Math 20-1 Chapter 2cayfordmathhelp.weebly.com/uploads/1/3/2/9/... · 2.4 – The Cosine Law The Cosine Law The cosine law relates the three side lengths of a triangle to the cosine

Math 20-1

Chapter 2 Trigonometry

General Outcome: Develop trigonometric reasoning.

Specific Outcomes: T1. Demonstrate an understanding of angles in standard position [0° to

360° ]. [R, V]

T2. Solve problems, using the three primary trigonometric ratios for angles from 0° to 360° in

standard position.

[C, ME, PS, R, T, V] [ICT: C6–4.1]

T3. Solve problems, using the cosine law and sine law, including the ambiguous case.

[C, CN, PS, R, T] [ICT: C6–4.1]

Mark Assignments

2.1: Page 83 – 87 # (1-7, 9, 12, 14, 17) (11, 13, 15)

2.2: Page 96 – 99 # (1-6, 8abc, 16) (7, 9, 11-13, 15, 19)

2.3: Page 108 – 111 # 1-5(ac), 6, 8ac, 10-14, 18

2.4: Page 119 – 125 # 1-2(ac), 4bdf, 7, 8, 10-12, 19, 20

Quiz 2 Date:

Chapter 2 Test Date:

Unit 1: Patterns Chapter 2 – Trigonometry

2.1 – Angles in Standard Position

In geometry, an angle is formed by two rays with a common __________________.

In trigonometry, angles are often interpreted as _________________ of a ray.

The starting position and final position are called the ____________________ and

______________ of the angle.

If the angle of rotation is ________________ the angle is _________________

(and vise versa).

On a Cartesian plane, you can generate an angle by rotating a ray about the origin. The

starting position of the ray, along the __________________________ is the

________________ of the angle. The final position, after a rotation about the origin,

is the _________________ of the angle.

An angle is AN ANGLE IN STANDARD POSITION if:

- _________________________________________________ and

- _________________________________________________.

Angles in standard position are ALWAYS shown on the Cartesian plane. The x-axis and

y-axis divide the plane into four quadrants.


For each angle in standard position, there is a corresponding acute angle called the

REFERERENCE ANGLE. The reference angle is ________________________

___________________________________. The reference angle is always

positive and measures between ___________.

Example 1 – Sketch an Angle in Standard Position

Sketch each angle in standard position. State the quadrant in which the terminal arm

lies.

a) 36°

b) 210°

c) 315°


Example 2 – Determine a Reference Angle

Determine the reference angle, 𝜃𝑅 for each angle 𝜃. Sketch 𝜃 in standard position and

label the reference angle 𝜃𝑅.

a) Θ = 130°

b) Θ = 300°

Example 3 – Determine the Angle in Standard Position

Determine the angle in standard position when an angle of 40° is reflected

a) in the y-axis

b) in the x-axis

c) in the y-axis and then in the x-axis

2.1 – Angles in Standard Position (Part B)

Review of Trigonometric Ratios:

There is a relationship between the sides and angles in EVERY right triangle:

𝑠𝑖𝑛𝜃 = 𝑐𝑜𝑠𝜃 = 𝑡𝑎𝑛𝜃 =


Special Right Triangles:

For angles of 30⁰, 45⁰, and 60⁰, you can determine the exact values of trigonometric

ratios.

Examples:

HERE ARE THE TWO TRIANGLES WITH EXACT VALUES …

****************MEMORIZE THESE!!!!!***************

Example 1 – Copy and complete the table without using a calculator. Express each

ratio using exact values.

𝜃 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 𝑡𝑎𝑛𝜃

30⁰

45⁰

60⁰


Example 2 – Find an Exact Distance

Allie is learning to play the piano. Her teacher uses a metronome to help her keep time.

The pendulum arm of the metronome is 10cm long. For one particular tempo, the

setting results in the arm moving back and forth from a start position of 60⁰ to 120⁰.

What horizontal distance does the tip of the arm move in one beat? Give an exact

answer.

You Try – The tempo is adjusted so that the arm of the metronome swings from 45⁰ to

135⁰. What exact horizontal distance does the tip of the arm travel in one beat?


2.2 – Trigonometric Ratios of Any Angle

Trigonometric Ratios on the Cartesian Plane:

Trigonometric ratios can be used to determine ANY ANGLE using a reference

triangle.


Example 1 – The point P(-8, 15) lies on the terminal arm of an angle, 𝜃, in standard

position. Determine the exact trigonometric ratios for sin 𝜃, cos 𝜃, and tan 𝜃.

Example 2 – Determine the exact value of …

a) cos135⁰ b) sin240⁰


Example 3 – Suppose 𝜃 is an angle in standard position with terminal arm in quadrant

III, and 𝑐𝑜𝑠𝜃 = −3

4. What are the exact values of 𝑠𝑖𝑛𝜃 and 𝑡𝑎𝑛𝜃?

Example 4 –Determine the values of 𝑠𝑖𝑛𝜃, 𝑐𝑜𝑠𝜃, and 𝑡𝑎𝑛𝜃 when the terminal arm of

quadrantal angle 𝜃 coincides with the positive y-axis, 𝜃 = 90°.

You Try! Use the diagram to determine the values of 𝑠𝑖𝑛𝜃, 𝑐𝑜𝑠𝜃 and 𝑡𝑎𝑛𝜃 for quadrantal

angles of 0⁰, 180⁰, and 270⁰. Organize your answers in a table as shown below.


KEY IDEAS:

Summary: Writing the primary trigonometric ratios in terms of x, y and r.

