Unit 7

“Right Triangles”

Academic Geometry

Fall 2013

Name_____________________________ Teacher__________________ Period______

YOU’RE RIGHT!

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Unit 7 at a glance

“Right Triangles”

This unit focuses on the investigation of the relationships between and within right triangles. Students will build on prior knowledge to solve for all missing measurements within right triangles.

Essential Questions What strategies can I use to identify patterns and develop geometric relationships? How do concrete models help me understand mathematics? How does geometry explain or describe a structure within the real world? How can it

help me solve a problem? In Unit 7, students will…

Derive and justify the Pythagorean Theorem and apply the Pythagorean Theorem and Pythagorean triples (including making connections to similar right triangles) to solve problems in a variety of contexts;

Extend the Pythagorean Theorem by applying the converse of the Pythagorean Theorem to classify triangles using side lengths and solve problems in both real-world and purely mathematical situations;

Investigate using patterns, develop, apply and justify triangle similarity relationships between the length of the attitude drawn to the hypotenuse of a right triangle and the length of the hypotenuse;

Apply and justify properties of altitudes of similar right triangles (including the Geometric Mean (Altitude) Theorem and Geometric Mean (Leg) Theorem) to solve problems in situations;

Investigate using patterns, develop, apply and justify 45-45-90 and 30-60-90 as similar triangles to solve problems in both real-world and purely mathematical situations;

Develop, justify, and apply the tangent ratio to solve problems in both real-world and purely mathematical situations;

Develop, justify, and apply the sine and cosine ratios to solve problems in both real-world and purely mathematical situations including angle of elevation and angle of depression.

Apply trigonometric ratios to solve problems in both real-world and purely mathematical situations including using inverse trigonometric functions when appropriate.

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Vocabulary

acute triangle

altitude of a triangle

angle of elevation

angle of depression

converse

cosine

denominator

geometric mean

hypotenuse

inverse

inverse cosine

inverse sine

inverse tangent

irrational numbers

isosceles triangle

leg of a right triangle

numerator

obtuse triangle

perfect square

principal root

pythagorean theorem

radical

radicand

rationalizing

right triangle

similar polygons

simplest radical form

sine

solving a right triangle

special right triangles

tangent

trigonometric ratio

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Radicals (Day 1) NOTES

Simplifying Radicals

A radical is another term for “square root”. For example “radical 4” means the same as

“square root of 4”. So if you are going to simplify “radical 4”, then this is the same as

taking the “square root of 4”. If no symbol is written in front of the radical symbol, then a

positive answer is presumed. This is called the principal root. If you were solving a

quadratic equation in the form then assume that both the positive and negative

solutions are desired. For example, the equation has two possible solutions for

….they are 2 and -2. A calculator will only give you the positive answer. For these

problems you need to know that there is also a negative answer and include it as part of

your solution.

√

When you are evaluating a radical that is a perfect square you get an integer, such as the

two examples above. Another example is √ , which is 9. When you evaluate a radical

that is not a perfect square, the result is a decimal. An example is √ . You get

6.32455532 (try it in your calculator). Even this is not an “exact” answer, since your

calculator only has the capacity to show you a certain amount of numbers. (This number

is called irrational.)

Sometimes we write the answers to radicals that are not perfect squares with more

radicals. This is called “simplifying a radical”. The object is to get a number under the

radical sign that does not have a perfect square factor.

Simplify 40 ….an easy way to do this is to factor it into all its prime factors (make a factor

tree).

“radical 4” or

“square root of 4”

is 2

This equation has

two solutions, 10

and -10

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Another example………Simplify √ .

Whenever you get factors that are “doubles”, all the doubles will multiply to a

perfect square which has an integer square root. When there are no more

“doubles”, you’re done and whatever is leftover stays under the radical. As a

shortcut, you could also factor out the largest prime number factor, if you

immediately know it.

EXAMPLE PROBLEMS

Simplify the radicals.

1. √ 2. √ 3. √

TRY IT WITH VARIABLES!

