do I have?
Adjacent and hypotenuse
Ask yourself:
What trig ratio uses adjacent and hypotnuse?
cosine
Set up the equation and solve:Cos 68 = 18 x
(x) (x)
(x)Cos 68 = 18
X = 18 cos 68
_____ _____cos 68 cos 68
X = 48.05 cm
Ok… we’ve found side lengths, now let’s find angle measures.
Refer to your table… what function will we use to find angle measures?
SIN-1
COS-1
TAN-1These are called INVERSE FUNCTIONS
42 cm
22 cm
θ
This time, you’re looking for theta. Ask yourself:In relation to the angle, what pieces do I have? Opposite and hypotenuse
Ask yourself:
What trig ratio uses opposite and hypotenuse? sine
Set up the equation (remember you’re looking for theta):
Sin θ = 22 42
Remember to use the inverse function when you find theta
THIS IS IMPORTANT!!
Sin -1 22 = θ 42
31.59°= θ
Let’s practice…
C
2cm
B 3cm A
Find an angle that has a tangent (ratio) of 2
3
Round your answer to the nearest degree.
Process:
I want to find an ANGLE
I was given the sides (ratio)
Tangent is opp
adj
TAN-1(2/3) = 34°
17 cm
22 cm
θ
You’re still looking for theta.
Ask yourself:
What trig ratio uses the parts I was given? tangent
Set it up, solve it, tell me what you get.
tan θ = 17 22
THIS IS IMPORTANT!!
tan -1 17 = θ 22
37.69°= θ
Your assignment
TrigWorksheets
(Kuta)