Transcript Document

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 ( right triangle).
Warm up
• Solve the equations:
• A) 0.875 = x/18
• B) 24/y = .5
• C) y/25 = .96
E.Q:
How can we find the sin, cosine, and the
tangent of an acute angle?
How do we use trigonometric ratios to
solve real-life problems?
Trig. Ratios
Name
“say”
Abbreviation
Abbrev.
Ratio of an
angle
measure
Sine
Cosine
tangent
Sin
Cos
Tan
Sinθ = opposite side cosθ = adjacent side
hypotenuse
hypotenuse
tanθ =opposite side
adjacent side
Three Trigonometric Ratios
• Sine – abbreviated ‘sin’.
– Ratio: sin θ = opposite side
hypotenuse
Θ this is the symbol for
an unknown angle
measure. It’s name is
‘Theta’.
• Cosine - abbreviated ‘cos’.
– Ratio: cos θ = adjacent side
hypotenuse
• Tangent - abbreviated ‘tan’.
– Ratio: tan θ = opposite side
adjacent side
Easy way to remember trig
ratios:
SOH CAH TOA
Let’s practice…
Write the ratio for sin A
B
Sin A = o = a
h c
c
Write the ratio for cos A
a
C
b
Cos A = a = b
h c
A
Write the ratio for tan A
Let’s switch angles:
Find the sin, cos and
tan for Angle B:
Sin B = b
c
Tan A = o = a
a b
Cos B = a
c
Tan B = b
a
Make sure you have a calculator…
I want to find
Use these calculator keys
sin, cos or tan
ratio
Angle measure
Set your calculator to ‘Degree’…..
MODE (next to 2nd button)
Degree (third line down… highlight it)
2nd
Quit
SIN
COS
TAN
SIN-1
COS-1
TAN-1
Let’s practice…
Find an angle that has a
tangent (ratio) of 2
C
2cm
B
3
Round your answer to the
nearest degree.
3cm
A
Process:
I want to find an ANGLE
I was given the sides (ratio)
Tangent is opp
adj
TAN-1(2/3) = 34°
Practice some more…
Find tan A:
Tan A = opp/adj = 12/21
24.19
A
12
Tan A = .5714
21
Find tan A:
Tan A = 8/4 = 2
8
4
A
Trigonometric Ratios
• When do we use them?
– On right triangles that are NOT 45-45-90 or
30-60-90
Find: tan 45
1
Why?
tan = opp
hyp
Using trig ratios in equations
Remember back in 1st grade when you had
to solve:
(6)12 = x (6)
What did you do?
6
72 = x
Remember back in 3rd grade when x was in
the denominator?
(x)12 = 6 (x)
What did you do?
x
__
__
12x = 6
x = 1/2
Ask yourself:
In relation to the angle,
what pieces do I have?
34°
15 cm
Opposite and hypotenuse
Ask yourself:
x cm
What trig ratio uses
Opposite and Hypotenuse?
SINE
Set up the equation and solve:
(15) Sin 34 = x (15)
15
(15)Sin 34 = x
8.39 cm = x
Ask yourself:
In relation to the angle,
what pieces do I have?
53°
12 cm
Opposite and adjacent
x cm
Ask yourself:
What trig ratio uses
Opposite and adjacent?
tangent
Set up the equation and solve:
(12)Tan 53 = x (12)
12
(12)tan 53 = x
15.92 cm = x
x cm
Ask yourself:
In relation to the angle,
what pieces do I have?
Adjacent and hypotenuse
68°
18 cm
Ask yourself:
What trig ratio uses
adjacent and hypotnuse?
cosine
Set up the equation and solve:
(x) Cos 68 = 18 (x)
x
(x)Cos
18
_____68 =_____
cos 68 cos 68
X = 18
X = 48.05 cm
cos 68
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
Sin -1 22 = θ
42
31.59°= θ
You’re still looking for theta.
θ
Ask yourself:
22 cm
17 cm
What trig ratio uses the parts I
was given? tangent
Set it up, solve it, tell me what you get.
tan θ = 17
22
tan -1 17 = θ
22
37.69°= θ
Using trig ratios in equations
Remember back in 1st grade when you had
to solve:
(6)12 = x (6)
What did you do?
6
72 = x
Remember back in 3rd grade when x was in
the denominator?
(x)12 = 6 (x)
What did you do?
x
__
__
12x = 6
x = 1/2
Types of Angles
• The angle that your line of sight makes
with a line drawn horizontally.
• Angle of Elevation Line of Sight
Angle of Elevation
Horizontal Line
• Angle of Depression
Horizontal Line
Angle of Depression
Line of Sight
SOA CAH TOA
SOA CAH TOA