4. √ 5. √ 6. √

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Do you see a SHORTCUT to simplifying radicals with variables?

One application of simplifying radicals is in solving quadratic equations of the type

with x2 = c. (You will see this again when solving Pythagorean Theorem problems.)

Write each solution as a simplified radical. Remember to include all possible solutions.

7. 8. 9.

Multiplying Radicals When multiplying radicals, simply multiply the radicands (the terms under the radical signs) together to get your new result under a new radical sign. Then simplify as you have previously learned. Examples:

1. 2 2 2 2 4 _____

2. 5 40 5 40 ____ __________

3. 12 8 _____ _____ _______ __________x x

4. 23 5 ________________________________________________

***Helpful hint: When multiplying terms with variables, you add the exponents of the like variables that

are being multiplied together. x = x1……..so x ∙ x = x1 ∙ x1 = x 1+1 = x2.

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Simplifying Radicals with Fractions (Part 1…perfect square in denominator)

General Rule: √

√

√

Examples:

1.

4 4

81 81

___________

2. 6

6

49m

____________

3. 24

__________________________200

Careful! Reduce

first!

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Radicals (Day 1) HOMEWORK

Simplify the Expression.

1) √ 2) √ 3) √

4) √ 5) √ 6) √

7) √ √ 8) √ √ 9) √

11

10) √

11) √

12) √

13) √ 14) √ 15) √

16) √ √ 17) √ √ 18) ( √ ) √

Solve each equation. Remember to include all possible solutions.

19) 20) 21)

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Radicals (Day 2) NOTES

Simplifying Radicals

Simplifying Radicals with Fractions (Part 2…Rationalizing the Denominator) Division of radicals: The square root of the quotient is equal to the quotient of the square roots.

√

√

√

Rationalizing the denominator means to remove all radicals in the denominator of a

fraction.

Simplify, if possible. Multiply the fraction by 1, but in the form of the fraction using the radical in the

denominator as both the numerator and denominator. Simplify, if possible.

√

√

√ √

√

√

√

√

√

√

EXAMPLE PROBLEMS

1) 3

5 2)

18

2 3)

8

3

Having a radical in the denominator is a big

“no-no”! You will have to rationalize!

It is okay to have a radical in the numerator.

This radical has been simplified.

This is equivalent to multiplying by 1, since √

√ .

1.

13

4) 5

15 5)

20 2

15 6)

4

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Solve for x. Simplify. Leave your answer in exact form (as a radical, not a rounded decimal).

***SPECIAL NOTE: It is customary to write the variable in front of the radical.

For example, write “ √ ” instead of “√ ” to avoid confusion.

7) 4

= 2 3

x 8)

5 3

2 x

9) 4 x

= x 10

10) 2 3

= 2 x

11)

6 3

2 x

12) 1 2

= 5 2x

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Radicals (Day 2) HOMEWORK

Simplify each expression.

1)

√ 2) √

3)

√

4) √

5)

√

√ 6)

√

Solve for x. Simplify.

7)

√ 8)

√

9)

√

√

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Mixed Problems. Simplify.

10) √

11) √

√ 12) √

13) √

14) √ √ 15) √

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7.1/7.2 NOTES

Pythagorean Theorem & Converse

The Pythagorean Theorem can be used to find the lengths of the sides of a right triangle.

Pythagorean Theorem: In a right triangle, the square of the lengths of the hypotenuse is

equal to the sum of the squares of the lengths of the legs.

2 2 2

2 2 2

c a b

hyp leg leg

EX: Find the length of the unknown side in each right triangle. Write the answer as a radical in simplified form.

(1) (2)

EX: Find the lengths of the sides of a square that has a diagonal length of 42 mm. Write as both a radical and decimal answer.

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Pythagorean Theorem:

IF it’s a right triangle, THEN 2 2 2c a b

Converse of the Pythagorean Theorem:

In any triangle, if c is the largest side and a and b are the other two sides,

IF 2 2 2c a b , THEN it’s a right triangle.

IF 2 2 2c a b , THEN it’s an acute triangle.

