Grade 11 Pre-Calculus Mathematics

[MPC30S]

Chapter 2

Trigonometry

Outcomes

T1, T2, T3

11P.T.1. Demonstrate an understanding of angles in standard position. 11P.T.2. Solve problems, using the three primary trigonometric ratios (sine, cosine, and tangent) for angles from 0° to 360° in standard position. 11P.T.3. Solve problems, using the cosine law and sine law, including the ambiguous case.

MPC30S Date: ____________________

Pg. #2

MPC30S Date: ____________________

Pg. #3

Chapter 2 – Homework

Section Page Questions

MPC30S Date: ____________________

Pg. #4

MPC30S Date: ____________________

Pg. #5

Outcome T1

Chapter 2 – Trigonometry 2.1 – Angles in Standard Position

Angles in Standard Position

• In geometry, an angle is formed by two rays with a common endpoint or ____________

• The starting position is called the _______________ arm.

• The final position is called the _______________ arm.

• If the angle of rotation is ___________________________________, then the angle is ____________________.

• If the angle of rotation is __________________, then the angle is ________________.

MPC30S Date: ____________________

Pg. #6

An angle in a coordinate plane is in ______________________________ if:

• _________________________________________

• _________________________________________

• Angles in standard position are always shown on the Cartesian Plane. The x – axis and the y – axis divide the plane into four ____________________.

MPC30S Date: ____________________

Pg. #7

Example #1 Sketch each angle in standard position. State the quadrat in which the terminal arm lies. a) 35°

b) 230°

c) 310°

d) −210°

MPC30S Date: ____________________

Pg. #8

Reference Angles

• For each angle in standard position, there is a corresponding acute angle (𝜃 < 90°) called the reference angle.

• A reference angle is the angle that is formed between the _______________________

and the ____________________.

• The reference angle is always ______________________ and measures between __________ and __________.

Quadrant I: Quadrant II: Quadrant III: Quadrant IV:

Example #2

Determine the reference angle, 𝜃., for each angle 𝜃. Sketch 𝜃 in standard position and label the reference angle 𝜃.. a) 140° b) 300°

MPC30S Date: ____________________

Pg. #9

Outcome T2

Chapter 2 – Trigonometry 2.2 – Part 1: Trigonometric Ratios of Any Angle

Suppose that 𝜃 is an angle in standard position, and P(𝑥, 𝑦) is any point on its terminal arm, at a distance 𝑟 from the origin. We can determine the length of 𝑟 using ______________________________________ Recall the three trigonometric ratios:

sin 𝜃 =oppositehypotenuse cos𝜃 =

adjacenthypotenuse tan 𝜃 =

oppositeadjacent

There may be re-written in terms of 𝑥, 𝑦 and 𝑟:

MPC30S Date: ____________________

Pg. #10

We can determine the exact value for any angle, 𝜃, where 0° < 𝜃 < 360° using a reference triangle. The chart below from your textbook summarizes the signs of the trigonometric ratios in each quadrant, and shows you where each reference triangle is located.

MPC30S Date: ____________________

Pg. #11

Example #1 Determine the exact value of sin 𝜃, cos𝜃 and tan 𝜃 for the angle whose terminal arm goes through the point P(2,−5). Also, determine the value of 𝜃. Example #2

Given sin 𝜃 = D

√F and 𝜃 is located in quadrant ΙΙ, determine the exact value of cos𝜃 and

tan 𝜃.

MPC30S Date: ____________________

Pg. #12

Example #3 The point P(𝑥, 𝑦) is on the terminal arm of 𝜃 = 35°. The distance, 𝑟, between P and the origin is 8 units. To the nearest tenth, determine the coordinates of P.

