unit 1 - trigonometry - booklet€¦ · title: microsoft word - unit 1 - trigonometry -...

Workplace 30

Unit 1 – Trigonometry

WA30.3 : Solve problems that involve the sine law and cosine law, excluding the ambiguous case. [CN, PS, V]

* Adapted from Pacific Educational Press MathWorks 12

WA 20 – Unit 1 – Slope and Rate of Change

Lesson 1 – Proportional Reasoning

Date: _______________


Proportion – A fractional statement of equality between two ratios or rates

Proportional Reasoning – The ability to understand and compare quantities that are related multiplicatively

Proportional Reasoning

1) Multiply by a denominator to x by its self

2) Divide to get x by itself

(If you forget, you can remember to cross multiply and divide)

Example 1:

3

4 16

x

2 16

5 x

5

5 25

x

Setting up a Proportion

1) Set up a ratio as a fraction

2) Set up a second ratio equal to the first fraction

3) If you are solving for an unknown, put x in the place of the unknown

** REMEMBER ** Keep your ratio the same on the top and bottom in both ratios



Date: _______________

Example 2: If a carpenter bonds two pieces of wood with epoxy resin, she must first mix the epoxy with a hardener. She mixes these materials in a ratio of 10 to 1, where there is 10 parts of epoxy to 1 part of hardener. This ratio can be written as 10:1 or as

a fraction, 10

1. If the carpenter wants to use 150 parts of epoxy, how many part of hardener would she need?



Date: _______________

Example 3: You are in charge of making punch for party. To make one batch of punch you need 2 cups of juice and 5 cups of sprite. How many cups of sprite do you need to make 28 cups of punch?

Example 4: John has found that he can arrange the work cubicles of his employees’ best if the ratio between the length and width of a room is 3:2. If a room is 6 meters long, how wide should the room be?



Date: _______________

Worksheet 1 Solve for x.

1. 40

10 50

x 2.

12 18

16 x

3. 56

64 8

x 4.

18 36

27 x

5. 3

2056 4

x 6.

3 15

12 x

7. 3

5 460

x 8.

25 40

200x



Date: _______________

9. On a bicycle with more than one gear, the ratio between the number of teeth on the front gear and the number of teeth on the back gear determines how easy it is to pedal. If the front gear has 30 teeth and the back gear has 10 teeth, what is the ratio of front teeth to back teeth?

10. Some conveyor belts have two pulleys. If one pulley has a diameter of 45 cm and another has a diameter of 20 cm, what is the ratio of the smaller diameter to the larger diameter?

11. For a silk screening project, Jan mixes a shade or orange ink. She uses a ratio of red in to yellow ink of 2:3 and yellow ink to white ink of 3:1.

a) How many mL of yellow ink would she need if she used 500mL of white ink?

b) How many mL of red ink would she need if she used 750 mL of yellow ink?

12. The ratio of flour to shortening in a recipe for piecrust is 2:1. If a baker makes 30 cups of piecrust, how many cups of flour and shortening does he use?

13. If a type of salami at the deli costs $1.59 per 100 g, how much will you pay for 350 g?

WA30 – Unit 1 – Trigonometry Name: _________________

Lesson 2 – Finding Length Using the Sine Law Date: __________________

Lesson 2 – Finding Length Using The Sine Law Labelling a Triangle 1) Label the vertex’ A, B, C 2) The side opposite side from the vertex the lower case of the angle it is opposite from

Example 1: Label the following triangle.

Acute Triangle – A triangle which has all acute angles (less than 90°).

The Sine Law – In any acute triangle: = =

Finding Ratio’s in a Triangle 1) Label all angles and sides 2) Write ratios of each side and angle 3) Make them equivalent

Example 2: In the following triangles label all equivalent ratios.

a) b)

Sketching a Triangle from a Ratio 1) Sketch a triangle 2) Label the degree’s 3) Label the corresponding side based on the ratio

Example 3: Sketch the triangle from the following ratios: °=

°=

.

°

46.5° 83.3°

50.4°




When Can you Use the Sine Law - When you know two sides and the angle opposite a known side.

