Chapter 2 –...

Chapter 2 – Trigonometry

1 ● Chapter 2 – Trigonometry

Pre-Calculus 11

Angles in Standard Position An angle consists of two rays joined together called the vertex. An angle in standard position (θ) is

one of the arm or ray is always on the positive x-axis (called the initial arm) and its vertex (corner

where the 2 rays meet) is always at the origin (0, 0) (called the terminal arm).

The graph is broken into 4 different Quadrant. Angles are measured from the initial arm to the

terminal arm. If rotated counter-clockwise the angle is ____________. If rotated clockwise the

angle is _________________.

Example 1: Without measuring, draw each angle, in standard position and state which quadrant the terminal arm lies in.

a) 175° b) 200° c) −75°

You Try…

page 83 #1-2, 3-4(alt. letters)

y

x

y

x

Chapter 2 – Trigonometry ● 2

Pre-Calculus 11

Reference Angles

The reference angle (θR) is the angle between the terminal arm and the CLOSEST x-axis. The reference angle is always positive (no matter the direction of rotation) and measures between 0° and 90°.

Example 2: Determine the quadrant and the reference angle for each angle in standard position?

a) 215° b) 300°


Pre-Calculus 11

Example 3: Determine the measure of the three other angles in standard position, 0° < θ < 360°, that have the reference angle of

a) 65° b) 15°

=

You Try…

page 83-84 #5-7

Special Triangles

Recall

Pythagoras Theorem and Sine, Cosine and Tangent of a right triangle

sin _______ cos ________

tan ________

a c

θ

b


Pre-Calculus 11

Example 1: Solve each triangle. Round to the nearest tenth when appropriate.

a) b)

Right triangles are pretty special in their own right. But there are two extra-special right triangles They are 30-60-90 triangles and 45-45-90 triangles.

45-45-90 Triangle

45sin _______ 45cos _______ 45tan _______

45°

45°

18 cm 30 cm 27 cm

55°

x r Y

X

S

t


Pre-Calculus 11

30-60-90 Triangle

Example 2: Determine the exact lengths for each of the following a) b)

30sin _______

30cos _______

30tan _______

60sin _______

60cos _______

60tan _______

30°

6 cm

x cm

45° 3 cm x cm


Pre-Calculus 11

Example 3: Determine the exact lengths for each of the following

a)

b)

i. What is DE?

ii. What is the exact vertical distance between A and C?

You Try…

Special Right Triangle Worksheet

AB = BC = 10

30 m


Pre-Calculus 11

Trigonometric Ratios of Any Angle Suppose θ is any angle in standard position, and P(x, y) is any

point on its terminal arm, at a distance r from the origin. Then

by the Pythagorean Theorem, ________________________

You can use the reference angle to determine the three primary trigonometric ratios in terms of x, y and r.

hypotneuse

oppositesin

hypotneuse

adjacentcos

adjacent

oppositetan

The chart below summarizes the signs of the trigonometric ratios in each quadrant

P(x, y)

θ

y

x

r


Pre-Calculus 11

The sign of the trigonometric ratios is determined by the quadrant that the terminal side is in. This means that if 0sin (sine ratio is positive), then θ must terminate in either quadrant _____ or _____ , or if 0cos (cosine ratio is positive), then θ must terminate in either quadrant _____ or _____. Finding the trigonometric Ratios of Angles Example 1: The point P(5, −3) lies on the terminal side of the angle θ. Determine the exact trigonometric ratios for sin θ, cos θ and tan θ.

You Try … The point P(−5, −12) lies on the terminal side of the angle θ. Determine the exact trigonometric ratios for sin θ, cos θ and tan θ.

Finding the exact value of a Trigonometric Ratios (using special triangles

Example 2: Determine the exact value of 120cos


Pre-Calculus 11

You Try… Determine the exact value of

Example 3: Given that 7

4cos and that θ terminates in Quadrant IV, determine the exact values of

the other trigonometric ratios.

Example 4: If 4

3cos

and

3

7tan , determine the exact value of sin .

You Try…

page 96 #1, 2(a-c), 3-6, 8, 11, 16


Pre-Calculus 11

The Trigonometric Ratios of the Quadrantal Angles

The quadrantal angles are any rotation angle which is a multiple of 90° (ie. 0°, 90°, 180°, 270°, 360°). Another way of describing them is to say that their terminal arm is on either __________________.

It is important to know that Quadrantal angles DO NOT form a triangle, but by the definition of angles in standard position we have…

siny

r

cosx

r

tany

x

siny

r cos

x

r tan

y

x

0

90

180

270

, 0r

0 , r

, 0r

0 , r


Pre-Calculus 11

Solving for more than one angle

Example 5: Without a calculator, solve for 2

3sin and 0 360 .

Example 6: Given 20

17sin where 3600 , determine the value(s) of θ to the nearest tenth

of a degree.

Note: sometimes the trigonometric ratio is given as a decimal number (for example: sin 0.8228 ), you will solve exactly like example #6.