MPC30S Date: ____________________

Pg. #13

Outcome T2

Chapter 2 – Trigonometry 2.2 – Part 2: The Unit Circle

The ______________________ is a circle with a radius of 1 and centered at the origin. Any point on the Unit Circle can be written 𝑃(𝜃) = (𝑥, 𝑦) and can be found using trigonometric ratio. Since the radius is one unit, then: 𝑥 = 𝑦 = Therefore, 𝑃(𝜃) = (cos𝜃 , sin 𝜃)

MPC30S Date: ____________________

Pg. #14

Special Triangles Applied to the Unit Circle 45° FAMILY à This family contains all angles that have a reference angle of 45°. Example: ________________________________, etc. sin 45° = ________ cos45° = ________ tan 45° = ________

MPC30S Date: ____________________

Pg. #15

60° FAMILY à This family contains all angles that have a reference angle of 60°. Example: ________________________________, etc. sin 60° = ________ cos60° = ________ tan 60° = ________

MPC30S Date: ____________________

Pg. #16

30° FAMILY à This family contains all angles that have a reference angle of 30°. Example: ________________________________, etc. sin 30° = ________ cos30° = ________ tan 30° = ________

MPC30S Date: ____________________

Pg. #17

MPC30S Date: ____________________

Pg. #18

Example #1 State the exact value for each of the following. State which quadrant each angle is found. State the corresponding reference angle.

a) cos 135° = Quadrant: __________ 𝜃.: __________

b) tan 330° = Quadrant: __________ 𝜃.: __________

c) sin 240° = Quadrant: __________ 𝜃.: __________

d) tan 120° = Quadrant: __________ 𝜃.: __________

e) sin 315° = Quadrant: __________ 𝜃.: __________

f) cos 210° = Quadrant: __________ 𝜃.: __________

g) tan 150° = Quadrant: __________ 𝜃.: __________

h) cos 330° = Quadrant: __________ 𝜃.: __________

i) cos120° = Quadrant: __________ 𝜃.: __________

j) sin 225° = Quadrant: __________ 𝜃.: __________

k) sin 150° = Quadrant: __________ 𝜃.: __________

l) tan 315° = Quadrant: __________ 𝜃.: __________

m) cos 240° = Quadrant: __________ 𝜃.: __________

n) sin 60° = Quadrant: __________ 𝜃.: _________

MPC30S Date: ____________________

Pg. #19

90° FAMILY à This family contains all intercepts. Example: ________________________________, etc. Example #2

For the following angles in standard position, 𝜃, give the exact value. Do not use a calculator. a) sin 270°

b) tan 90°

c) cos180°

d) tan 270°

e) tan 360°

MPC30S Date: ____________________

Pg. #20

MPC30S Date: ____________________

Pg. #21

Outcome T2

Chapter 2 – Trigonometry 2.2 – Part 3: Trigonometric Equations

When asked to solve a trigonometric equation, make sure to following the following steps. Equations Involving Exact Values Example #1

Solve the trigonometric equation 2 sin 𝜃 + 1 = 0 for 𝜃, where 0° ≤ 𝜃 ≤ 360° Example #2

Solve the trigonometric equations for 𝜃, where 0° ≤ 𝜃 ≤ 360° a) √2 cos 𝜃 = −1 b) tan 𝜃 = −1

Step 1) Isolate your trig function. Step 2) Determine 𝜃. (the reference angle) Step 3) Determine which quadrant(s) 𝜃 is found in by looking at the sign of the ratio. Step 4) Calculate 𝜃 for each angle.

MPC30S Date: ____________________

Pg. #22

Equations Involving NON-Exact Values Example #3

Solve the trigonometric equation 4 sin 𝜃 + 1 = 0 for 𝜃, where 0° ≤ 𝜃 ≤ 360° Example #4

Solve the trigonometric equations for 𝜃, where 0° ≤ 𝜃 ≤ 360° a) 3 tan 𝜃 = −1 b) cos 𝜃 = 0.444

Step 1) Isolate your trig function. Step 2) Determine 𝜃. (the reference angle) by taking the inverse trig function of the positive ratio. Step 3) Determine which quadrant(s) 𝜃 is found in by looking at the sign of the ratio. Step 4) Calculate 𝜃 for each angle.

MPC30S Date: ____________________

Pg. #23

Outcome T3

A

aB

bc

C

Chapter 2 – Trigonometry 2.3 – Part 1: The Sine Law

When solving for missing sides in right angle triangles, we can use SOHCAHTOA and Pythagorean Theorem. However, we need additional rules when solving for missing sides and angles with all other triangles (non-right triangles). Sine Law: Is a relationship between the sides and angles in any triangle. Let ∆ABC be any triangle, where 𝑎, 𝑏, and 𝑐 represent the measures of the sides opposite ∠𝐴, ∠𝐵 and ∠𝐶, respectively. Then, Note: Side a is ________________ angle A, side b is opposite angle B, and side c is opposite angle C. Note: The sum of the angles in a triangle add to ______________

sin𝐴𝑎

=

OR

MPC30S Date: ____________________

Pg. #24

A

B

C23.1°

110°59.6 m

Example #1 Determine the measures of side a and b and angle C.

