Foundations and Pre-Calculus 10

Unit 2 - Trigonometry FP10.4 Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles

*Adapted from Pearson’s Foundations and Pre-Calculus Mathematics 10

F10 – Unit 2 – Trigonometry Key Terms

Date: _______________

Key Terms Trigonometry – The study of the properties and applications of triangles. Similar Triangles – Triangles that have the same shape but can have different sizes. Corresponding Sides - Matching sides in a similar triangle. Corresponding Angles – Matching angles in a similar triangle. Proportional Reasoning – Using ratio’s to solve for missing variables. Pythagorean Theorem – The rule that states that, for any right triangle, the are of the square on the hypotenuse is equal to the sum of the areas of the squares on the legs. Hypotenuse – The side across from the right angle. Adjacent Side – Side that is touching the angle. Opposite Side – The side opposite to the angle. Tangent Ratio – For an acute A in a right triangle, the ratio of the length of the side opposite A to the length of the side adjacent to A. Written as tan A. Acute Angle – An angle that is less than 90º Angle of Inclination: the acute angle between the horizontal and a line or line segment Primary Trigonometric Ratios – sin, cos, tan Sin Ratio – For an acute A in a right triangle, the ratio of the length of the side opposite A to the length of the hypotenuse. Written as sin A. Cosine Ratio – For an acute A in a right triangle, the ratio of the length of the side adjacent to A to the length of the hypotenuse. Written as cos A. Solving a Triangle – means to determine the measure of all the angles and all the lengths of all the sides in the triangle. Angle of Depression – the angle between the horizontal and the line of sight from an observer. Angle of Elevation – The angle between the horizontal through eye level and a line of sight to a point above eye level.

F10 – Unit 2 – Trigonometry Unit Checklist

Date: _______________

Unit 2 - Checklist

Lesson 1 – Video Lesson 1 - Worksheet Lesson 1 – Portfolio

Lesson 2 – Video Lesson 2 Homework: Page 75 #3, 4, 5, 8, 10, 14

Lesson 2 – Portfolio Questions

Lesson 3 – Video Lesson 3 Homework: Page 82 #3-8

Lesson 3 – Portfolio Questions

Quiz Review: Page 88 # 1-5 Quiz

Lesson 4 - Video Lesson 4 - Page 95 #4, 5, 6, 7, 8, 10, 12

Lesson 4 – Portfolio Question

Lesson 5 – Video Lesson 5 – Page 101 #3, 4, 5, 6, 7, 11

Lesson 5 – Portfolio Question

Lesson 6 – Video Lesson 6 Homework: Page 111 #3, 4, 6, 10

Lesson 6 – Portfolio Question

Lesson 7 – Video Lesson 7 Homework: Page 118 #3, 6, 8, 10

Lesson 7 – Portfolio Question

Test Review: Page 124 #1, 3, 5, 6, 11, 12, 13, 15, 16, 18, 19, 23

Portfolio – Post Questions Portfolio – Big Questions Portfolio – Unit Summary

Test

F10 – Unit 2 – Trigonometry Lesson 1 – Similar Triangles

Date: _______________

Lesson 1 – Similar Triangles Corresponding Sides - Matching sides in a similar triangle. Corresponding Angles – Matching angles in a similar triangle. For similar triangles ABC and A′B′C′:

Corresponding Angles Corresponding Sides

A = _____; AB = _____

B = _____; BC = _____

C = _____ AC = _____

Similar Right Triangles We know Right Triangles will be similar if one of the following cases is true: 1) One pair of corresponding acute angles is equal In Triangles PQR and P′Q′R’: P = _____ R = _____ 2) The ratio of two side lengths is equal

QP

PQ =

RP

PR or

QP

PQ =

RQ

QR or

RQ

QR =

RP

PR

** Set up proportion, simplify the fraction **

Example 1: Why would the following triangles be similar?

20º

20º

10

20

2

4

F10 – Unit 2 – Trigonometry Lesson 1 – Similar Triangles

Date: _______________

Pythagorean Theorem - The rule that states that, for any right triangle, the area of the square of the hypotenuse is equal to the sum of the areas of the squares on the legs. Hypotenuse – The side directly opposite from the 90° in a right triangle. We write: 𝑙 + 𝑙 = ℎ

1) Substitute values in for each variable 2) Isolate unknown variable 3) Solve for that variable

Example 2: In ∆BCD, determine the length of BD to the nearest tenth of a centimeter.

Example 3: In ∆EFG, determine the length of EF to the nearest tenth of a centimeter.

F10 – Unit 1 – Trigonometry Name: ____________________ Lesson 1 – Similar Triangles Worksheet

Lesson 1 – Worksheet

1. Which triangles below are similar to ΔXYZ? Explain how you know.

a) b) c) d) e) f)

F10 – Unit 1 – Trigonometry Name: ____________________ Lesson 1 – Similar Triangles Worksheet

2. Solve each equation.

a) 12 = 3

m b) 10 =

2

t c)

2.1

s = 5 d)

9

v = 1.1

3. Solve each equation.

a) 10 = v

3 b) 24 =

u

8 c)

r

6.3 = 6 d)

t

12 = 0.4

4. Determine the length of each side AB to the nearest tenth of a centimeter.

a) b) c) d)

F10 – Unit 2 – Trigonometry Lesson 2 – Tangent Ratio

Date: _______________

Lesson 2 - The Tangent Ratio Using Tangent Ratio to Determine the Measure of an Angle 1) Determine which side is opposite to the angle 2) Determine which side is adjacent to the angle 3) Find the tangent ratio 4) Find the tangent ratio as a decimal 5) Use 1tan to find the measure of the angle Example 1: Find the tangent ratio for G and F. Find the measure of G and F Example 2: Find the measure of A to the nearest degree.