You Try…

page 96-97 #2d, 7, 9, 10, 13, 19

Step 1: Determine which quadrants the solutions will be in by looking at the sign (+ or −) of the given ratio.

Step 2: Solve for the reference angle, either by special triangle or inverse trigonometric |ratio| (absolute value).

Step 3: Sketch out which quadrants will the terminal arm will be (from step 1) in with the reference angle (step 2).

Step 4: determine the standard position angle.


Pre-Calculus 11

The Sine Law Recall:

Example 1: Solve for x.

Method 1: Using Primary Trigonometric Ratios

Method 2: Using Sine Law

The Sine Law for non-right triangles

In any triangle

c

C

b

B

a

A sinsinsin

or

C

c

B

b

A

a

sinsinsin

The ratio of the sine of any angle to its opposite side is the same for all three angles in any triangle.

B

C A

a

b

c

Where does the Sine Law come from? See this YouTube Video

https://www.youtube.com/watch?v=4_yQZgjFYy8

x

20° 55°

6.4


Pre-Calculus 11

Example 2: Determine the indicated lengths in the triangles below using sine law:

Example 2: Determine the size of the missing angles in the diagram:

12 cm y

x

42° 61°

18 km

8.7 km

P

N M 162°

When solving for an angle, don’t forget to apply inverse sine (sin−1)


Pre-Calculus 11

Example 3: Solve the triangle KGB with 37K , 84G and cmk 12 .

Example 4: Pudluk’s family and his friend own cabins on the Kalit River in Nunavut. Pudluk and his friend wish to determine the distance from Pudluk’s cabin to the store on the edge of town. They know that the distance between their cabins is 1.8 km. Using a transit, they estimate the measures of the angles between their cabins and the store, as shown in the diagram. Determine the distance from Pudluck’s cabin to the store, to the nearest tenth of a kilometre.

You Try…

page 108-109 #1(c, d), 3, 4(b-d), 5, 10, 12

Store


Pre-Calculus 11

The Ambiguous Case When solving a triangle, you must analyse the given information to determine if a solution exists. If

you are given the measures of two angles and one side (ASA), then the triangle is uniquely defined.

However, if you are given 2 sides and an angle opposite one of those sides (SSA), the triangle may not

be unique. It is possible that there is _______________,_________________, or

______________ with the given measurements.

In triangles where A is acute No triangle possible

Condition

Aba

ha

sin

Example

Exactly one triangle

Aba

ha

sin

Or

ba

Two possible triangles

baAb

bah

sin

or


Pre-Calculus 11

If A is obtuse and ba , then there is one solution. If ba then there is no solution.

Example 5: For each of the following, determine whether there is no solution, one solution or two solutions.

a) In ABC with 128A , cma 36 and cmb 45

b) In PQR with 36P , 24.8p cm and 23.4q cm Example 6: In ABC , 34A and cmb 250 . Determine the range of values of a for which there is

a) one triangle b) no triangle c) two triangles

If A is an acute angle, then Find sinh b A

ba = one solution

ha = one solution (right Δ)

a < h = no solution

h < a < b = two solutions

If A is an obtuse angle, then

ba = no solution

a b = one solution


Pre-Calculus 11

Solving a Triangle with Two Possible Solutions Example 7: Solve the triangle ABC with 40A , cma 24 and cmb 30 . You Try… page 108-109 #4(a),6, 8, 9, 11, 13


Pre-Calculus 11

The Cosine Law The Cosine Law is a rule relating the three sides of a triangle with the cosine of one of the angles.

For any triangle, ABC , Abccba cos2222

Or

Cabbac

Baccab

cos2

cos2222

222

Determining Side Length using Cosine Law Example 1: Determine the distance x in the diagram below.

A

C

B

b a

c

x

35 km

26 km

21°

Where does the Cosine Law come from? See this YouTube Video

https://www.youtube.com/watch?v=lPP-pABvwdA


Pre-Calculus 11

Determining Angles using Cosine Law To solve for the angle in cosine law, you must do some algebra or memorize the following formula.

bc

acbA

2cos

2221

Example 2: Determine the size of A . You Try… page 119-122 #1, 2, 4, 6 Example 3: An aircraft-tracking station determines the distance from a helicopter to two aircraft as 50 km and 72 km. The angle between those two distances is 49°. Determine the distance between the two aircraft.

22 cm

10 cm

14 cm

A

B

C


Pre-Calculus 11

Example 4: Solve the triangle ABC with 36B , cma 14 and cmc 21 . Which one to use? Use Sine Law if you are given the measurement of

two angles and one side.

Two sides and an opposite one of the given sides (check for the ambiguous case) Use Cosine Law if you are given the measurement of

When all 3 sides

Two sides with an angle in between.


Pre-Calculus 11

You Try… page 119-122 #3, 4(a, c, e), 5, 10, 15

FIN! Enjoy the rest of your summer.

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