MFM2PUnit 4 – Trigonometry

Name: ___________________________

Date

Topic / Learning Goal

Learning Goal Achieved?

Fri. Mar 21

Pythagorean Theorem

· I can use Pythagorean Formula to find a missing side of a right triangle

Mon. Mar 24

Labelling a Right Triangle

· Use a reference angle to label the opposite and adjacent side

· Use the right angle to label the hypotenuse

· Create the sin, cos and tan ratio

Tues. Mar 25

Primary Trig Ratios to solve for sides

· I can use the primary trigonometry ratios to solve for a missing side

Wed. Mar 26

Solving missing angle

· I can solve a missing angle using a trig ratio

Thurs. Mar 27

Which method is best? / review

· I can choose the appropriate method to solve a missing side or angle in a right triangle

Fri. Mar 28

Elevation and Depression

· I can label the angle of elevation and depression and use them to solve real-world trig problems

Mon. Mar 31

Clinometers/Real World Problems

· I can create a clinometer and use it to find the height of a tall object.

Tues. Apr 1

Real World Problems

· I can use a trig ratio to solve real-world trigonometry problems

Wed. Apr 2

REVIEW!

Thurs. Apr 3

UNIT TEST - TRIGONOMETRY

Date _________________

MFM 1P1

Pythagorean Theorem

What is a right triangle?

The longest side of a right triangle is called the

1. Determine the length of the unknown side.

a)b)c)

15cm

8cm

15mm

3mm

15m

6.3m

2. When Arnold swims laps in his rectangular swimming pool, he swims along the diagonal so he doesn’t have to turn around so often. Find the distance Arnold travels by swimming once along the diagonal.

3.2m

9.8m

3.A 5m ladder is leaning against a wall. The top of the ladder touches the wall 4m above the ground. How far is the base of the ladder from the wall? (Draw a diagram!!)

Name __________________

Date ___________________

MFM 2P1

Pythagorean Theorem Assignment

1) Determine the length of the missing side in each of the following. Round decimal answers to 1 decimal place. Show all of your work on a separate piece of paper.

a)b)c)

7 mm

9 mm

15 cm

10 cm

3 cm

11 cm

25 m

d)e) f)

18 m

12 cm

5 m

30 m

9 cm

The following problems MUST be answered in correct form. All answers must include proper units. need not copy the diagrams, but if a diagram is not given in the question, you MUST draw one. All answers are to be rounded to 1 decimal place.

2)

You wish to strengthen a door by

nailing on a diagonal brace as shown.

x How long should the brace be?

97 cm

20 cm

How far is it from the cabin across the lake to the beach?

3)

Beach

Cabin

2.4 km

3 km

Ranger Station

4) Bob has let out 35 m of kite string when he observes that his kite is directly above Betty. How high is the kite?

35 m

Bob Betty

19 m

5) A ladder which is 8.5 m long leans against a wall. The foot of the ladder is 2.3 m from the base of the wall [include a diagram].

a) How far up the wall does the ladder reach?

b) The top of the ladder slips 1 m down the wall. How far will the foot of the ladder be from the base of the wall now? [You will need a second diagram]

Labelling Right Triangles & Intro to Trigonometry

Trigonometry is simply the study of __________________________________ measurements.

In grade 10, we will be dealing ONLY with __________________________ (90°) triangles.

Trigonometry looks at ______ of the three sides in relation to an "indicated" angle.

Before we can jump into trigonometry we need to know the proper names for the sides in a triangle

How we label a right triangle DEPENDS ON WHERE THE REFERENCE ANGLE is… BUT the hypotenuse is always across the 90 degree angle.

Since angle A is the “REFERENCE” angle, how would you label the sides of the triangle below?

C

BB

A

START with the hypotenuse, then label the opposite, and the adjacent sides relative to each marked reference angle.

a)b) c)d)

Label the sides of the triangles below, then complete the following statements.

a) In triangle JKL….

· the length of the hypotenuse is _______________

· the length of the opposite side is ______________

· the length of the adjacent side is ______________

We can write ratios using the lengths of each side. These ratios have special names:

TOA

CAH

SOH

The Primary Trig Ratios compare an angle in a right angle triangle to the ratio of two of its sides. We use them to solve for missing angles or a side.

Let’s learn how to use our calculators. Find the value of the following rounded to 4 decimal places.

STEPS

If using the table (back side)

If using scientific calculator

1. Find angle (under angle column)

2. Look under appropriate column (sine, cosine or tangent)

3. Write value with 4 decimal places

1. Put in DEGREE mode

2. Press trig (sin/cos/tan) button

3. Type angle value

4. Press enter

5. Round to 4 decimal places

Practice:a) sin 28º = b) cos57º = c) tan89º =

Trigonometric Table

Note: all angles are measured in degrees

Practice on Trig. Ratios Using the Table

When you can’t find the exact decimal in the table, choose the closest match!

Fill in the chart below using the 2 given sides and the reference

angle. Note: The reference angle is the marked angle.