- When you know two angles and any side

Determine the Length of a Side using the Sine Law: 1) Set up the Sine Law with your side lengths in the numerator 2) Fill in all values known from the triangle 3) If you know two angles find the third angle by taking 180° subtract the known angles 4) Determine which ratio you need to use (only can have one unknown) 3) Multiply by your denominator to isolate your variable 4) Solve for your variable Example 1: a) In ∆𝐺𝐻𝐽, determine the length of GH to the nearest tenth of a centimetre. b) Determine the length of side h to the nearest tenth of a centimetre. Example 2: A triangle has angles measuring 80° and 55°. The side opposite the 80° angle is 12.0 m in length. Determine the length of the side opposite the 55° angle to the nearest metre.

b a

c BA

C

j 7.8 cm

h J-

G

H

51⁰ 44⁰



Lesson 2 – Worksheet A) Find the length in each triangle:



B) Sketch the following triangles using the Sine Ratio

1) =. 2) .

=.

3. =. 4) .

=.



1.

2.

3.

4.

5.



6.

7.

8.

9.

10.


Lesson 3 – Finding Measures Using the Sine Law Date: __________________

Lesson 3 – Finding Measures Using the Sine Law

The Sine Law – In any acute triangle: = = Determine the Measure of an Angle Using the Sine Law: 1) Set up the Sine Law with your angle in the numerator 2) Fill in all values known from the triangle 3) Determine which ratio you need to use (only can have one unknown) 4) Multiply by your denominator to isolate your variable 4) Solve for your variable using sin-1 to find the angle’s measure Example 1 - In ∆𝑀𝑁𝑃, determine the measure of <N to the nearest degree.

Solve a Triangle – Means to find all missing sides and angles Example 2 – Solve the following triangle.

b a

c BA

C

11.8 cm

10.0cm N

M

P

92⁰

5 cm 8 cm

FD

E

80⁰



Lesson 3 – Worksheet 1)

2)

3)

4)

5)



Solve Each Triangle

WA30 – Unit 1 –Trigonometry Name: _________________

Lesson 4 – Solving Problems Using the Sine Law Date: __________________

Lesson 4 – Solving Problems Using the Sine Law


or

= =

Solving Problems With Sine Law 1) Read the problem all the way through 2) Underline information that is important to solving the problem 3) Circle what the question is asking 4) Draw a diagram 5) Use the Sine Law to solve the problem

Example 1: Toby uses chains attached to hooks on the ceiling and a winch to lift engines at his father’s garage. The distance between the two chains attached at the ceiling is 2.8 m. One chain is 1.9 m long and the other is 2.2 m long. The angle made between the two chains at the winch is 86°. Determine the angle each chain makes with the ceiling.

Measuring Direction – Directions are often stated in terms of north and south on a compass. They are written as degree’s east or west from the North direction or the south direction.

Example 2: Draw a compass showing the directions N30°E and S45°W.

Example 3:: The captain of a small boat is delivering supplies to two lighthouses, as shown. His compass indicates that the lighthouse on his left (lighthouse A) is located N30°W and the lighthouse to his right (lighthouse B) is located N50°E. He knows from his map that the two lighthouses are 12 km apart, and he is 9 km away from lighthouse A. Determine how much farther away the captain is from lighthouse B.

b a

c B

c A

C


Lesson 4 – Solving Problems Using the Cosine Law Date: __________________

Lesson 4 – Worksheet 1. A post is supported by two wires (one on each side going in opposite directions) creating an angle of 80° between the wires. The ends of the wires are 12m apart on the ground with one wire forming an angle of 40° with the ground. Find the lengths of the wires. 2. 3 friends are camping in the woods, Bert, Ernie and Elmo. They each have their own tent and the tents are set up in a Triangle. Bert and Ernie are 10m apart. The angle formed at Bert is 30°. The angle formed at Elmo is 105°. How far apart are Ernie and Elmo? 3. Two scuba divers are 20m apart below the surface of the water. They both spot a shark that is below them. The angle of depression from diver 1 to the shark is 47° and the angle of depression from diver 2 to the shark is 40°. How far are each of the divers from the shark?



4. Two observers are standing on shore ½ mile apart at points F and G and measure the angle to a sailboat at a point H at the same time. Angle F is 63° and angle G is 56°. Find the distance from each observer to the sailboat. 5. Points A and B are on opposite sides of the Grand Canyon. Point C is 200 yards from A. Angle B measures 87° and angle C measures 67°. What is the distance between A and B? 6. A 4m flag pole is not standing up straight. There is a wire attached to the top of the pole and anchored in the ground. The wire is 4.17m long. The wire makes a 68° angle with the ground. What angle does the flag pole make with the wire?