F10 – Unit 2 – Trigonometry Lesson 2 – Tangent Ratio

Date: _______________

Angle of Inclination: the acute angle between the horizontal and a line or line segment. Steps for Word Problems Using Angle of Inclination: 1) Read the question ALL the way through 2) Draw a diagram to understand what the question is asking 3) Determine your opposite and adjacent sides 4) If you are missing an opposite or adjacent side, use Pythagorean Theorem to find it. 5) Find tangent ratio 6) Solve to find the angle of inclination Pythagorean Theorem: 𝑙 + 𝑙 = ℎ Example 3: A guy wire is fastened to a cell-phone tower 8.5 m above the ground. The wire is anchored to the ground 14 m from the base of the tower. What angle, to the nearest degree, does the wire make with the ground? Example 4: A support cable is anchored to the ground 5 m from the base of a telephone pole. The cable is 19 m long. It is attached near the top of the pole. What angle, to the nearest degree, does the cable make with the ground?

F10 – Unit 2 – Trigonometry of Right Triangles Lesson 3 – Tangent Ratio - Lengths

Date: _______________

Lesson 3 – Tangent Ratio - Lengths Using the Tangent Ratio to Find Lengths

1) Set up tangent ratio 2) Determine which value is the variable you want to solve for 3) Isolate your variable 4) Solve for variable

Example 1: Find the length of BC to the nearest tenth of a centimeter. Example 2: Find the length of PQ to the nearest tenth of a centimeter. Example 3: A ladder leans on a wall, as shown in the diagram below. How far up the wall does the ladder reach? Give your answer to the nearest tenth of a meter.

F10 – Unit 2 – Trigonometry of Right Triangles Lesson 4 – The Sine and Cosine Ratio

Date: _______________

Lesson 4 – The Sine and Cosine Ratio

Finding the Measure of an Angle Using Sin 1) Determine your values of opposite, adjacent, and hypotenuse 2) Substitute values into ratios 3) Find ratio as a decimal 4) Use sin-1 or cos-1 to solve for the measure of the angle Example 1: In DEF determine the Sine and Cosine ratio for <D. Using each ratio solve for the measure of the angle.

F10 – Unit 2 – Trigonometry of Right Triangles Lesson 4 – The Sine and Cosine Ratio

Date: _______________

Example 2: Find the measures of each angle using the Sine and Cosine ratios. Angle of Elevation – The angle formed between the horizontal and the line of sight when looking upward. Angle of depression – the angle formed between the horizontal and the line of sight when looking down. Example 3: An observer is sitting on a dock watching a float plane in a Vancouver harbour. At a certain time, the plane is 300 m above the water and 430 m from the observer. Determine the angle of elevation of the plane measured from the observer to the nearest degree.

F10 – Unit 2 – Trigonometry of Right Triangles Lesson 5 – The Sine and Cosine Ratio - Length

Date: _______________

Lesson 5 – The Sine and Cosine Ratio - Length Using Cosine to Solve for Lengths 1) Determine which values you are given, label hypotenuse, adjacent and opposite 2) Set up cosine ratio 3) Determine which value is the variable you want to solve for 4) Isolate your variable 5) Solve for variable Example 1: Find the length of RS to the nearest tenth of a meter. Example 2: Find the length of the hypotenuse to the nearest tenth of a centimeter. Example 3: The angle of a cable from a point 12.5 metres from its base is 52˚. How long is the cable? Example 4: A surveyor makes the measurements shown in the diagram to find the distance between two observation towers on the opposite sides of a river. How far apart are the towers? Give the answer to the nearest meter.

F10 – Unit 2 - Trigonometry Lesson 6 – Applying the Trigonometric Ratios

Date: _______________

Lesson 6: Applying the Trigonometric Ratios Solve Triangle – Find all the misisng measures and all missing lengths in a triangle.

** Remember: The sum of all angles in a triangle is 180° ** Example 1: Solve the following triangle. Example 2: Solve the following triangle to the nearest tenth of a metre. Example 3: Solve the following triangle to the nearest tenth of a metre.

F10 – Unit 2 - Trigonometry Lesson 7 – Solving Problems

Date: _______________

Lesson 7 - Solving Problems Involving More than One Right Triangle

Solving Problems with More Than One Triangle 1. Fill in all unknown lengths and angles 2. Use the unknowns you fill in to find the missing side/angle you need

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

Example 2: Jason is lying on the ground midway between two trees, 100 m apart. The angles of elevation of the tops of the trees are 13 degrees and 18 degrees. How much taller is one tree than the other? Give the answer to the nearest tenth of a meter.

F10 – Unit 2 - Trigonometry Lesson 7 – Solving Problems

Date: _______________

Example 3: For a small plane, V, the angle of depression of a sailboat is 21 degrees. The angle of depression of a ferry on the other side of the plane is 52 degrees. The plane is flying at an altitude of 1650 m. How far apart are the boats, to the nearest meter? Example 4: A surveyor stands at a window on the 9th floor of an office tower. He measures the angles of elevation and depression of the top and the base of a taller building. The surveyor sketches this plan of his measurements. Determine the height of the taller building to the nearest tenth of a meter.