A&W Math 11 Working With Trigonometric Ratios Notes

Trigonometry is the study of the relationship between side lengths and angle measures in Right-

Angle Triangles.

There are 3 different ‘ratios’ used in trigonometry they are: Sine, Cosine, and Tangent

The triangle sides have 3 names: _______________ , _________________ , ________________

When labeling the sides of triangles it all depends on where the relevant angle is.

The Trigonometric Ratios:

𝑠𝑖𝑛𝜃 =𝑂𝑝𝑝

𝐻𝑦𝑝 𝑐𝑜𝑠𝜃 =

𝐴𝑑𝑗

𝐻𝑦𝑝 𝑡𝑎𝑛𝜃 =

𝑂𝑝𝑝

𝐴𝑑𝑗

Find the 3 trigonometric ratios for the following triangle:

sinө =

cosө =

tanө =

Given the trigonometric ratios, label the following triangles:

𝑠𝑖𝑛(37°) =3

5 𝑐𝑜𝑠(53°) =

1.5

2.5 tan(37°) =

6

8

ө ө

ө

ө

5

7

9

37°

53° 37°

53°

37° 53°

Using your calculator: Make sure your calculator is in ‘degree’ mode

To solve tan 15° punch the __________ button then ________ then ___________

tan 15° =

Try these ones: sin 61° = cos(8) =

tan 22 = 2sin(78°) =

Finding Angles:

When you want to find an angle using any trigonometric ratio you use the Inverse function

- Push the 2nd or Shift button on your calculator

- Then push sin, cos, or tan

It will show on your calculator like this sin-1 , cos-1 , or tan-1

Try the following: (Don’t forget to use the inverse function)

1. tan(A) = 4

3 A = 2. sinө =

8

7 ө =

3. tanx = 4

5 x = 4. sinB = 0.2543 B =

5. cos(z) = 2

1 z = 6. cosα = 0.08 α =

Example 1. Find the the indicated trigonometric ratio (as a fraction), and calculate ө to one decimal place.

a) tan 𝜃 = b) cos 𝜃 =

ө = ө =

ө

5.2 cm

7 cm

ө 3.9 m

10.1 m

Example 2. Find the missing side of the following triangles to 1 decimal place. a) b) Example 3. Calculate the length of CD Example 4. A women is standing on one side of a deep ravine. The angle of depression to the bottom of the far side of the ravine is 57° and the ravine is 72 m across. How deep is the ravine? Assignment: Pg. 200 #1-2 (#1 – find ratio as a fraction, not 4 decimal places) Pg. 202 #3-5 Pg. 205 #6-8

5.7 cm

6 cm

x

11.3 mm

7.5 mm

z

A&W Math 11 4.1 – Solving for Angles, Lengths, & Distances Notes

Remember:

a2 + b2 = c2 𝑠𝑖𝑛𝜃 =𝑂𝑝𝑝

𝐻𝑦𝑝 𝑐𝑜𝑠𝜃 =

𝐴𝑑𝑗

𝐻𝑦𝑝 𝑡𝑎𝑛𝜃 =

𝑂𝑝𝑝

𝐴𝑑𝑗

Example 1. A rancher who lives near 100 Mile House, BC, has a piece of property shown on the plan below. He is planning to build a new fence, using 3 strands of barbed wire, around the perimeter. How much wire will he need for the fence? Example 2. Soo-Jin is installing carpet in a den. Using the floorplan below, calculate the area of the carpet Soo-Jin will need to buy.

Example 3. Ethan buys a used portable conveyor for stockpiling landscaping materials such as gravel in his maintenance yard. The conveyor has a length along the belt of 12 m, and makes a maximum angle of 15° with the ground.

a) Sketch the conveyor. Label your diagram with the information you have.

b) What is the maximum height of gravel stockpile that this conveyor can make?

c) The gravel stockpile is in the shape of a cone and makes an angle of 37° with the ground. What volume of material can Ethan store in the stockpile?

Example 4. Miranda is installing a pipe with a bend in it. The first part of the pipe is 6.2 m long and makes an angle of 18° below the horizontal. The second part of the pipe is 3.4 m long and makes an angle of 42° below the horizontal. How many metres below the horizontal is the end of the pipe? Example 5. A roller coaster at an amusement park heads upwards at an angle of 40° for 50 m along the slope, then drops right back to ground level over a horizontal distance of 30 m. What is the angle of descent, indicated by x? Assignment: Pg. 210 #1-5

A&W Math 11 4.2 – Solving Complex Problems – Part I Notes

Remember:

a2 + b2 = c2 𝑠𝑖𝑛𝜃 =𝑂𝑝𝑝

𝐻𝑦𝑝 𝑐𝑜𝑠𝜃 =

𝐴𝑑𝑗

𝐻𝑦𝑝 𝑡𝑎𝑛𝜃 =

𝑂𝑝𝑝

𝐴𝑑𝑗

Example 1. Calculate x, y, and h, for the following diagram. Example 2. Calculate CD to the nearest tenth of a centimeter.