Lesson 5 – Finding Lengths Using the Cosine Law Date: __________________

Lesson 5 – Finding Lengths Using the Cosine Law Cosine Law – In any acute triangle, and can solve problems where the Sine Law does not apply.

c2 = a2 + b2 – 2abCosC

a2 = b2 + c2 – 2bcCosA

b2 = a2 + c2 – 2acCosB

When Can you Use the Cosine Law - When you know all three sides in a triangle

- When you have two sides and the contained angle

Using the Cosine Law to Determine the Length of a Side

1) Write your equation so that the unknown side is your first variable

2) Substitute the given measures

3) Solve for your missing length, make sure to take the square root of both sides

Example 1: Determine the length of BC to the nearest tenth of a centimeter.

Example 2: In Triangle ABC side AC is 32m, angle A is 58° and side AB is 40m. Determine the length of side CB.

C B

A

c

a

b

17 cm

a C-

B

A

10 cm

31⁰



Lesson 5 – Worksheet


Lesson 6 – Finding Measures Using the Cosine Law Date: __________________

Lesson 6 – Finding Measures Using the Cosine Law Cosine Law – In any acute triangle, and can solve problems where the Sine Law does not apply.




Using the Cosine Law to Determine the Measure of an Angle

1) Write the Cosine statement that includes the Cos of your missing angle

2) Substitute the given measures

3) Solve for your missing angle

Example 1: The diagram below shows the plan for a roof. The building code requires that the angle formed at the peak of a roof to fall within a range of 70° to 80° so that snow and ice will not build up. Will this plan pass the local building code.

Example 2: In ∆KMN, KM = 16 cm, KN = 11 cm, and MN = 6 cm; determine the measure of ∠M to the nearest degree.

C B

A

c

a

b

19.5’ 10’

20’



Lesson 6 – Worksheet 1)

2)

3)

4)



Solve Each Triangle



Lesson 7 – Solving Problems Using the Cosine Law Cosine Law – In any acute triangle, and can solve problems where the Sine Law does not apply.




Solving Problems Using the Cosine Law

1) Read the problem all the way through 2) Underline any important information to help you solve the problem 3) Circle what the question is asking

4) Draw a diagram 5) Set up the Cosine Law 6) Solve for your answer

Example 1: A three pointed star is made up of an equilateral triangle and three congruent isosceles triangles. Determine the length of each side of the equilateral triangle knowing the lengths of the isosceles triangle is 60 cm and the contained angle is 20°.

Example 2: The pendulum of a grandfather clock is 100.0 cm long. When the pendulum swings from one side to the other side, the horizontal distance it travels is 9.6 cm. Determine the angle that the pendulum swings.

C B

A

c

a

b



Lesson 7 – Worksheet 1. Two ships are sailing from Halifax. The Nina is sailing due east and the Pinta is sailing 43° south of east. After an hour, the Nina has travelled 115km and the Pinta has travelled 98km. How far apart are the two ships? 2. To estimate the length of a lake, Caleb starts at one end of the lake and walks 95m. He then turns and walks on a new path, which is 120° to the direction he was first walking in, and walks 87m more until he arrives at the other end of the lake. Approximately how long is the lake?

3. Two ships leave port at 4 p.m. One is headed at a bearing of N 38 E and is traveling at 11.5 miles per hour. The other is traveling 13 miles per hour at a bearing of S 47 E. How far apart are they when dinner is served at 6 p.m.?



4. The distance on a map from the airport in Miami, FL to the one in Nassau, Bahamas is 295 kilometers due east. Bangor, Maine is northeast of both cities; its airport is 2350 kilometers from Miami and 2323 kilometers from Nassau. What bearing would a plane need to take to fly from Nassau to Bangor?

5. After the hurricane, the small tree in my neighbor’s yard was leaning. To keep it from falling, we nailed a 6-foot strap into the ground 4 feet from the base of the tree. We attached the strap to the tree 3½ feet above the ground. How far from vertical was the tree leaning?

6. You are heading to Beech Mountain for a ski trip. Unfortunately, state road 105 in North Carolina is blocked off due to a chemical spill. You have to get to Tynecastle Highway which leads to the resort at which you are staying. NC-105 would get you to Tynecastle Hwy in 12.8 miles. The detour begins with a 18 veer off onto a road that runs through the local city. After 6 miles, there is another turn that leads to Tynecastle Hwy. Assuming that both roads on the detour are straight, how many extra miles are you traveling to reach your destination?

unit 1 - trigonometry - booklet€¦ · title: microsoft word - unit 1 - trigonometry -...

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