IF 2 2 2c a b , THEN it’s an obtuse triangle.

EX: Determine if the side lengths represent those of an acute, right, or obtuse triangle.

(3) √ , 24, 30 (4) 14, 7, √ (5) 8, 10, 12

A Pythagorean triple is a set of three positive integers a, b, and c, that satisfy the equation2 2 2c a b . Other triples are found by multiplying each integer by the same factor.

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7.1/7.2 Homework

Determine if the given side lengths belong to an acute, right, or obtuse triangle.

1) 4, 6, 9 2) 4.5, 6, 7.5 3) 5, √ , 8 For each of the following, (1) draw a diagram and (2) solve the problem. Write answers in decimal form and in simplest radical form. 4) A 12 foot rope is fastened to the top of a flagpole. The rope reaches a point on the

ground 6 feet from the base of the flagpole. What is the height of the flagpole? 5) Arlo and Janie biked 12 miles directly east and then 6 miles directly north. How far

are they from their starting point?

6) Square ABCD has a perimeter of 25 centimeters. What is the length of its diagonal ̅̅ ̅̅ ? 7) Suppose that the radius of Earth is approximately 3960 miles. If a

satellite is orbiting at about 7 miles above the Earth, what is the approximate length of x? Note: The figure is not drawn to scale.

x 7 mi

Earth

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8) If C is the center of the circle and

AB √ cm, then what is the length of the circle’s radius?

9) Use the Pythagorean Theorem to find the distance between point A and point B. Write your answer in decimal and radical form.

10) You have a garden in the shape of a right triangle with the dimensions shown. a) Find the perimeter of the garden.

b) You are going to plant a post every 15 inches around the garden. How many posts do you need?

c) You plan to attach fencing to the posts around the garden. If each post costs $1.25 and each foot of fencing costs $0.70, how much will it cost to enclose the garden?

11) Kim walked diagonally across a rectangular field that measured 100 ft. by 240 ft.

Which expression could be used to determine how far Kim walked?

A √

B √ √

C

D √

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7.3 NOTES Similarity in Right Triangles

Theorem 7.5 If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.

Term Definition

GEOMETRIC MEAN

The geometric mean of two positive numbers a and b is the

positive number x that satisfies

.

So and √ .

Example 1: Find the geometric mean of 6 and 12. Example 2:

Find the geometric mean of

and 25.

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Parachute Man

Scenario #1-

b a

-------------------- c -------------------------

Scenario #2-

b a

--------------------- c ------------------------

Scenario #3-

b a

x 𝑥 𝑎 𝑐

𝑥 𝑎 𝑎 𝑏

or

y 𝑦 𝑏 𝑐

𝑦 𝑏 𝑏 𝑎

or

h

ℎ 𝑏 𝑎

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Examples:

1) Solve for x.

6

-------------------- 14 -----------------------

2) Solve for y.

3) Solve for y.

4 5

x

y

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7.3 HOMEWORK Find the geometric mean between each pair. 1) 24 and 48 2) 16 and 18 3) 4 and 25 4) 6 and 20

Find the value(s) of the variables. 5) 6)

7) 8)

9) 10)

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Summary: In a _______°-_______°-_______° ____________________________ triangle, the length of the hypotenuse is ___________ times the length of one leg. ***The ratio of the sides is _______ - _______ - _______.

7.4 (Day 1) NOTES

Special Right Triangles (45°-45°-90°)

Discovery Activity 1) Here are 5 different squares. For each square, draw a diagonal and use the

Pythagorean Theorem to find the length of each diagonal. Write the length of the diagonal in simplified radical form.

Length of side Length of Diagonal

(simplified radical form) Work (Pythagorean Theorem)

1 unit

2 units

3 units

4 units

5 units

2) According to the data in the table, what should be the length of the diagonal of a

20-unit by 20-unit square? ___________

How long is the diagonal of a square with sides that are 65 units? _________

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EXAMPLES: (Write all answers in the simplest radical form.)

