unit 6 trigonometry - ms threatful's classes · unit 6 – trigonometry 19 6.7 solving right...

Workplace Math 10

Name: ________________________________________________________________________________ Block: ______________

Unit 6 – Trigonometry

6.1 The Pythagorean Theorem 6.2 Introduction to Trigonometry 6.3 The Sine Ratio 6.4 The Cosine Ratio 6.5 The Tangent Ratio 6.6 Determining the Appropriate Trigonometric Ratio 6.7 Solving Right Triangle Problems

Workplace Math 10 Unit 6 – Trigonometry

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6.1 The Pythagorean Theorem A right triangle (one with a _________ angle) has two ______________ that form the right angle. The side

opposite the right angle is called the _________________________________.

The sides of a right triangle form a relationship known as the ____________________________________________.

In this theorem, the area of the square on the hypotenuse is equal to the sum of the areas of the

squares on the legs. The formula for this is __________________________________, where 𝑎 and 𝑏 are the

_____________ and 𝑐 is always the length of the ________________________________.

Pictorially Symbolically


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Examples Ex 1. Find the length of side 𝑐.

Ex 2. Find the length of the unknown side. Round to the nearest tenth of a metre.

6.1 Practice

1. Use the Pythagorean theorem to find each missing side length to the nearest tenth. a) b)


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c) d)

2. A path is being constructed between the corners of a park. What is the length of the path?

3. Find the height of the kite to the nearest tenth of a metre.

4. Luc is building a shed. What should the measure of the diagonal be so that the 12 ft wall is perpendicular to the 10 ft wall? Round to the nearest tenth.


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5. A 10 foot ladder is leaning against the side of a house. If the base of the ladder is 5 feet from the wall, how far up the wall does the ladder reach, to the nearest tenth?

6. The size of a flat screen TV is given by the length of the diagonal of the screen. What is the size of this TV, to the nearest inch?

7. Bobby placed a 12 ft ladder against a pillar as shown. Does the pillar form a right angle with the ground? How do you know?

8. Challenge question: Find the length of AB in this box (from the back bottom left corner to the front top right corner) to the nearest metre.


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6.2 Introduction to Trigonometry

Trigonometry deals with relations between the sides and angles of triangles. In this course, we will

only deal with right triangles (those with a _________ angle). We name the sides of a triangle according

to the non-right angle that we know or want to find, which we often label as ____________________________.

The names of the sides of the triangle The names of the sides of the triangle in relation to the angle at A are: in relation to the angle at B are:

Ex 1. Label the hypotenuse as well as the opposite and adjacent sides in relation to angle theta.

a) b)

The special trigonometric functions, __________________________, _______________________________, and

___________________________________ help us to find a missing angle or side of a right triangle. They are

simply one particular side of the triangle divided by another (called a _________________).


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The primary trigonometric ratios are: sin 𝜃= cos 𝜃 = tan 𝜃 = where 𝜃 is the measure of the angle. Ex 2. Triangle 1 Triangle 2 Triangle 3

a) Find sin 37 ° (the sine ratio for the 37° angle) for each of the three triangles above. What do

you notice?

b) Use your calculator to find sin 37 °. Round to four decimal places.

c) Find cos 37 ° (the cosine ratio for the 37° angle) for each of the three triangles above. What do you notice?

d) Use your calculator to find cos 37 °. Round to four decimal places.

e) Find tan 37 ° (the tangent ratio for the 37° angle) for each of the three triangles above. What do you notice?

f) Use your calculator to find tan 37 °. Round to four decimal places.


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Before calculators, mathematicians used tables containing all of the trigonometric ratios. Your

calculator has been programmed with all of these values so you don’t have to use the table

We know what each ratio should be for a particular angle, so we can find missing angles in right

triangles. Since we are going backwards from the ratio to get the angle, this requires using the

________________________ trigonometric ratios: _________________, ________________ and _________________.

Ex 3. Find each angle to the nearest degree. First use the table above, then use your calculator.

a) cos 𝜃 = 0.7431 b) tan 𝐴 = 19.1 c) sin 𝐵 = 0.4


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6.2 Practice


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x

6.3 The Sine Ratio

Recall the formula for the sine ratio: Examples Ex 1. Calculate the length of the indicated side. Round to one decimal place.

a) b) c) d) Ex 2. Find the measure of the indicated angle. Round to the nearest degree.

a) b)

x


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6.3 Practice


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15. In this roof truss, what is the height, h, to the nearest tenth of an inch? (Recall 1 ft = 12 in)

16. How long is the wheelchair ramp, to the nearest tenth of a metre?

17. The Leaning Tower of Pisa, in Italy, leans because the ground underneath it is unstable. When these measurements were taken, what was p, the angle of the tilt of the tower? Round to the nearest degree.

18. How long is a wire that is attached 4.2 meters up a pole if it makes an angle of 52° with the

ground? Round to one decimal place.


