Name: __________ Period GH

UUNNIITT 99:: TTRRIIGGOONNOOMMEETTRRYY

I can define, identify and illustrate the following terms:

Sine Cosine Tangent θ (Theta) Angle of elevation Angle of depression

Cosecant Secant Cotangent Law of Sines Law of Cosines

Dates, assignments, and quizzes subject to change without advance notice.

Monday Tuesday Block Day Friday

2

3

Intro to Trig Ratios

4/5

8-3: Solving Triangles

& 8-5: Law of Sines

and Law of Cosines

6

8-4: Problem Solving

9

Review

10

TEST

11/12

The Unit Circle

13

UNIT CIRCLE

MINI-TEST

Tuesday, 1/3/12

8-2: Trigonometric Ratios

� I can write a trig ratio.

� I can set up an equation using trig ratios.

� I can set up an equation using trig ratios to solve a real world problem.

PRACTICE: Introduction to Trigonometry Worksheet

Wednesday, 1/4/11 or Thursday, 1/5/12

8-3: Solving Right Triangles

� I can use a calculator to find the decimal value of a given trigonometric ratio.

� I can use a calculator to find the angle for a given trigonometric ratio.

� I can solve a trigonometric equation.

� I can solve for missing sides and angles using the Law of Sines and Law of Cosines

PRACTICE: Pg 529 #23-43 (odd), 44-50 & Pg 538 #30-35, 67-70 & Pg 555 #27-37 (odd)

Friday, 1/6/12

8-4: Angles of Depression & Elevation / Problem Solving

� I can set up and solve a real world problem using trigonometric equations.

PRACTICE: Applications of Trigonometry Worksheet

Monday, 1/9/12

Review

PRACTICE: Review Worksheet

If you need more practice, try pg 573 – 574, #12 – 23. (Answers are in the back.)

Tuesday, 1/10/12

���� Test 17: Trigonometry Score:

Wednesday, 1/11/12 or 1/12/12

Unit Circle

� I can write the ordered pair for the Unit Circle.

� I can solve equations using the Unit Circle.

Friday, 1/13/12

���� Mini Test: Unit Circle Score:

Introduction to Trigonometry

The field of mathematics called Trigonometry is the study of ___________ triangles and the ratios between

the sides.

There are 3 of these relationships that we study:

• Sine is the ratio of the ________________ side to the _________________.

• Cosine is the ratio of the ________________ side to the _________________.

• Tangent is the ratio of the __________________ side to the ____________________ side.

• The ________________ NEVER changes, but ____________ and _____________ are dependent on the

_________ used. The _________ angle is NEVER used.

The three sides of the triangles are referred to as Hypotenuse (H), Adjacent (A), and Opposite (O).

Label each side of each triangle using angle W as your reference.

Ex 1. Ex 2. Ex 3.

1. 2. 3.

To help you remember these relationships, you can use the phrase ________ _______ ________.

Where: S: sine (sin) O: opposite

C: cosine (cos) A: adjacent

T: tangent (tan) H: hypotenuse

The trigonometric ratios are written in an equation form. The Greek letter __________________ (θ ) is often

used to represent an angle. Our angles will always be measured in _____________.

Sineθ

= Cosineθ

= angent θ

Τ =

W

P

W M M

W P

W

W

Y Z W

Y Z

�Turn the page. Worksheet continues.

Use the triangle at the right to determine the following ratios. Be sure to simplify your answers!

Ex 4. sin 40° = Ex 5. sin θ =

Ex 6. cos 40° = Ex 7. cos θ =

Ex 8. tan 40° = Ex 9. tan θ =

Use the triangle at the right to determine the following ratios. Be sure to simplify your answers!

4. sin 40° = 5. sin 50° =

6. cos 40° = 7. cos 50° =

8. tan 40° = 9. tan 50º =

Set up equations using trig ratios that could be used to solve for the variable.

10. 11. 12.

13. 14. 15.

16. 17. 18.

17

w

38°

x

24

62°

z 50°

35

17

25

w° 28

z°

51

32

14

x°

40º

n 10

m

θ

40º

n 10

30

31 25

θ k

30°

10

d

14

18°

�Turn the page. Worksheet continues.

