Download Lesson 7.1 – Pythagorean Theorem

Lesson 7.1 – Pythagorean Theorem
Warm-Up: “Remembering Pythagoras”
 What relationship is ∆ABC displaying?
B
 What type of triangle is this relationship true for?
A
C
 What is side “c” called?
Ex:1
Solve for each unknown side. Round to the nearest tenth. Show at least three vertical steps.
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Ex:2
The size of a television/monitor is the length of the diagonal
measurement.
In Mrs. Lee’s family room there is a corner shelving unit. One shelf
is 40” wide and 30” tall. What is the largest TV Mrs. Lee could buy
to fit in this shelf?
Ex:3
Paul is a carpenter. Explain how he can use the Pythagorean Theorem to prove that two
pieces of wood form a true corner. Provide a diagram to help your explanation.
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7.1 Practice Questions
1.
Solve for each unknown side. Round to the nearest tenth. Show at least three vertical steps.
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2.
To solve for the unknown side “x” for the triangle at the right,
Simon starts by writing: x2 = 222 + 262. Should Simon use this
approach to solve for “x”? Explain.
3.
Emily is building a model of a ramp. She uses a 23 cm
piece of wood for the ramp slope and an 11 cm piece of
wood for the vertical drop. How long should the
horizontal piece of wood be, to the nearest tenth?
4.
Gary joins a 4 ft piece of lumber with a 5 ft piece of lumber to create a corner. The diagonal
measures 6 ft. Has Gary created a perfect corner? Explain.
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Lesson 7.2 – Introduction to Trigonometry
Warm-Up #1: “You Tube”:
Watch the first two minutes of the following You Tube clip: “Introduction to Trigonometry”
http://www.youtube.com/watch?v=YxsEYWHb4i8
Now label the “opposite”, “adjacent” and “hypotenuse” sides for the following two right triangles,
in relation to angle X. (think about the water spray bottle in the video clip)
X
X
Warm-Up #2: “Similar Triangles”:
What must be true for triangles to be similar to each other?
“What’s My Ratio?” Activity
You will be given a separate handout with two triangle sets and a protractor:
1.
2.
3.
4.
5.
6.
7.
Measure each angle A and label it on the triangle.
Measure the horizontal and vertical sides of each triangle. Count each square as 1 unit.
Label each side length on the triangle.
Use the Pythagorean Theorem to calculate the length of each hypotenuse to one
decimal place. Use the back of the triangle handout for the work needed. Label each
hypotenuse on the triangle.
Use a ruler and draw scale versions of each triangle for the missing sizes. (S, M, L)
Repeat steps #1-3 for the four new triangles.
For all six triangles, label the opposite (OPP), adjacent (ADJ), & hypotenuse (HYP) sides.
Within each set, are the triangles similar? Explain.
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8.
Complete the following table for the triangles in Set #1.
Calculate each ratio to 2 decimal places
Measure
<A
Opposite
Side
Length
Adjacent
Side
Length
Hypotenuse
Side Length
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑎𝑑𝑎𝑗𝑎𝑐𝑒𝑛𝑡
Small ∆
Medium
∆
Large ∆
9. What do you notice about each of the three ratios for the triangles?
10. What can you now conclude about the three ratios for similar right triangles?
Note: Each of the three ratios have their own name…we will introduce their names next
class…stay tuned!!!
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7.2 – Practice Questions
1.
To confirm our conclusion from today’s lesson:
 complete the following table for the triangles in Set #2.
Calculate each ratio to 2 decimal places
Measure
<A
Opposite
Side
Length
Adjacent
Side
Length
Hypotenuse
Side Length
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑎𝑑𝑎𝑗𝑎𝑐𝑒𝑛𝑡
Small ∆
Medium
∆
Large ∆
What do you notice about each of the three ratios for the triangles?
2. How can the third angle in the six triangles be calculated? Perform the calculation for each set
of triangles.
3. If the third angle is our “angle of reference”, how does the opposite (OPP), adjacent (ADJ), &
hypotenuse (HYP) side labelling change? Draw a diagram.
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7.3 – Trigonometric Ratios Introduction
Warm-Up: “Triangle Data”
Transfer the ratio values only from last class for triangle set #1 to the table below.