LABEL THE SIDES OF EACH TRIANGLE:

HYPOTENSUE, OPPOSITE, ADJACENT

Trig Ratio

Ratio of Side Names

Numeric Ratio

Ratio in Decimal Form

Size of Reference angle in Degrees (look in chart!)

Can You Fill in the Missing Pieces?

Fill in the chart below using the given information.

LABEL THE SIES OF EACH TRIANGLE:

HYPOTENSUE, OPPOSITE, ADJACENT

Trig Ratio

Ratio of Side Names

Numeric Ratio

Ratio in Decimal Form

Size of Reference angle in Degrees

TAN

OPP

HYP

18

19.5

FINDING A MISSING SIDE

Now, we need a way to solve for a missing side.

Let’s look at a ratio equationsin A = __opp___

hyp

When we are finding the missing side… we use the RATIO from the chart.

Practice: Finding the ratio given the angle

(a) sin 75° = ______________(b) tan 32° = ______________ (c) cos 65° = ______________

(d) tan 45° = ______________ (e) cos 45° = ______________(f) sin 90° = ______________

(g) cos 6° = ______________(h) tan 82° = ______________ (i) sin 56° = ______________

Examples: Finding Missing Side

1. Determine the length of x to the nearest tenth of a metre.

STEPS

1. Label the sides

2. Choose SOH CAH TOA

3. Fill in what you know

4. Find the decimal

5. Cross multiply

6. Include units

CROSS MULTIPLICATION NOTE

Solve for unknown:

1. = 2. =

Let’s use this with trig examples *CALCULATOR MUST BE IN DEGREE MODE!!!

Example:Sin25 =

1. Type sin 25 into calculator to find

decimal(or use trig table!)

2. Cross multiply

3. Solve for unknown

What do we do when the unknown is on the bottom?

Example:Cos 37 =

1. Put Cos 37 as fraction over 1

2. Find Cos 37 using calculator

3. Cross multiply

4. Solve for unknown

MORE MISSING SIDES

Let’s use SOH CAH TOA to help us find the side of a right angled triangle when we are given one side and one angle (other than the right angle). Round answers to 1 decimal place. Diagrams are not drawn to scale.

43º

12 cm

x

x

58º

15 m

a) b)

51º

20 cm

x

32º

x

9 m

c) d)

28 cm

60º

x

y

e)

How to Find a Missing Angle of a Triangle

STEPS

1. Label the sides

2. Choose SOH CAH TOA

3. Fill in what you know

4. Find the ratio (divide top by bottom)

5. Find the angle in the chart

a) b)

X

11mm

25mm

X

22cm

33cm

c) d)

X

18mm

9mm

21

33

X

· Label the sides of each triangle below

· Compete the statements

· Circle the correct ratio

· Find the angle

a)

x

14 cm

10 cm

I have the ________________

and the ______________

Circle one: SOH CAH TOA

Find the angle

b)

20 cm

x

9cm

I have the ________________

and the ______________

Circle one: SOH CAH TOA

Find the angle

c)

11.2m

7.3 m

x

I have the ________________

and the ______________

Circle one: SOH CAH TOA

Find the angle

EXTRA PRACTICE: FINDING A MISSING ANGLE

Solve for the angle θ, to one decimal place.

Solving Trigonometry Problems

Date: ________________

MFM 2P

Angles of elevation and depression are always with reference to the line _______________ to the ground.

Elevation

Depression

Example 1:

Joe needs to build a ramp to load cars onto a transport truck. The truck is 1.5 m off the ground and the angle of elevation of the ramp must be 55º. How long must Joe make the ramp?

Example 2:

A tunnel 2 035 m below the peak of a mountain is being build. The angles of elevation to the peak are 40º and 48º. To estimate costs, the length of the tunnel must be found. Find the length of the tunnel. This is a mult-step problem.

Example 3:

From a cliff 106 m high, two ships are spotted. The angle of depression to the nearer ship is 27º. The angle of depression to the further ship is 12º.

a) How far away from the base of the cliff is the nearer ship, A?

b) How far away from the base of the cliff is the further ship, B?

Example 4:

Given the diagram below, determine the height of the larger building. This is a multi step problem.

55º

32º

51.5 m

1. Determine the unknown side

24m

37◦

x

Answer:

2. Determine the unknown side.

3.

4.

5. Determine the missing angle.

x

3cm

20cm

Answer:

6. What is the best ratio to use (SOH CAH or TOA) to find x:

x

20◦

4m

5. A tow truck raises the front end of a car 0.85 m above the ground. The car is 3.85 m long. What angle does the car make with the ground?

· 6. A 7.9 m ladder is leaning against a 5.2 m high wall. How far away from the base of the building is the ladder?

7. 7. Dalton is standing infront of a pole at an angle of 60 degrees. He notices that Chloe is standing behind the pole at a 70º angle with the ground. If the pole is 10m tall, how far is Dalton from Chloe?

Chloe

Dalton

10m

70◦

60◦

hypotenuse

opposite

=

q

sin

hypotenuse

adjacent

=

q

cos

adjacent

opposite

tan

=

q

x

25

cm

35

º