Geom Unit 8 Teacher Notes 2019 PDF

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UNIT 8: Right Triangle Trigonometry

Geometry 2018-2019

Assignments for Geometry Unit 8 Right Triangles and Trigonometry

Assignment (Due the next class meeting) Unit 7 Test HW: Radicals Review Worksheet

Day Thursday Friday

Date 1/31/19 (A) 2/01/19 (B)

Monday Tuesday

2/04/19 (A) Notes: 8.1 Special Right Triangles 2/05/19 (B) HW: 8.1 Worksheet

Wednesday Thursday

2/06/19 (A) Notes: 8.2 The Sine, Cosine, and Tangent Ratios 2/07/19 (B) HW: 8.2 Worksheet

Friday Monday

2/08/19 (A) Notes: 8.3 The Inverse Sine, Cosine, and Tangent Ratios 2/11/19 (B) HW: 8.3 Worksheet

Tuesday Wednesday

2/12/19 (A) 2/13/19 (B)

Notes: 8.4 Solving Right Triangles HW: 8.4 Worksheet

Thursday Friday

2/14/19 (A) 2/15/19 (B)

Unit 8 Practice Test HW: Unit 8 Practice Test STUDY!!!

Tuesday Wednesday

2/19/19 (A) 2/20/19 (B)

Unit 8 Test 9.0 Circle Vocabulary HW: Plate Activity

NOTE: You should be prepared for daily concept checks. HW reminders:  You will receive a 2% bonus grade for completing all of your homework.  If you cannot solve a problem, get

help before the assignment is due.

 Help is available before school, during lunch, or during IC.  For extra practice, visit www.khanacademy.org, Mathguy.us or Youtube 1

UNIT 8: Right Triangle Trigonometry

Geometry 2018-2019

Note: Round all answers in this unit to one decimal place, unless otherwise specified.

8.1 Special Right Triangles Essential Question: What can you say about the side lengths associated with special right triangles?

Note: When solving for the side lengths of a right triangle, we notice a patterns in the side lengths that appear more often than others. These are called Common Pythagorean Triplets.

Examples:

2

UNIT 8: Right Triangle Trigonometry

Geometry 2018-2019

45°-45°-90° Triangles: Isosceles Right Triangles

Examples: Find the value of x and y. Simplify any radical answers. 1.

2.

3.

4.

3

UNIT 8: Right Triangle Trigonometry

5.

Geometry 2018-2019

6.

A student claims that if you know one side length of an isosceles right triangle, then you know all the side lengths. Do you agree or disagree? Explain.

B

Explain how to find y in the right triangle at right. D

1 y

A

1

C

YOU TRY: 7.

8.

4

UNIT 8: Right Triangle Trigonometry

Geometry 2018-2019

30°-60°-90° Triangles

Examples: Find the value of x and y. Simplify any radical answers. 1. 2.

3.

4.

5.

6.

YOU TRY: 7.

8. 5

UNIT 8: Right Triangle Trigonometry

9.

Geometry 2018-2019

A student drew a right triangle with a 60 ° angle and a hypotenuse of length 10. Then he labeled the other side lengths as shown. Correct how you can tell just by glancing at the side lengths that the L student made an error. Then explain the error.

10

J

10.

A square room has a perimeter of 36 feet. How long is the diagonal of the room?

11.

A right triangle has a hypotenuse of 15 feet. How long could each of the sides be?

12.

In the right triangle, x and y What is the length of side x ? A.

2

B.

4

C.

2 √3

D.

3 √2

5

K

represent unknown side lengths.

6

UNIT 8: Right Triangle Trigonometry

Geometry 2018-2019

8.2: The Sine, Cosine, and Tangent Ratios Essential Question: How can I find the tangent, sine, and cosine ratios for an acute angle of a right triangle?

TRIG FUNCTIONS Trigonometry is the study of the relationships between the sides and angles of right triangles. The legs are called adjacent or opposite depending on which acute angle is being used. The hypotenuse is always the longest side sin A

=

cos A =

tan A =

An easy way to remember ratios is to use SOH-CAH-TOA.

7

UNIT 8: Right Triangle Trigonometry

Geometry 2018-2019

A few important points: 

Always reduce ratios (fractions) when you can.



Use the Pythagorean Theorem to find the missing side (if there is one).



If there is a radical in the denominator, rationalize the denominator.

Trigonometric Ratios

Example 1: Find the requested fraction. sin ∠ A=¿

cos ∠ A=¿

tan ∠ A=¿

sin ∠C=¿

cos ∠ C=¿

tan ∠ C=¿

Note: We only find trig functions for the _______________ angles of a right triangle! Example 2: Find the following values by using your calculator. Round to 4 decimal places. Your calculator must be in Degrees! a . sin 18 °

b . cos 68 °

c.

tan 32 °

d.

tan

80

°

Steps for using trig functions to solve for a variable: 1) Decide what trig function matches your picture. 2) Make an equation. Trig function angle = fraction. 3) Solve for x by using inverse operations.

Examples: Round to the hundredth.

8

UNIT 8: Right Triangle Trigonometry

3.

Geometry 2018-2019

4.

5.

7.

8.

YOU TRY! 6.

YOU TRY! 9. What is the value of x in the right triangle to the right? Choose all that apply. A)

16 sin 35°

B)

sin 35 ° 16

C)

16 cos 35°

D)

16 sin 35 °

Angle of Elevation Angle of Elevation is the “line of sight” angle made between an object and the ground. What is the angle of elevation in the picture shown?