MPC30S Date: ____________________

Pg. #25

45°

18 cm

14 cm

F

E

D

Example #2 Determine the measures of angles D and E and side d. Example #3

Boats are anchored at positions J, K and M on a lake. Boats J and K are 80 m apart and boats J and M are 110 m apart. Angle K is 120°. What is the angle J? How far is it from K to M?

MPC30S Date: ____________________

Pg. #26

MPC30S Date: ____________________

Pg. #27

Outcome T3

Chapter 2 – Trigonometry 2.3 – Part 1: The Ambiguous Case of the Sine Law

Ambiguous Case of the Sine Law: If in a triangle you are given __________________ ________________________________________________, from the given information the solution for the triangle is not clear. With this information, you may form ________________________, ________________________, or _________________. We call this the __________________________________. For any DABC with height ℎ = 𝑏 sin𝐴, and given ∠𝐴 and the lengths of a and b, • If ∠𝐴 is an acute angle, there are four possibilities to consider:

i. ∠𝐴 is acute and 𝑎 ≥ 𝑏 ii. ∠𝐴 is acute and 𝑎 = ℎ ___________________________ ___________________________ iii. ∠𝐴 is acute and 𝑎 < ℎ iv. ∠𝐴 is acute and ℎ < 𝑎 < 𝑏

___________________________ ___________________________

MPC30S Date: ____________________

Pg. #28

• If ∠𝐴 is an obtuse angle, there are two possibilities to consider

i. ∠𝐴 is obtuse and 𝑎 > 𝑏 ii. ∠𝐴 is obtuse and 𝑎 < 𝑏or𝑎 = 𝑏

___________________________ ___________________________

Example #1

Given ∆ABC, where 𝑎 = 2 cm, 𝑏 = 6 cm and ∠𝐴 = 30°. Determine all possible measures for the sides and angles.

MPC30S Date: ____________________

Pg. #29

Example #2 Given ∆ABC, where 𝑎 = 4 cm, 𝑏 = 6 cm and ∠𝐴 = 30°. Determine all possible measures for the sides and angles.

MPC30S Date: ____________________

Pg. #30

Example #3 Given ∆ABC, where 𝑎 = 3 cm, 𝑏 = 6 cm and ∠𝐴 = 30°. Determine all possible measures for the sides and angles.

MPC30S Date: ____________________

Pg. #31

Example #4 Given ∆ABC, where 𝑎 = 24 cm, 𝑏 = 42 cm and ∠𝐴 = 30°. Solve this triangle.

MPC30S Date: ____________________

Pg. #32

MPC30S Date: ____________________

Pg. #33

Outcome T3

Chapter 2 – Trigonometry 2.4 – The Cosine Law

There are certain triangles where the ___________________ will not work (we don’t have a SSA triangle). In these cases, we must use the ________________________. Example #1

Determine the measure of side a.

𝑎! = 𝑏! + 𝑐! − 2𝑏𝑐(cos𝐴)

MPC30S Date: ____________________

Pg. #34

Example #2 In ∆𝐵𝑇𝑊, 𝑡 = 9 cm, 𝑤 = 7.8 cm and ∠𝐵 = 112°. Solve this triangle.

MPC30S Date: ____________________

Pg. #35

When we are given a SSS triangle, we can re-write the cosine law to isolate the cosine ratio. Example #3

In ∆𝐵𝑇𝑊, 𝑏 = 5.8 cm, 𝑡 = 3.5 cm and 𝑤 = 6.7cm. Determine the measure of ∠𝐵. Example #4

In ∆𝐴𝐵𝐶, 𝑎 = 6 cm, 𝑏 = 9 cm and 𝑐 = 4 cm. Determine the measure of the smallest angle.

cos𝐴 =𝑏! + 𝑐! − 𝑎!

2𝑏𝑐