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6.4 The Cosine Ratio Recall the formula for the cosine ratio: Ex 1. Calculate the length of the indicated side. Round to one decimal place.

a) b) Ex 2. Find the measure of the indicated angle. Round to the nearest degree. 6.4 Practice


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13. How far from the base of a house is a 40-foot ladder if the angle it makes with the ground is 72°?

Round to one decimal place.

14. A 20.3 m long cable used to support a tower is anchored to the ground 12.5 meters from the

tower’s base. What angle does the cable make with the ground? Round to the nearest degree.


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6.5 The Tangent Ratio

Recall the formula for the tangent ratio:

Examples Ex 1. Find each missing measure to one decimal place.

a) b) 6.5 Practice


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9. A guy wire is attached to a tower at a point that is 10 m above the ground. The wire is

anchored 21 m from the base of the tower. What angle, to the nearest tenth of a degree, does the guy wire make with the ground? Draw a diagram and show your work.

10. What is the height of the roller coaster, to one decimal place?


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6.6 Determining the Appropriate Trigonometric Ratio

We can use this acronym to remember the formulas for the three trigonometric ratios:

You will need to identify what you have and what you need in any given question to determine which

of the three to use.

Examples Ex 1. Find the length of the indicated side or the measure of the indicated angle to one decimal place.

a) b)

c) d)


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6.6 Practice


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6.7 Solving Right Triangle Problems

If Juliet is standing at the window and looking down at Romeo, the angle that Juliet’s line of sight makes with the horizontal is known as the angle of _____________________________________________.

The angle that Romeo’s line of sight makes with the horizontal is the angle of ________________________________________________________________.

Examples Ex 1. Determine the angle of elevation to the top of a 5-meter tree at a point 3 meters from the base

of the tree. Ex 2. The angle of depression to a boat from the top of a 150m cliff is 20°. How far is the boat from

the base of the cliff?


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6.7 Practice

1. A plumber is laying drainage pipe for a septic system. For every 300 cm of horizontal distance, there should be a 2.5 cm drop in height for the water to run. At what angle to the surface must she lay the pipe? Round to one decimal place.

2. After an hour of flying, a jet has travelled 300 miles, but strong winds have blown the jet off course. The instruments in the cockpit show that the jet is 48 miles west of the planned flight path. By how many degrees is the jet off course? (Round to one decimal place).

3. Roxanne is sitting on the ground, at R, watching fireworks. The diagram shows one of the fireworks exploding directly above her head. a) At what height above ground does it explode, to the nearest yard?

b) About how many yards does it travel before exploding?


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4. Glenda is a forester. She uses a clinometer, a device that measures angles of elevation, to sight the top of a tree at 48°. Her eyes are 1.6 m above the ground, and she is 7.2 m from the tree. How tall is the tree, to one decimal place?

5. Arthur, a golfer, hit his golf ball 230 yd from the tee.

a) How far is the hole from the tee, to the nearest yard?

b) How many yards is the ball from the hole?

6. A building code states that for stairs, the steepest angle is 72 cm of rise for each 100 cm of run. What is the steepest angle for building stairs, to the nearest degree?


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ANSWERS

Section 6.1

1. a) 10 cm b) 7.5 yd c) 7.9 cm d) 13.9 ft 2. 45 m 3. 110.1 m 4. 15.6 ft 5. 8.7 ft 6. 46 in 7. No, since 𝑎2 + 𝑏2 ≠ 𝑐2 8. 10 m

Section 6.2 1. 0.8824 2. 0.2800 3. 1.3333 4. 0.7241 5. 3.4286 6. 0.7241 7. 0.7536 8. 0.9455 9. 0.9205 10. 0.0000 11. 0.2756 12. 8.1443 13. 23° 14. 43° 15. 52° 16. 64° 17. 7° 18. 50° Section 6.3 1. 8.2 2. 5.3 3. 20.8 4. 17.8 5. 19.4 6. 22.7 7. 10.9 8. 25.4 9. 77° 10. 69° 11. 53° 12. 22° 13. 27° 14. 18° 15. 42.2 in 16. 28.7 m 17. 4° 18. 5.3 m

Section 6.4 1. 8.8 2. 19.1 3. 39.9 4. 7.5 5. 14.3 6. 25.0 7. 66° 8. 60° 9. 46° 10. 33° 11. 69° 12. 41° 13. 12.4 ft 14. 52° Section 6.5 1. 67.2 2. 5.0 3. 25.6 4. 41.8 5. 29° 6. 30° 7. 40° 8. 27° 9. 25.5° 10. 29.8 m Section 6.6 1. 24° 2. 53° 3. 22° 4. 46° 5. 23° 6. 66° 7. 74° 8. 64° 9. 21.7 10. 20.8 11. 32.2 12. 3.4 13. 15.9 14. 3.9 15. 30.0 16. 4.9

Section 6.7 1. 0.5° 2. 9.2° 3. a) 75yd b) 85 yd 4. 9.6 m 5. a) 228 yd b) 32 yd 6. 36°

unit 6 trigonometry - ms threatful's classes · unit 6 – trigonometry 19 6.7 solving right...

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