Example 3. Eloise is standing on the edge of a building. She can see that the angle of elevation to the top of the next building is 46° and the angle of depression to the bottom of the building is 62°. If the building is 30 m away, how tall is it? Example 4. For a research project, Jordan measures the depth of erosion of certain cliffs near Alberta’s Dry Island Buffalo Jump Provincial Park. In the past, Cree hunters stampeded animals over these cliffs as a hunting technique. Eighty years ago, a surveyor recorded that the cliff was 49.07 m tall. Jordan stands 25 m away from the base of the cliff. The ground and the base of the cliff form a right angle. He finds that the angle of elevation is 62.9°.

a) How tall is the cliff now?

b) How many centimetres has the cliff’s height lost to erosion?

c) What was the angle of elevation 80 years ago?

Example 5. An extension ladder must be used at an angle of elevation of 62°. At its shortest length, it is 15 feet long. Fully extended, it has a length of 27 feet.

a) How much higher up a building will it reach when it is fully extended, compared to its shortest length?

b) How much farther from the house must the base be when it is fully extended, compared to its shortest length?

Assignment: Pg. 219 #4-7 Pg. 223 #1-3

A&W Math 11 4.2 – Part II – Problems in Three Dimensions Notes

Remember:

a2 + b2 = c2 𝑠𝑖𝑛𝜃 =𝑂𝑝𝑝

𝐻𝑦𝑝 𝑐𝑜𝑠𝜃 =

𝐴𝑑𝑗

𝐻𝑦𝑝 𝑡𝑎𝑛𝜃 =

𝑂𝑝𝑝

𝐴𝑑𝑗

Example 1. From the top of a 90-ft. observation tower, a fire ranger observes one fire due west of the tower at an angle of depression of 5o, and another fire due south of the tower at an angle of depression of 2o. How far apart are the fires to the nearest foot? The diagram is not drawn to scale.

Example 2. An architectural firm has been asked to design a hotel that will hang over the edge of a canyon and be built partially into a cliff. Before they design the building, they need to know the height of the cliff. It is difficult to measure the height of the cliff by physical means because of the irregular shape of the rock, and the bulk of the cliff is in the way to get a line of sight from point A. However, there is a line of sight from point B across the edge of the canyon, and the workers were able to make the following measurements.

a) What is the distance from point B to point C at the bottom of the cliff?

b) What is the height of the cliff?

Example 3. Tim looks straight north and sees the top of a building 40 m distant at an angle of elevation of 20°. He looks straight west from the same position and sees the top of a building 80 m distant at an angle of elevation of 14°.

a) Which building is taller, and by how much?

b) What is the straight-line distance between the two buildings?

c) What is the angle of elevation from the top of the lower building to the top of the higher building?

Assignment: Pg. 222 #8-9 Pg. 225 #4-5

A&W Math 11 Scale Drawings Notes

Architects regularly use scale drawings when they design houses and buildings. A scale is a ratio that compares the measurements used in the drawing to the actual measurements. Scale drawings or models are similar to the actual drawing or figure (the sizes are proportional). Ratio for scale drawings is

Scale Measurement : Actual Measurement or

𝑆𝑐𝑎𝑙𝑒 𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡

𝐴𝑐𝑡𝑢𝑎𝑙 𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡

Example 1: If a model car measures 2 cm for every 1 ft of the actual car, the scale ratio is 2 𝑐𝑚

1 𝑓𝑡. Knowing this ratio we can set up a proportion to figure out how long the actual car

would be if the scale model was 36 cm.

We can use cross multiplication to solve this proportion and find the actual length of the car.

(2 𝑐𝑚)(𝑥 𝑓𝑡) = (1 𝑓𝑡)(36 𝑐𝑚) Now divide by 2 cm to find x. Example 2: A photograph measures 4.5 inches wide and 6 inches long. If you have the photograph enlarged to fit a frame 33¾ inches wide, what is the widest the photograph can be?

2 𝑐𝑚

1 𝑓𝑡=

36 𝑐𝑚

𝑥 𝑓𝑡

Example 3: The smallest mammal, Kitti’s hog-nosed bat, has a head-body length of

17

50 inches and a wingspan of about 5

1

10 inches. A scale drawing of this little bat is made

showing it in full flight. The wingspan on the drawing is 15 inches. What should the length of the bat be in the drawing? Example 4: A beluga that is actually 4.2 m long is represented in a chindren’s picture book with the following picture.

a) Measure the drawing and write a scale statement for the picture.

b) An Alligator is drawn at the same scale. In the drawing, it is 5.9 cm long. How long is the actual alligator?

c) How tall will an ostrich be in the picture if it is actually 1.9 m tall?

Example 5: Lorne is building a doghouse using a set of scale plans that do not include dimensions. To work out the amount of plywood he needs, Lorne must work out what the real dimensions will be. So that the doghouse is big enough for his dog, he would like the doghouse to be 5 feet long.

a) What scale factor will Lorne use to enlarge the plan?

b) What will the measurements of the actual doghouse be?

c) Lorne’s dog is about 2¼ feet tall. How tall and wide do you think the door in the front wall of the doghouse should be?

Example 6: Mrs. Lewis wants to have a new house built. She used graph paper to sketch some thoughts for a possible floor plan for her house. The bold outline in the figure below represents the outline of the first floor of the house. (Each square represents 9 inches in real life). If Mrs. Lewis intends to have the dining room floored with wood flooring that comes in 1 foot squares which cost $2.70 each, what will it cost to buy enough squares to cover her dining room floor? Assignment: Pg. 241 #1-6