1) The following triangle was formed by cutting a square along its diagonal. What is the length of the hypotenuse?

2) The following triangle was formed by cutting a square along its diagonal. What is the length of the leg (the side of the square)?

3) Find the length of a leg.

4) Calculate the length of the hypotenuse.

5) Calculate the length of the hypotenuse.

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7.4 (Day 1) HOMEWORK

Find the unknown side measures. Write answers in simplest radical form.

1)

2) 3)

4)

5) 6)

7)

8) 9)

y

3

y

y

8 2

x

y

14 x

y

8

x

x

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10) The blades of a helicopter meet at right angles and are all the same length. The distance between the tips of two consecutive blades is 36 ft. Find the length of one of the blades.

11) A ladder is leaning against a house. The base of the ladder is 4 ft from the house and

makes a 45° angle with the ground. How long is the ladder?

12) An isosceles right triangle has legs of length √ . What is the length of the hypotenuse? Draw a diagram and label the measures. Record your answer and fill in the bubbles on the grid.

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7.4 (Day 2) NOTES

Special Right Triangles (30°-60°-90°)

Discovery Activity 1) Here is an equilateral triangle. What are the angle measures? If you draw an altitude,

what happens?

2) Here are 3 equilateral triangles. For each triangle, draw an altitude and the resulting right triangle. Write the length of the short leg, and then use the Pythagorean Theorem to find the length of the longer leg. Write this length as a simplified radical.

3)

Length of side (hypotenuse) Length of

shorter leg Length of longer leg

(Pythagorean Theorem, simplified radical)

Hypotenuse = 2 units

Hypotenuse = 8 units

Hypotenuse = 20 units

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4) What pattern do you notice? According to the data in the table, what should be the

lengths of the legs of a 30°-60°-90° triangle with a hypotenuse of 100 units?

Short leg _________, long leg _________

Examples: 1) The following right triangle is half of an equilateral triangle. What are the lengths of the

short leg and long leg?

2) Given the length of the short leg in the following triangle is 11 cm, what are the lengths of the long leg and hypotenuse?

Summary: In a _______°-_______°-_______° ____________________________ triangle, the length of the hypotenuse is _______ times the length of the short leg, and the long leg is ________ times the length of the short leg.

***The ratio of the sides is _______ - _______ - _______.

Note: The ___________ leg is opposite the _____° angle,

and the ___________ leg is opposite the ______° angle.

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For each of the following, find the two missing side lengths. No decimals!

3)

4)

5)

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7.4 (Day 2) HOMEWORK

Find the missing side lengths. Write answers in simplest radical form. 1)

2) 3)

4)

5) 6)

7)

8)

9) 60

10

x y

x

y

x

y

17

13

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Problem Solving. Make a sketch and label appropriately. 10) A ladder leaning against a house makes a 60 angle with the ground. If the ladder is 12

feet long, how far from the house is the base of the ladder? 11) An equilateral triangle has a perimeter of 18 cm. How long is the altitude? 12)Each half of the drawbridge is about 284 feet

long, as shown below. How high does a seagull (who is on the end of the drawbridge) rise when the angle with measure x° is 30°? 45°? 60°?