You can also take the reciprocal of each trigonometric function.

The Reciprocal Trigonometric Ratios are as follows:

• Reciprocal of Sine Function:

Cosecant (csc) is the ratio of the ________________ side to the _________________.

Cosecant is also: 1

cscsin

θθ

=

• Reciprocal of Cosine Function:

Secant (sec) is the ratio of the ________________ side to the _________________.

Secant is also: 1

seccos

θθ

=

• Reciprocal of Tangent Function:

Cotangent (cot) is the ratio of the ________________ side to the _________________.

Cotangent is also: 1

cottan

θθ

=

Use the triangle at the right to determine the following ratios. Be sure to simplify your answers!

Ex 9. csc θ = 1. csc 40° =

Ex 10. sec θ = 2. sec 40° =

Ex 11. cot θ = 3. cot 40° =

Set up equations using trig ratios that could be used to solve for the variable.

19. 20. 21.

csc w° = csc x° = csc z° =

sec w° = sec x° = sec z° =

cot w° = cot x° = cot z° =

θ

40º

5 4

3

30

40

w° 21

z°

35

25

20

x°

50 15

28

Name:_________________________ Period _____ GH

Trigonometry Applications

Set up equations that could be used to solve each problem.

Step 1: Draw a picture

Step 2: Label picture

Step 3: Pick the best trig ratio

Step 4: Set up equation

Ex. 1 Angie looks up at 25 degrees to see an airplane flying toward her. If the plane is flying at an

altitude of 3.5 miles, how far is it from being directly above Angie?

Ex. 2 A six foot vertical pole casts a shadow of 11 feet. What is the angle of elevation with the ground?

Ex. 3 Lauren is at the top of a 15 meter tall lookout tower. She looks down at an angle of depression of

25° and sees Evan coming toward her. How far is Evan from the base of the tower?

1. What is the angle of elevation if you stand 850 feet away from a cliff that is 400 feet high and look at

the top?

2. The string of a flying kite makes an angle of 63° with the ground. If all 250 feet of string are out, and

there is no sag in the string, how high is the kite?

3. Tal’s hill at Minute Maid Park has an elevation of 30°. If the hill has a six foot vertical rise, how

long is its hypotenuse?

�Turn the page. Worksheet continues.

25°

3.5 miles

Angle of depression – looking down from

a horizontal line

Angle of elevation – looking up from a

horizontal line

4. Joey is putting up an antenna. At the 30 foot mark, he attaches a 50 foot guy wire. What angle does

the guy wire form with the antenna?

5. A person at the top of a cliff 100 feet tall sees Gilligan’s boat. His sighting of the boat is at an angle

of depression of 10°. How far is the boat from the base of the cliff?

6. A 24 foot ladder is leaned against a wall at 55° with the ground. How far away from the wall is the

base of the ladder?

7. A 32 in. bat is leaning against a fence. If the bat is 15 in. away from the base of the fence, what angle

is formed between the ground and the bat?

8. A plane takes off at an elevation of 20°. In its path, 500 feet away from the takeoff point, is a 170-ft

tall tower. Will the plan clear the tower? If yes, by how much?

9. Ana knows that she is one mile from the base of a tower. Using a protractor she estimates an angle

of elevation to be 3°. How tall is the tower to the nearest foot? (1 mile = 5280 feet)

10. The base of an isosceles triangle has a length of 16cm. and the vertex angle measures 68º. What is

the length of each leg? Round to the nearest tenth of a cm.

11. Matt hiked to the top of the smaller cliff shown below. From the top, he could see the bottom of the

large cliff at an angle of depression of 25°. He could see the top of the large cliff at an angle of

elevation of 20°. Find the height of each cliff (x and y).

x

120 meters

25o y 20

o

Trigonometry Test Review

Solve each problem. You will need to use your own paper. Please remember to round side lengths to the

nearest hundredth and angles to the nearest degree.

1. Find x and y 2. 3.

Solve for x.

4. 5. 6.

7. A tree casts a shadow of 28 m. The elevation of the sun is 49°. How tall is the tree?

8. Shane is 61 feet high on a ride at an amusement park. The angle of depression to the park entrance is

42°, and the angle of depression to his friends standing below is 80°. How far from the entrance are

his friends standing?