Calculate each ratio to 2 decimal places
Measure
<A
Opposite
Side
Length
Adjacent
Side
Length
Hypotenuse
Side Length
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑎𝑑𝑎𝑗𝑎𝑐𝑒𝑛𝑡
Small ∆
Medium
∆
Large ∆
Naming the Ratios:
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
Since we discovered that all similar right triangles have the same values for each of the ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒,
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
, and 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ratios; each ratio is given it’s own name:
Sine =
𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
Cosine =
𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
Tangent =
𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆
𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕
“sin”
“cos”
“tan”
Place each name above the appropriate table column.
To remember which sides belong with which trigonometric ratio, use “SOH
CAH TOA”:
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Using Your Calculator for Trigonometry:
Always ensure your calculator is in “degree” (DEG) mode to perform trigonometry calculations.
1.
Calculate the following values to two decimal places.
a) sin79o ________
2.
cosA = 0.97
tanA = 6.31
Calculate each angle, to the nearest degree.
a) sinB = 0.47
4.
c) tan81o _________
Reverse It:
What if we know the value of the ratio (in decimal form) but not the angle?
For example:
sinA = 0.98
3.
b) cos15o ________
b) cosC = 0.816
c) tanD = 0.4453
Determine sinA, cosA, and tanA for angle A of set #1. What do you notice?
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7.3 - Practice Questions
Always ensure your calculator is in “degree” (DEG) mode to perform trigonometry calculations.
1.
2.
3.
Calculate the following trigonometric ratios, to two decimal places.
a) sin63o
_________
b) cos47o
__________
c) tan58o
_________
d) sin19o
__________
e) tan66o
__________
f) cos45o
__________
g) sin60o
__________
h) cos30o
__________
i) tan86o
__________
j) sin90o
__________
Determine the measure of each angle, to the nearest degree.
a) sinA = 0.97
__________
b) cosB = 0.27
__________
c) tanC = 1.325
__________
d) sinD = 0.084
__________
e) tanE = 0.56
__________
f) cosF = 0.56
__________
g) sinG = 0.56
__________
h) cosh = 0.22
__________
i) tanX = 2.48
__________
j) sinJ = 0.3719
__________
Perform the following calculations, to the nearest tenth.
a)
12sin75o
__________
b) 29cos40o
__________
c)
34tan51o
__________
d) 14.6sin73o
__________
__________
f)
__________
h)
e)
g)
72
sin30°
9.8
tan21°
164
cos67°
65.1
sin40°
__________
__________
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4.
Use ∆MNP at the right to answer the following
questions.
a) In relation to <M, label the “opposite”, “adjacent”, and “hypotenuse” sides.
Revisit lesson 4.2 if needed.
b) State each ratio as a fraction and then as a decimal rounded to two decimal places.
Think: “SOH CAH TOA”
Trigonometric Ratio
sinM
Fraction
Decimal
cosM
tanM
c) Use the sine ratio above to determine the measure of <M, to the nearest degree.
d) Use the cosine ratio above to determine the measure of <M, to the nearest degree.
What do you notice?
e) What is the measure of <N? How do you know?
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Lesson 7.4 – Solving for Angles
Warm-Up: “SOH CAH TOA”
1. State sin C, cos C, and tan C for ∆ABC.
2. How can we use the above ratios to determine the measure of <C?
Therefore, when given two sides of a right triangle, an angle can be calculated with the appropriate
trigonometric ratio and the inverse trig button on our calculators.
Ex:1
a) Calculate the measure of <A.
b) What is the measure of the third angle?
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Ex:2
a) Calculate the measure of <A.
b) What is the measure of the third angle?
Ex:3
Determine the measure of <D and <F in ∆DEF.
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7.4 Practice Questions
1.
Determine the measure of each angle to the nearest degree.
a)
sinA = 0.756
2.
Determine the measure of <D.
3.
Use ∆KLM for this question.
b) cosB =
3
4
c) tanC =
3.5
1.1
a) Determine the measure of <M.
b) Determine the measure of <K.
c) Determine the length of side KL, to the nearest tenth.
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4.
Determine the measure of <F using two different trigonometric ratios.
5.
a) Calculate:
sin30o = ____________
b) Draw a right triangle with a 30o angle.
c)
Label the two sides of the above triangle that must be true according to your
calculation in part a).
d) Calculate the length of the third side in the triangle. Label the measure of the third
angle.
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Lesson 7.5 – Solving for Sides
Warm-Up: “Calculator Review”
1.
