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Elizabeth Pawelka
Geometric Probability
Geometry
Lesson Plans
Chapter 9: Right Triangle Trigonometry
Sections 9-1 and 9-2: Trigonometric Ratios
4/16/12
Warm-up (15 mins)
4/11/12
p.1
Elizabeth Pawelka
Geometric Probability
4/11/12
p.2
Statement of Objectives (5 mins)
The student will be able to find sine, cosine and tangent ratios for acute angles of right triangles.
(include picture of my fish: Trig and Tangent)
Teacher Input (50 mins)
Investigation: Trigonometric Ratios
Materials: protractors, rulers, paper, pencil
Elizabeth Pawelka
Geometric Probability
4/11/12
p.3
Divide class into groups of four and assign each group a different angle measure (30, 45, 60). Have
each member of the group draw a right triangle (labeled ABC) with their angle measure as one of the
other angles (A). Stress that the triangles should be different sizes.
Measure the sides (in cm) of each triangle and fill in this chart:
___
Leg
Leg
Hypotenuse Verify that
degree
opposite
adjacent
=c
a2 + b2 = c2
Triangle <A (BC) to <A
=a
(AC) = b
1
2
3
4
a
c
b
c
Compare the ratios. They should be the same within a group of the same angle measures.
Angle
30
45
60
(sin)
a
c
s
= 0.5
2s
s
= 0.707
s 2
s 3
= 0.866
2s
(cos)
b
c
s 3
= 0.866
2s
s
= 0.707
s 2
s
= 0.5
2s
(tan)
a
b
s
= 0.577
s 3
s
= 1.0
s
s 3
= 1.732
s
a
b
Elizabeth Pawelka
Geometric Probability
4/11/12
Trigonometric Ratios (Right Triangles)
Sine of ∠A =
leg opposite  
=>
hypotenuse
Cosine of ∠A =
sin A =
leg adjacent t o  
=>
hypotenuse
opposite
hypotenuse
cos A =
adjacent
hypotenuse
S=
O
H
C=
A
H
p.4
Elizabeth Pawelka
Tangent of ∠A =
Geometric Probability
leg opposite  
=>
leg adjacent t o  A
tan A =
opposite
adjacent
4/11/12
T=
O
A
“SOH CAH TOA”
 Studying Our Homework Can Always Help To Obtain Achievement
 Some Of Her Children Are Having Trouble Over Algebra
 Some Old Hippy Caught Another Hippy Tripping On Acid
 Some Old Horse Caught Another Horse Taking Oats Away
 Saddle Our Horses Canter Away Happily to Other Adventures
 Some Old Helmets Can Allow Head Trauma On Accidents
Have them make up their own as an “exit ticket.”
p.5
Elizabeth Pawelka
Geometric Probability
4/11/12
p.6
Example 1: Find the following ratios for this triangle
sin P = ___________ (6/10)
cos P = ___________ (8/10)
tan P = ____________ (6/8)
sin Q = ____________ (8/10)
cos Q = ___________ (6/10)
tan Q = ____________ (8/6)
Example 2: Find sine, cosine, and tangent of angles A and B in each right triangle. Leave in fraction
form.
sin A =
sin B =
cos A =
cos B =
tan A =
tan B =
sin A = 12/15 sin B = 9/15
cos A = 9/15 cos B = 12/15
tan A = 12/9 tan B = 9/12



sin A =
cos A =
tan A =
sin A = 7/25
cos A = 24/25
tan A = 7/24
sin B =
cos B =
tan B =
sin B = 24/25
cos B = 7/25
tan B = 24/7
sin A =
cos A =
tan A =
sin B =
cos B =
tan B =
sin A = 4/5
cos A = 3/5
tan A = 4/3
sin B = 3/5
cos B = 4/5
tan B = 3/4
What do you notice about the relationship of sine and cosine in angles A and B? (sin A = cos B)
1
What do you notice about the relationship of tangents in angles A and B? (tan A =
) “Inverse
tan B
relationship”
What relationship do angles A and B have? Complementary angles = 90 degrees
Reference Table in back of the book – p. 731. To emphasize that the sine, cosine, and tangents are
unique to an angle, point out that sin 30 = 0.500 and challenge them to find another one with sine = 0.5
Elizabeth Pawelka
Geometric Probability
4/11/12
p.7
Now introduce using the calculator and point out that they can use the table if they don’t have a
calculator.
Using your Calculator:
First ensure the calculators are set to use degrees for angles:
Elizabeth Pawelka
Geometric Probability
4/11/12
p.8
Elizabeth Pawelka
Geometric Probability
4/11/12
p.9
Do examples of finding a missing side given one side and an angle.
Find appropriate trig function for what you are given and then find that function of your angle and
divide
Example 3: Find missing side
x is opposite ∠A (500) and you are given the hypotenuse, 15.
x
So, use the sine function: sin 500 =
15
(sin 500) * (15) = x
(.766)(15) = x
11.49 = x
Challenge:
 What is CA? (cos 50 = CA/15 => CA = 9.64)
 Check your answer(s) using Pythagorean Thm
(11.492 + 9.642 = 152 => 132 + 93 = 225)
Example 4: Find missing sides. Round to nearest tenth.
use tan(54) = t/10
tan(54) * 10 = t
13.8 = t
use sin(49) = 9/s
sin(49)s = 9
s = 9/sin(49)
s = 11.9
use cos(63) = c/14
cos(63) * 14 = c
6.4 = c
Introduce Inverse functions
“So, if we input an angle, we can find the sine/cosine/tangent. Could we do it the other way around?
E.g. if we had the sine/cosine/tangent, can we find the angle?” Compare to “undoing” functions such as
addition/subtraction and multiplication/division.
Show calculator and where to find it: 2nd function:
Elizabeth Pawelka
Geometric Probability
Then do examples of finding angle given two sides.
Example 5: Find m∠x to the nearest whole number
4/11/12
p.10
Elizabeth Pawelka
Geometric Probability
4/11/12
p.11
Since you’re given the hypotenuse and opposite side, use
inverse sine function: m∠x = sin-1(15/20)
m∠x = 48.59 ≈ 49
Example 6: Find missing angle measurements to the nearest whole number
= sin-1(12/24) = 30
= tan-1(10/15) = 33.69 = 34
= cos-1(25/28) = 26.765 = 27
Elizabeth Pawelka
Geometric Probability
4/11/12
p.12
Unit Circle (if time allows)
(measures rounded to two decimal places)
Use Geogebra file: sincos.ggb to show that, if the hypotenuse is always 1 (unit circle), then the sin is the
y value (height of y axis side) and the cos is x value (length of the x axis side)
Sine Box: http://www.ies.co.jp/math/products/trig/applets/sinBox/sinBox.html
Cosine Box: http://www.ies.co.jp/math/products/trig/applets/cosbox/cosbox.html
Closure (5 mins)
Today you learned to find sine, cosine and tangent ratios for acute angles of right triangles.
Tomorrow you’ll learn to identify the angle of elevation and depression and use them with trigonometric
ratios.
Homework (H)
 p. 472, # 1-16, 27 - 29
 p. 479, # 1-17, 22 -25
Homework (R)
 p. 472, # 1-16, 28
 p. 479, # 1-17, 24