9. A 30 foot tree broke from its base and fell against a house. If the tree hit the house 18 feet above the

ground, what angle is the tree forming with the house?

10. A freeway entrance ramp has an elevation of 15°. If the vertical lift is 22 feet, what is the distance

along the ramp?

11. Lauren is at the top of a 15 m lookout tower. From an angle of depression of 25°, she sees Evan

coming toward her. How far is Evan from the base of the tower?

12. Rosalinda has a rocket that can travel 1500 feet before exploding. On the 4th

of July, she lights the

rocket at an elevation of 75°. How high will the rocket be when it explodes?

13. Mark has two sticks, 25 in. and 20 in. If he places them end to end perpendicularly, what two acute

angles would be formed when he added the hypotenuse?

42’

71o 67

o

y

x

12 16

52° θ

x 15

29

θ 23º

x

42°

12x 2x +8

78°

18x2

6x

37°

14x – 3

11x

Name: Period:

The Unit Circle

� We’re going to be using special right

triangles, so first, remind yourself of the

patterns of 45-45-90 triangles and 30-60-

90 triangles:

� Use special right triangles to fill in the lengths of the missing sides. NO DECIMALS.

� Now use what you know about trigonometry to write the following sine and cosine ratios for the

angles shown in these triangles. SIMPLIFY THOSE RATIOS!

sin 30° =

cos 30° =

sin 45° =

cos 45° =

sin 60° =

cos 60° =

� Next we’ll take one of those triangles and put it on a coordinate

plane:

What would the x- and y- coordinates of point A (formed by a 30° angle) be? (Hint: Use the lengths of the sides of the

triangle.)

� Do the same for a 45° angle and a 60° angle:

If you look at the three points we

have traced out so far, they all lie on a

circle centered at the origin.

What is the radius of this circle?

45°

45°

60°

30°

45°

1

60°

1

30°

1

30°

1

(0,0)

A

1

(0,0)

B

45°

1

(0,0)

C

60° 1

The circle is known as the unit circle, since its radius is one unit. It is an integral part of trigonometry.

You will cut out the triangles on the next page to help fill in the coordinates of this unit circle. Use what

you know about reflections to help!

HINTS FOR QUADRANT II:

The angles in quadrant II are 120°, 135°, and 150°.

Look at the 120° point and the 60° point.

What should be true about their y-coordinates?

What should be true about their x-coordinates?

( ) ,

( ) ,

( ) ,

( ) ,

( ) ,

( ) ,

( ) ,

( ) ,

( ) ,

( ) ,

( ) ,

( ) ,

( ) ,

( ) ,

( ) ,

( ) ,

� Use the work you did with the triangles on the front page to fill in the following table:

What relationship do you notice

between the coordinate on the unit

circle and the values of sine and

cosine?

� Look at the point shown below, which corresponds to 90°.

What are the coordinates of this point? So what is the value of sin 90°? What is the value of cos 90°?

What would be the coordinates of the point that corresponds to 0°? What is sine of 0°? What is cosine of 0°?

� More practice: Use the unit circle you just made – your answers should be exact, NO DECIMALS.

1. sin 120° =

2. cos 180° =

3. cos 135° =

4. sin 315° =

5. cos 240° =

6. sin 330° =

(Hint: Write the sides lengths inside each triangle)

Angle

measure

Coordinate

on unit circle

Value of

sine

Value of

cosine

30°

45°

60°

60° 45° 30°

The unit circle is powerful because it can be used to solve trigonometric equations with

exact answers rather than decimals that have to be rounded.

For example:

6sin 45

x° =

From the unit circle,

2sin 45

2° =

2 6

2 x=

Substitute

2 12x = Cross-multiply

2 2 Divide to isolate x

12 2 12 26 2

22 2x = ⋅ = =

Simplify

More importantly, the unit circle lets you solve these trig equations without a calculator,

which you will often be expected to do in future math classes.

Solve:

cos3018

x° =

20sin 315

n° =

5cos 210

m° =

You should either memorize the unit circle or be able to use

triangles to recreate it at a moment’s notice.