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The Story of SOHCAHTOA

Once upon a time, a long time ago, there lived a poor girl on a farm. Her name was Wilma, she lived alone. She lived a humble life that didn’t involve too much change. She started each day tending her onion and carrot garden and ended each day with her favorite passion. She loved dancing. Every night, as the sun began to set, she would retire into her cottage, turn on her CD, and dance the night away. Meanwhile, up at the castle, Prince Fred was wondering what the rest of the world was like. He hopped on his handsome steed and set out to find out what others in the Kingdom did with their days. He made his rounds and bumped into a million different people, but nothing made much of an impression with him until he heard a sound. He rode over to Wilma’s cottage and listened to the unfamiliar rhythm he heard. He knocked on the door and asked what she was doing. She explained that she was dancing to music. He asked her if she would be willing to teach him how to dance and she accepted immediately. Fred was a very slow learner when it came to dancing. It appeared that no matter how hard Wilma tried to get him into the right step, he would somehow loose the beat and wind up stepping on her toe. He left at midnight, but asked if she would mind if he came back for another lesson. Needless to say, the dancing lessons went on, and they fell in love. Fred soon proposed marriage to Wilma and she, of course, accepted. King Fred and Queen Wilma continued their dancing hobby every Friday night. The only problem for Queen Wilma was that no matter how hard she tried to teach King Fred, he remained terribly clumsy. And whenever they went dancing, he would always step on Queen Wilma’s big toe. Queen Wilma learned to live with this slight flaw in their true love relationship. But this big-toe stomping was no laughing matter. So each Saturday morning, after one of their big dancing dates, Wilma would wake up early and sneak out of the castle to her favorite pond. There she would sit for an hour, soaking her big toe until the swelling went down. As the years passed by, the whole kingdom would watch the King and the Queen’s ritual of dancing and soaking with delight. Even after the King and the Queen died, their love story lived on in the hearts of the kingdom. Of course, like all good legends, time changed the facts, as well as the names. King Fred was remembered as King Klutz and Queen Wilma was remembered as Queen Sohcahtoa (soak a toe a).

You can determine the height of objects using TRIGNOMETRIC RATIOS.

A TRIGNOMETRIC RATIO is a ratio of the lengths of two sides of a right triangle.

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7.5/7.7 (Day 1) NOTES Trigonometric Rations (Finding Sides)

A trigonometric ratio is the ratio of the lengths of two sides in a right triangle. You will use these to find either missing side lengths or the acute angle measures in a right triangle. The three trig ratios are: sine, cosine, and tangent. Given below with acute angle A,

Sine sin A = opposite leg to A

hypotenuse

= SOH

Cosine cos A = adjacent leg to A

hypotenuse

= CAH

Tangent tan A = opposite leg to A

adjacent leg to A

= TOA

Use the diagrams and your calculator to complete the chart below. Label the diagrams with respect to angle A. Your calculator must be in “degree mode”.

Example 1: Find the tan K and the cos J as both a fraction and decimal.

tan K = ________________________________________ cos J = ________________________________________

Trigonometric Ratio

Abbreviation Definition Ratio from

Picture Decimal

sine A sin A opp A

hyp

cosine A cos A adj A

hyp

tangent A tan A

opp A

adj A

B

opposite hypotenuse

leg

C adjacent leg A

The adjacent leg is the

one touching the angle. The opposite leg

is the one across

from the angle.

A

B C

5

12 13

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Example 2: Find the values of x and y.

(a)

(b)

Angle of Elevation/Depression

When you look up at an object, the angle your line of sight makes with a horizontal line is called the angle of elevation. If you look down at an object, the angle your line of sight makes with a horizontal line is called the angle of depression.

Example 3: Leo is sitting in a seat on top of a 200-foot Ferris wheel looking down at his brother Jason on the ground at an 80° angle of depression. How far is Jason from the base of the Ferris wheel? Hint: Draw and label a diagram.

Steps: 1. Label the sides with respect to the

given angle. 2. Use the appropriate trig ratio to

write a proportion. 3. Solve the proportion for x.

Something to remember: Sines, cosines, and tangents are just

NUMBERS! So you can multiply and divide with

y

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7.5/7.7 (Day 1) HOMEWORK

Find the indicated trigonometric ratio as a fraction and as a decimal rounded to three decimal places.

1) sin M 2) cos L 3) tan M

Use your calculator to find the value of each ratio to three decimal places.

4) sin 12 = ____________ 5) cos 32 = _______________

Use a trig function to write an equation for each x and y. Then solve.

6) 7) 8) 9) 10) 11)

M

K L

30 34

16

36

8 x

J L

K

64

x

15

x

5 50

y

7.2

70

36

y

9

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12) A handicap ramp has a vertical rise of 24 inches. If the ramp makes an angle of 8 with the ground, how long is the ramp in feet? Draw a picture and label it. Write an equation and solve it.

