Essential Question? How can we use triangles, especially right triangles, to solve problems?
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Transcript of Essential Question? How can we use triangles, especially right triangles, to solve problems?
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Essential Question?
How can we use triangles, especially right triangles, to solve problems?
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Properties of Rational Exponents
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Warm upInverse Variation:
Boyle’s Law states that when a sample of gas is kept at a constant temperature, the volume, V varies inversely with the pressure, P exerted on it.
•Write an equation for Boyle’s Law
•If V = 20 Liters at 500 psi, what is V if pressures is 800 psi
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Trigonometric Ratios
A RATIO is a comparison of two
numbers. For example; boys to girls cats : dogs
right : wrong.
Trigonometry – study of the measurement of sides and angles in triangles
In Trigonometry, the comparison is between sides of a right triangle.
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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’.
A
C Bopposite
hypotenuse
θ
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Three Trigonometric Ratios
Θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’.
A
C B
• Cosine - abbreviated ‘cos’.
Ratio: cos θ = adjacent side
hypotenuse
ad
jace
nt
hypotenuse
θ
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Three Trigonometric Ratios
Θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’.
A
C B
• Tangent - abbreviated ‘tan’.
Ratio: tan θ = opposite side adjacent
side
opposite
ad
jace
nt
θ
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Easy way to remember trig ratios:
SOH CAH TOA
Three Trigonometric Ratios• Sine – abbreviated ‘sin’.
– Ratio: sin θ = opposite side
hypotenuse
• Cosine - abbreviated ‘cos’. – Ratio: cos θ = adjacent side
hypotenuse
• Tangent - abbreviated ‘tan’. – Ratio: tan θ = opposite side
adjacent side
Θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’.
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Trig. Ratios
Name
“say”Sine Cosine tangent
Abbreviation
Abbrev.Sin Cos Tan
Ratio of an angle measure
Sinθ = opposite side
hypotenuse
cosθ = adjacent side
hypotenuse
tanθ =opposite side
adjacent side
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Make sure you have a calculator…
I want to find Use these calculator keys
sin, cos or tan
ratio
SIN
COS
TAN
Angle measure
SIN-1
COS-1
TAN-1
To set your calculator to ‘Degree’…..
•Press MODE (next to 2nd button)
•Degree (third line down… highlight it by pressing Enter
•2nd Quit Clear
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Let’s practice…
B
c
a
C b A
Sin Θ =
13
12
5
OppositeHypotenuse
Cos Θ = AdjacentHypotenuse
Tan Θ = OppositeAdjacent
Sin A= Sin B =
Cos A= Cos B =
Tan A= Tan B =
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Lesson 4.4 (I)Lesson 4.4 (I)
35sin6x
4.3x
25cos4x
6.3x
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55tan/7x
9.4x
65sin/16x
7.17x
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)4/1(tan 1x
ox 0.14
)2/1(cos 1x
ox 60
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• Ex 1) How do we find the angle measure?
C A
B
Θo
18 cm
12
2nd Cos(12/18) = Cos-1(12/18)
= 48.2o
1) What is given?
2) What trig ratio?
3) What is asked for?
Find measure of <B?
Hypotenuse
Adjacent
Cos Θ = adj/hyp
Find angle Θ =
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Using trig ratios in equations
Remember when you had to solve:12 = x What did you do? 6
(6) (6)
72 = x
What if x is in the denominator? 12 = 6 What did you do? x
(x) (x)
12x = 6__ __
12 12 x = 1/2
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Ex 2) Let’s practice…
B
C A
Process:
1)Identify what is given
2)Which trig ratio, sin, cos, or tan will work with what is given
3)Plug in and solveX cm
40o
7.6 cm
Process:
1)Hyp = 7.6
<A = 40o and opposite = x
2) Sin = opposite/hypotenuse
3) solve:
7.6 cm
X cmSin 40o =
7.6 x Sin 40o X =
X = 4.9 cm
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Ex 3) Let’s practice…
B
c A
Process:
1)Identify what is given
2)Which trig ratio, sin, cos, or tan will work with what is given
3)Plug in and solveX cm
36o 18 cm
Process:
1)Hyp = 18
<B = 36o and adjacent = x
2) Cos = adjacent/hypotenuse
3) Solve:
18 cm
X cmCos 36o =
18 x Cos 36o X =
X = 14.6 cm
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• Ex 4) Let’s practice
c A
X cm
30o
18 cm
Process:
1)Hyp = x
<A = 30o and adjacent = 18
2) Cos = adjacent/hypotenuse
3) Solve:
18 cmX cmCos 30o =
Cos 30o =
Cos 30o =
18 cm
X cmX cm
Cos30o and x have to change places – Swith and divide!
X cm
X cmCos 30o = 18 cm Cos 30o
X cm = 18 cm
Cos 30o = 20.8 cm
B
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Practice some more…
Ex 5) Find tan A:
C A
B
48o
5.8
x
C A
B
54o
Ex 6) What trig function would find x?
18x
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Toolkit
Trig RatiosUnknown will be in one of
three places:
Sin Θ = Angle Θ
Cos Θ = Numerator:
Tan Θ = Denominator:
OppositeHypotenuse
AdjacentHypotenuse
OppositeAdjacent
2nd trig(ratio) = angle
xgivenTrig angle =
givenx
Trig angle =
Multiply
Switch and divide
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Warm upGoogle
1)When and where did Pythagoras live?
2)How old is the Great Pyramid of Egypt?
3)Is it an equilateral triangle? What is the base length?
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QuizDraw and label each triangle and find what is
asked for below:
1.Let side c = 15 ft. and side b = 9 ft. Find angle A and side aA
BC
c
a
b