Angles and their measure, Solving Right Triangle and Trigonometric Ratios

week 1

April 13, 2023PRECACULUS, Mathematics(2), MATH-112, McGraw Hill

Basic Sciences Department1

Outlines:- 6.1: Angles and Their Measure 6.2: Solving Right Triangles 6.3: Trigonometric Functions: A Unit Circle Approach

Objectives

At the end of this lecture, the student should be able to:

Define the concept of angle and identify its sign and type . Convert from decimal degree (DD) to degree-minute-second form (DMS)

and vise versa. Convert the measure of angle from radian to degree and vise versa. Evaluate the trigonometric ratios associated with an acute angle of a right

triangle. Solve the right triangle if we given two sides or one acute angle and a side. Find the coordinates of the circular point on the unit circle or circle with

radius and find the values of all six trigonometric functions by using this point.

Evaluate the trigonometric functions to four significant digits(three using a calculator in radian mode.

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Angles and Their Measure

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Section 6.1

6.1 .Angles and Their Measure

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Degree and Radian Measure

Angles

Converting Degrees to Radians and vice versa

6.1 .Angles and Their Measure

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AnglesAn angle is formed by rotating a ray , called the initial side of the angle , around its endpoint until it coincides with a ray , called terminal side of the angle.

The vertex

Angle or angle or

6.1 .Angles and Their Measure

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Sign of Angle:

Counterclockwise

Clockwise

Positive angle

Negative angle

6.1 .Angles and Their Measure

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Two different angles that have the same initial and terminal sides, are called coterminal.

6.1 .Angles and Their Measure

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An angle is said to be in standard position if its vertex is at the origin and the initial side is along the positive axis.

If the terminal side of an angle in standard position along the coordinate axis, the angle is said to be a quadrantal angle.

Degree and Radian Measure

Definition 1: Degree Measure

A angle formed by one complete rotation is said to have a measure of 360

degrees (36). A positive angle formed by of a complete rotation is said to have a

measure of 1 degree (1). The symbol denotes degrees

A angle formed by one complete rotationA angle formed by complete rotation 60 °

1

A angle formed by complete clockwise rotation − 90 °

Example:

6.1 .Angles and Their Measure

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Types of Angles:

6.1 .Angles and Their Measure

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Two positive angles are complementary if their sum is .

Two positive angles are supplementary if their sum is .

6.1 .Angles and Their Measure

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Degree = 60 minute , minute = 60 second . Thus:

1 minute = degree .

Converting from decimal degree (DD) to degree-minute-second form (DMS) and vise versa:

A degree can divided using :

1 (Decimal notation ( Example : )

2 (degree-minute-second ( Example : )

1 second = minute = degree .

6.1 .Angles and Their Measure

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(A) Convert to DD form.

(B) Convert to DMS form.Solutions:

Example 1: From DMS to DD and Back

¿¿)A (

¿21 . 787°

)B ( 105° (0 . 183 ×60)  ′

¿105° 10 . 98  ′

6.1 .Angles and Their Measure

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Matched Problem 1:

Solutions:

(A) Convert to DD form.(B) Convert to DMS form .

¿¿)A (

¿)B( ❑°(×)  ′

Definition 2: Radian Measure

6.1 .Angles and Their Measure

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6.1 .Angles and Their Measure

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The circumference of a circle of radius is so the radian measure of a positive angle formed by one complete rotation is:

= radians

Note that, if is negative angle, its radian measure is given by .

6.1 .Angles and Their Measure

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What is the radian measure of a central angle opposite an arc of 24 meters in a circle of radius 6 meters?

Solutions:

Example 2: Computing Radian Measure

𝑠𝑟

=

=4 radians.

Matched Problem 2:

Solutions:

What is the radian measure of a central angle opposite an arc of 60 feet in a circle of radius 12 feet?

6.1 .Angles and Their Measure

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=

………… =radians.

6.1 .Angles and Their Measure

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What is the radian measure of ?

¿2𝜋 𝑟

2¿𝜋𝑟𝑟

s=C2

θ=𝑠𝑟

𝜋

=

6.1 .Angles and Their Measure

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Converting Degree to Radians and Vice Versa

.

.

.