13) Use the diagram to find the distance across the suspension bridge. 14) The distance from the point directly below a kite to the point where the kite is

anchored to the ground with a string is 84 feet. The angle of elevation along the string to the kite is 65 . Determine the height of the kite, to the nearest tenth of a foot.

15) A lookout tower is 43 m tall. The angle of depression from the top of the tower to a

forest fire is 5 . How far away from the base of the tower is the fire? 16) A flagpole casts a shadow 4.6 m long. The angle of elevation of the sun is 49 . How

tall is the flagpole?

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7.5/7.7 (Day 2) NOTES Trigonometric Functions (Finding Angles)

The INVERSE trigonometric functions are used to find ANGLE MEASURES from given sides of a right triangle. The table below shows the format for setting up equations with inverse trigonometric functions to find angle measures. Given below with acute angle A,

To Find SIDE Measures To Find ANGLE Measures

Sine sin A = opposite

hypotenuse Inverse Sine

1Aopposite

m sinhypotenuse

Cosine cos A = adjacent

hypotenuse Inverse Cosine

1Aadjacent

m coshypotenuse

Tangent tan A = opposite

adjacent

Inverse Tangent

1Aopposite

m tanadjacent

Examples: Find the value of x.

1) 2) 3)

B

opposite hypotenuse leg

C adjacent leg A

The adjacent leg is the

one touching the angle. The opposite leg

is the one across

from the angle.

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SOLVING A RIGHT TRIANGLE To solve a right triangle means to find the measures of ALL its missing sides and angles. You can solve a right triangle if you know either of the following:

1. Two side measures 2. The measures of one side and one acute angle

If using the Pythagorean Theorem or special right triangle ratios is not possible based on the given information, you use SIN, COS, or TAN to calculate side measures. You use SIN-1 (inverse SIN), COS-1 (inverse COS), or TAN-1 (inverse TAN) to calculate angle measures. Example 1: Solve the right triangle (given two side measures).

Example 2: Solve the right triangle (given one side and one angle).

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7.5/7.7 (Day 2) HOMEWORK

Use the diagram to find the indicated measurement. Round to the nearest tenth.

1)

2)

3)

Use the inverse trig functions to find the indicated angle measure in each right triangle.

4) 5) 6)

7) 8) 9)

10) Solve the triangle.

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For each problem, make a diagram (if needed), and label the sides. Write a proportion using the most appropriate trig ratio and solve the proportion. Be sure to answer the question.

11) A clinometer is an instrument used to measure angles of

elevation, slope, or incline. Sam uses a homemade version to estimate the height of a tall tree. He stands about 60 feet away from the tree and looks through the clinometer and records the angle of elevation. After calculating using trig functions, he determines the tree to be about 91 feet tall. What angle did he record?

12) Bobby wants to set his viewscope so that he

can see the tops of the mountains. The mountains are known to be about 3000 feet in elevation. He stands at a point known to be about a mile (5280 feet) away , At what angle should he set his viewscope?

13) A lighthouse is built on the edge of a cliff near

the ocean, as shown in the accompanying diagram. From a boat located 200 feet from the base of the cliff, the angle of elevation to the top of the cliff is 18° and the angle of elevation to the top of the lighthouse is 28°. What is the height of the lighthouse, x, to the nearest tenth of a foot?

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UNIT 7 PERFORMANCE TASK Trigonometric Ratios

Each student will create an original word problem that demonstrates using a trig triangle in a real world situation. This project should be done on a blank sheet of paper. It should show creativity and use of colors. Pictures may be hand drawn or may be taken from a magazine or other periodical. Ink, crayons, color pencil are accepted (regular pencil is not). This project is worth a quiz grade. Example:

Miss Bauer went for a bicycle ride in Chicago. She biked 115 feet to the middle of the

bridge where she stopped to take a picture. After she rode off the bridge, the bridge

opened up to let sailboats go through. The bridge opened up 50 degrees. How high did the

bridge open?

sin = Opposite Hypotenuse

sin

sin ft.

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