Transcript Trigonometric Ratios

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).
CCSS: G.SRT.7
EXPLAIN and USE the relationship
between the sine and cosine of
complementary angles.
Standards for Mathematical
Practice
• 1. Make sense of problems and persevere in solving
them.
• 2. Reason abstractly and quantitatively.
• 3. Construct viable arguments and critique the
reasoning of others.
• 4. Model with mathematics.
• 5. Use appropriate tools strategically.
• 6. Attend to precision.
• 7. Look for and make use of structure.
• 8. Look for and express regularity in repeated reasoning.
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
Indirect Measurement
opposite
tan 59 =
adjacent
h
tan 59 =
45
h
45 tan 59 = h
74.9  h
59
45 ft
SOA CAH TOA
SOA CAH TOA
Solving a right triangle
• Every right triangle has one right angle,
two acute angles, one hypotenuse and two
legs. To solve a right triangle, means to
determine the measures of all six (6) parts.
You can solve a right triangle if the
following one of the two situations exist:
– Two side lengths
– One side length and one acute angle
measure
E.Q
• How do we use right triangles to solve
real life problems?
Note:
• As you learned in Lesson 9.5, you can use
the side lengths of a right triangle to find
trigonometric ratios for the acute angles of
the triangle. As you will see in this lesson,
once you know the sine, cosine, or tangent
of an acute angle, you can use a
calculator to find the measure of the angle.
WRITE THIS DOWN!!!
• In general, for an acute angle A:
– If sin A = x, then sin-1 x = mA
– If cos A = y, then cos-1 y = mA
– If tan A = z, then tan-1 z = mA
The expression sin-1 x is read as “the inverse sine
of x.”
• On your calculator, this means you will be
punching the 2nd function button usually in
yellow prior to doing the calculation. This is to
find the degree of the angle.
C
Example 1:
• Solve the right
triangle. Round the
decimals to the
nearest tenth.
3
B
2
c
HINT: Start by using the Pythagorean Theorem.
You have side a and side b. You don’t have the
hypotenuse which is side c—directly across from
the right angle.
A
C
Example 1:
3
B
(hypotenuse)2 = (leg)2 + (leg)2
2
c
c2 = 32 + 22
c2 = 9 + 4
Pythagorean Theorem
Substitute values
Simplify
c2 = 13
c = √13
c ≈ 3.6
Simplify
Find the positive square root
Use a calculator to approximate
A
Example 1 continued
• Then use a calculator to find the measure
of B:
2nd function
Tangent button
2
Divided by symbol
3 ≈ 33.7°
Finally
• Because A and B are complements,
you can write
mA = 90° - mB ≈ 90° - 33.7° = 56.3°
The side lengths of the triangle are 2, 3
and √13, or about 3.6. The triangle has
one right angle and two acute angles
whose measure are about 33.7° and
56.3°.
Ex. 2: Solving a Right Triangle (h)
• Solve the right
triangle. Round
decimals to the
nearest tenth.
g
H
25°
You are looking for
opposite and
13
hypotenuse which is
the sin ratio.
J
h
G
sin H =
opp.
hyp.
h
13 sin 25° =
Set up the correct ratio
13
Substitute values/multiply by reciprocal
13
13(0.4226) ≈ h
5.5 ≈ h
Substitute value from table or calculator
Use your calculator to approximate.
Ex. 2: Solving a Right Triangle (g)
• Solve the right
triangle. Round
decimals to the
nearest tenth.
cos G =
H
g
25°
You are looking for
adjacent and
hypotenuse which is
the cosine ratio.
13
adj.
hyp.
g
13 cos 25° =
13
13(0.9063) ≈ g
11.8 ≈ h
Set up the correct ratio
13
Substitute values/multiply by reciprocal
Substitute value from table or calculator
Use your calculator to approximate.
J
h
G
Using Right Triangles in Real Life
• Space Shuttle: During its
approach to Earth, the
space shuttle’s glide
angle changes.
• A. When the shuttle’s
altitude is about 15.7
miles, its horizontal
distance to the runway is
about 59 miles. What is
its glide angle? Round
your answer to the
nearest tenth.
Solution:
• You know opposite
and adjacent sides. If
you take the opposite
and divide it by the
adjacent sides, then
take the inverse
tangent of the ratio,
this will yield you the
slide angle.
Glide  = x°
altitude
15.7
miles
distance to runway
59 miles
tan x° =
opp.
Use correct ratio
adj.
tan x° =
15.7
Substitute values
59
Key in calculator 2nd function,
tan 15.7/59 ≈ 14.9
 When the space shuttle’s altitude is about 15.7 miles, the
glide angle is about 14.9°.
B. Solution
Glide  = 19°
altitude
h
• When the space
shuttle is 5 miles from
the runway, its glide
angle is about 19°.
Find the shuttle’s
altitude at this point in
its descent. Round
your answer to the
nearest tenth.
distance to runway
5 miles
tan 19° =
opp.
adj.
tan 19° =
h
Substitute values
5
5 tan 19° =
h
5
 The shuttle’s altitude is
about 1.7 miles.
Use correct ratio
5
Isolate h by
multiplying by 5.
1.7 ≈ h Approximate using calculator