𝒔𝒊𝒏𝜽 = 𝒄𝒐𝒔𝜽 = 𝒕𝒂𝒏𝜽 =

x and y change sign (+/-) depending on which quadrant you are in.

r is the length of the terminal arm of the angle and is always positive.

Reference Triangles are always drawn using the reference angle!

You can use the CAST Rule to verify the sign on your trig ratio.

Review 1: The point P(-5, -12) lies on the terminal arm of an angle in standard

position. Determine the exact trigonometric ratios for sin 𝜃, cos 𝜃, and tan 𝜃.

Review 2: Suppose 𝜃 is an angle in standard position with terminal arm in quadrant

III, and 𝑡𝑎𝑛𝜃 =1

5. What are the exact values of 𝑠𝑖𝑛𝜃 and 𝑐𝑜𝑠𝜃?


Quadrantal Angles are angles in standard position whose terminal arms lies on one

of the axes. (ie: 0°, 90°, 180°, 270°, 360°, …)

Review 3:Determine the values of 𝑠𝑖𝑛𝜃, 𝑐𝑜𝑠𝜃, and 𝑡𝑎𝑛𝜃 for a quadrantal angle of 360°.

SOLVING FOR ANGLES GIVEN THEIR SINE, COSINE OR TANGENT:

1. Determine which quadrants the solution(s) will be in using the

_____________________ and the _____________________.

2. Solve for the ________________________.

3. Sketch the reference angle in the appropriate quadrant. Use the diagram to

determine the measure of the related angle in standard position.

Example 1 – Solve an Angle Given Its Exact Sine, Cosine or Tangent Value

Solve for θ.

a) 𝑠𝑖𝑛𝜃 = 0.5, 0° ≤ 𝜃 < 360°

b) 𝑐𝑜𝑠𝜃 = −√3

2, 0° ≤ 𝜃 < 180°

c) 𝑡𝑎𝑛𝜃 = −√3, 0° ≤ 𝜃 < 360°


Example 2 – Given the following trigonometric ratios for θ, where 0° ≤ 𝜃 < 360°,

determine the measure of θ, to the nearest tenth of a degree.

b) 𝑐𝑜𝑠𝜃 = −0.6753 b) 𝑠𝑖𝑛𝜃 = −7

12

2.3 – The Sine Law

OBLIQUE TRIANGLES: _____________________________________

Use this diagram of a scalene triangle in Quadrant I.

Use ∆ABC to write an expression for h in terms of b and a

trigonometric ratio for <A.

Use ∆BDC to write an expression for h in terms of a and a

trigonometric ratio for <B.

Write an equation that relates the two expressions for h.

Any triangle can be positioned with one vertex at the origin, another vertex on the

positive x-axis, and the third vertex in Quadrant I or II.

C

A B D

b a h

x

y


The Sine Law

You can use the sine law to solve for a side or an angle in any triangle.

Example 1 – Determine an Unknown Side Length

Pudluk’s family and his friend own cabins on the Kalit River in Nunavut. Pudluk and

has 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 communications tower

near the store, as shown in the diagram.

a) Determine the distance from Pudluk’s cabin to

the store, to the nearest tenth of a kilometre.

b) Determine the distance from Pudluk’s friend’s cabin to the store.

Example 2 – Determine an Unknown Angle Measure

a) In ∆PQR, <P = 36⁰, p = 24.8 m, and q = 23.4 m. Determine the measure of <Q, to

the nearest degree.


b) In ∆LMN, <L = 64⁰, l = 25.2 cm, and m = 16.5 cm. Determine the measure of <N,

to the nearest degree.

The Ambiguous Case:

Sometimes when you are given two sides and an angle to solve, an ambiguous case

may occur. An ambiguous case means that ______________________________.

There are three possible outcomes:

_______________________________________________________

_______________________________________________________

_______________________________________________________

Check Out Page 105 in the Textbook!

The only time you can have the AMBIGUOUS CASE is when you are given

____________________________________________________.

Look back at Examples 1 & 2! Could any of these be an ambiguous case?


Example 3 – Use the Sine Law in an Ambiguous Case

In ∆ABC, <A = 30⁰, a = 24 cm, and b = 42 cm. Determine the measure of the other sides

and angles. Round your answers to the nearest unit.


2.4 – The Cosine Law

The Cosine Law

The cosine law relates the three side lengths of a triangle to the cosine of one of its

angles.

Proof for the Cosine Law:

Using the Sine & Cosine Laws:

When to use the SINE LAW: When to use the COSINE LAW:

*Use the sine and cosine laws together to solve any triangle*

(find all the missing sides and angles).


Example 1 – Determine a Distance

A surveyor needs to find the length of a swampy area near Fishing Lake, Manitoba. The

surveyor sets up her transit at a point A. She measure the distance to one end of the

swamp as 468.2 m, the distance to the opposite end of the swamp as 692.6 m and the

angle of sight between the two as 78.6°. Determine the length of the swampy area, to

the nearest tenth of a metre.

Example 2 – Determine an Angle

The Lion’s Gate Bridge has been a Vancouver landmark since it opened in 1938. It is the

longest suspension bridge in Western Canada. The bridge is strengthened by triangular

braces. Suppose one brace has a side lengths 14 m, 19 m, and 12.2 m. Determine the

measure of the angle opposite the 14-m side, to the nearest degree.


Example 3 – Solve a Triangle

In ∆ABC, a = 11, b = 5, and <C = 20°. Sketch a diagram and determine the length of the

unknown side and the measures of the unknown angles, to the nearest tenth.

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Math 20-1 Chapter 2cayfordmathhelp.weebly.com/uploads/1/3/2/9/... · 2.4 – The Cosine Law The Cosine Law The cosine law relates the three side lengths of a triangle to the cosine