9

UNIT 8: Right Triangle Trigonometry

Geometry 2018-2019

Examples: 10. For the picture shown, the height of the balloon is 24 feet. Find the length of the hypotenuse of the triangle (round answer to nearest tenth). Can you solve this in a different way?

11. An airplane makes a 15 degree angle of elevation from the runway when it takes off. airplane pictured below is 2,000 feet along the ground from its take-off point. Find the height, h, of the airplane (round answer to nearest foot).

12.

A 12-foot ladder is leaning up against the side of a house. The ladder makes an angle of the ground. How far up the side of the house does the ladder reach? A. 10.6 ft

B. 13.6 ft

C. 5.6 ft.

62°

with

D. 25.6 ft

YOU TRY: 13.

Find the requested fraction.

sin ∠ A=¿

cos ∠ A=¿

tan ∠ A=¿

sin ∠C=¿

cos ∠ C=¿

tan ∠ C=¿

14.

What is cos x ° A.

40 41

in the triangle? C.

41 9

10

UNIT 8: Right Triangle Trigonometry

9 41

B.

15.

B.

16.

9 40

D.

Which angle has a cosine of ∠A

A.

∠ B

C.

Geometry 2018-2019

3 5

?

∠C D.

None of the abov

In the right triangle, x and y represent unknown side lengths. What are the lengths of sides x∧ y ? Leave answers in simplest radical form.

REVIEW: In the figure, what is the distance a ball travels when thrown from second base to home plate? A 50 feet . B.

100 feet

C . D .

25 √2 feet 50 √2 feet

8.3 Inverse Trig Functions 11 Essential Question: How can I calculate the angle if I know the sine, cosine, or tangent of that acute angle?

UNIT 8: Right Triangle Trigonometry

Geometry 2018-2019

Trigonometric Ratios x =cos x =tan x =¿ sin ¿

HOW TO FIND MISSING ANGLES (INVERSE FUNCTIONS) To find an angle on your calculator, you must choose the correct trig function according to the sides labeled, −1 and you must use the inverse trig function buttons on your calculator. They might look like this: sin , Arcsin, etc… ***Make sure your calculator is in degree mode*** Example 1: Use your calculator to find the angle (round to nearest whole angle): a. b. cos ∠ B=0.5 tan x=1.33

c.

sin ∠ A=.8990

d.

cos x=0.397

Example 2: Use your calculator to find x: Note: decide if you should use the trig function or its inverse. sin 60=x cos x=.75 b. a.

c.

tan 30=x

d.

tan x=1

12

UNIT 8: Right Triangle Trigonometry

Geometry 2018-2019

Example 3: Find x in each picture. Trig function angle = fraction. a.

b.

c.

More Examples: Solve for the variable(s) in each triangle. Write your answer as simplest radical, when possible. 4.

5.

Example 6:

What is the value of A if

A.

30 °

C.

45 °

B.

3 sin A= √ 2

?

60 ° D.

Not here

Example 7: A 15-foot ladder is leaning against a building. If the ladder hits the building at a height of 10 feet, find the angle of elevation.

Example 8: A skateboard ramp is 3.5 feet high and 6 feet long along the horizontal. To the nearest degree, what is the measure of the angle that the ramp makes with a horizontal line? 13

UNIT 8: Right Triangle Trigonometry

Geometry 2018-2019

A.

27 °

B.

60 °

C.

30 °

D.

63 °

Example 9:

Name as many ways as possible to find the measure of ∠ N ?

8.4 Solving Right Triangles

Essential Question: How can I find unknown angles or sides of a right triangle if I know two side lengths or one side length and the measure of one acute angle?

WHAT DOES IT MEAN TO SOLVE A RIGHT TRIANGLE?

Example 1:

Find ALL of the missing sides and angles in each picture. Remember: Trig function angle = fraction. a.

∠C= BC = AB=

b.

∠C= ∠ A= AC

14

UNIT 8: Right Triangle Trigonometry

c.

∠ A=

Geometry 2018-2019

d.

AB =

BC =

∠ A=

AB=

∠C=

Example 2: One of the support wires for a radio tower is 100 feet long. One end of the wire is 40 feet from the base of the tower, as shown in the diagram. Find the measure of the angle, x, that the support wire makes with the ground. Round to the nearest tenth.

15

UNIT 8: Right Triangle Trigonometry

Geometry 2018-2019

Example 3: A lighthouse, which is 18 feet high, stands on a cliff that is 150 feet above sea level. The distance from the top of the lighthouse to a sailboat on the ocean is 360 feet. What is the angle of elevation, x, from the sailboat to the top of the lighthouse? Round your answer to the nearest degree.

right triangle… If I know… 2 sides 2 sides A side and an angle 2 angles

I should use… the Pythagorean Theorem inverse trig function trig function sum of 180 degrees

to find the… 3rd side related angle related side 3rd angle

YOU TRY: Solve all the missing parts to each of the right triangles. 1.

2.

D

O 4 C

G

11

O

13 W

Example 4: A person is standing at ground level with the base of the Empire State Building in New 16

UNIT 8: Right Triangle Trigonometry

Geometry 2018-2019

York City. The angle formed by the ground and a line segment from his position to the top of the building is 48.4 ° . The height of the Empire State Building is 1472 feet. Find the distance that he is standing from the base of the Empire State Building to the nearest foot. 8 feet 1968 feet A. C. B.

1307 feet

D.

2217 feet

17...