Calculate the following to two decimal places.
a) cos72o
Q:
b) tan18o
2.
State the three trigonometric ratios for <M.
3.
Determine the measure of <M, to the nearest degree.
4.
How many triangle sides are needed to solve for an angle?
c) sin29o
What if we know the angle but need to solve for a side length?
A:
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Ex:1
Determine the length of side a, to the nearest tenth.
Ex:2
Determine the length of side e, to the nearest tenth.
Ex:3
Determine the length of side q, to the nearest tenth.
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7.5 Practice Questions
1.
Determine the length of side r, to the nearest tenth.
2.
Determine the length of side r, to the nearest tenth.
3.
Determine the length of side d, to the nearest tenth.
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4.
a) Determine the length of side j, to the nearest tenth.
b) Determine the length of side MN, to the nearest tenth. Discuss a second method to
calculate MN.
c) Determine the measure of <M.
5.
a) Determine the height of the cliff, to the nearest metre.
b) Determine the distance the rope is from the base of the cliff, to the
nearest metre.
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7.6 SOH CAH TOA Practice
Warm-Up: “SOH CAH TOA”
1. State the three trigonometric ratios for angle K.
2. Calculate the measure of angle K.
3. Determine the length of side LM, to the nearest tenth.
4. Determine the length of side MK, to the nearest tenth.
** Continue practicing SOH CAH TOA with today’s assignment. **
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7.7 – Trigonometry Applications (Day 1)
 Trigonometry has many applications: navigation, construction, engineering, design, etc.
 Problem Solving Steps:
1. Read the problem carefully to understand the situation. Re-read it.
2. Draw a right triangle diagram with the known measurements labeled. Place a variable(s) on the
measurement(s) being solved for.
3. Set-up one of the primary trigonometric ratios that involve the unknown measurement.
4. Solve the ratio for the unknown measurement.
5. Check your answer is reasonable.
6. Answer the question with a concluding statement.
Ex:1
Rory is unloading his ATV from his truck, down
a ramp. Determine the angle the ramp makes
with the ground, to the nearest degree.
Ex:2
Determine both the vertical height (h) and the
horizontal height (d) of the hot air balloon, to the
nearest tenth.
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Ex:3
Kerry is looking up at a ceiling cloud. Her eye level,
angle of elevation, and horizontal distance are shown.
Calculate the height of the ceiling cloud to the nearest
tenth.
Ex:4
Safety regulations state that a wheelchair ramp must have a slope between 12 and 5.
1
2
Calculate the minimum slope angle and maximum slope angle for wheelchair ramps, to the
nearest degree.
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7.7 – Practice Questions
1.
Jacob parked his car illegally and is having it towed.
Calculate the angle the front of his car is at while being
towed, to the nearest degree.
2.
Two buildings are 51 m apart. Sarah views from the top of one
building to the bottom of the other with a 64o angle of
depression. How tall are the buildings? (to the nearest metre)
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3.
An 18 m ladder leans up against one building as shown
and then flipped and leaned against a second building
as shown.
a) Determine the distance between the two buildings
using the Pythgaorean Theorem, to the nearest
tenth.
b) Determine the distance between the two buildings using trigonometry, to the nearest
tenth.
4.
A tunnel runs through a 850 m tall mountain. Chris stands at one end of the tunnel and
views the top of the mountain at a 64o angle of elevation. Kevin stands on the other side of
the tunnel and views the top of the mountain at a 79 o angle of elevation. Determine the
length of the tunnel, to the nearest metre. Start with a diagram!
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7.8 – Unit 7 Review
1.
Label each unknown side x. Determine the length of side x, to the nearest tenth.
2.
a) State the three trigonometric ratios for angle C.
b) State the three trigonometric ratios for angle B.
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3.
Determine the measure of each unknown side, to the nearest tenth.
4.
Determine the measure of angle A , to the nearest degree.
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5.
Stephen’s car is being towed to the local garage.
Determine the angle the rear of his car is being
towed at, to the nearest degree.
6.
From the top of one building to the top of another, a
41o angle of elevation is noted. The shorter building is
10 m tall. The buildings are 18 m apart. Determine the
height of the taller building, to the nearest tenth.
7.
A land surveyor is standing at point S. She measures with
her laser transit angles to points A and B as shown. The also
measures her distance from points A and B as shown.
Determine the distance between points A and B, to the
nearest metre.
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