Solutions:

Example 3: Radian - Degree Conversions

)A (

¿𝜋

180°  ×  (75°)

¿1 .31

6.1 .Angles and Their Measure

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¿5𝜋12Exact

Three significant digits

)B (

¿ 180°

𝜋×  (5)

¿286 .5  °

¿ 900°

𝜋Exact

Four significant digits

)C= (

Change to DD first

¿¿¿ 41 .2°

θ rad=¿

¿𝜋

180°  ×  (41 .2°)

¿0 .72  °To two decimal places

6.1 .Angles and Their Measure

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Matched Problem 3:

Solutions:

6.1 .Angles and Their Measure

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Solving Right Triangles

Section 6.2

Trigonometric Ratio.

Solving Right Triangles.

Evaluation of Trigonometric Ratio.

6.2. Solving Right Triangles

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6.2. Solving Right Triangles

1¿𝛼+𝛽=90°

A right triangle is angle with one

2) Pythagorean Theorem: =

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Satisfied that:

it is impossible to solve the sides. If only the angles of right triangle are known,

Why?

If we are given

Two sides One acute angle and a sideT

hen

It is possible to solve the remaining three quantities. (This process is called solving the right triangle)

Trigonometric Ratios.

6.2. Solving Right Triangles

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Therefore

If two right triangles have the same acute .

Then, the triangles are similar and ratios of corresponding sides are equal

𝑏𝑐

¿ 𝑏′

𝑐 ′

𝑎𝑐

¿ 𝑎′

𝑐′

𝑏𝑎

¿ 𝑏′

𝑎′

𝑐𝑏

¿ 𝑐′

𝑏′

𝑐𝑎

¿ 𝑐′

𝑎′

𝑎𝑏

¿ 𝑎′

𝑏′

These six ratios, the trigonometric ratios, are called sine, cosine, tangent, cosecant, secant, and cotangent.

Trigonometric Ratios

6.2. Solving Right Triangles

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sin θ=𝑏𝑐

cos θ=𝑎𝑐

tan θ=𝑏𝑎

csc θ=𝑐𝑏

sec θ=𝑐𝑎

cot θ=𝑎𝑏

Right Triangle Ratios

sin θ=OppHyp

cos θ=AdjHyp

tan θ=OppAdj

csc θ=HypOpp

sec θ=HypAdj

cot θ=AdjOpp

HypotenuseOpposite

Adjacent

SOHCAHTOA

Reciprocal Relationships

6.2. Solving Right Triangles

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For : csc θ

Complementary Relationships

)

¿1

sin θ sec θ¿1

cosθ cot θ¿1

tanθ

For :

)

)

)

The trigonometric ratios, cosine, cosecant , and cotangent are sometimes referred to as the cofunctions of sine , secant , and tangent, respectively.

6.2. Solving Right Triangles

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Evaluation of Trigonometric Ratio.

sin 45°=1

√2

cos 45°=¿1

√2¿

tan 45°=1

csc 45°=√2sec 45°=√2

cot 45°=1sin 60°=√3

2

cos 60°=¿12¿

tan 60°=√3

csc 60°=¿2

√3¿

sec 60°=2cot 60°=¿

1

√3¿

Exact Values of the Trigonometric Functions( Standard angles)

How?

How? Why?

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Evaluate to four significant digits.(A) (B) (C)

Solutions:

Example 1: Calculator Evaluation

First, make certain that the calculator is set in degree mode .

)A (

)B (

11 . 43cos (26+ 42

60)

°

¿0 .8934

6.2. Solving Right Triangles

By   the   calculator

)C (1

sin 34°

¿1 .788

By   the   calculator

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6.2. Solving Right Triangles

Matched Problem 1:

Solutions:

Evaluate to four significant digits.

(A) (B) (C)

)A (

)B (

)C (

15 minutes Break

Solving Right Triangles.

Solve the right triangle with feet and

Solutions:

Example 2: Right Triangle Solution

First, draw a figure and label the parts .

Solve For : =

6.2. Solving Right Triangles

or   use csc 𝛽=𝑐𝑏

α ∧𝛽 are complementary

Solve For : sin 𝛽=𝑏𝑐

sin 32 . 2°=𝑏

6 .25 𝑏=6 .25 sin 32 . 2° 𝑏=3 .33 feet

Solve For : cos 𝛽=𝑎𝑐

cos 32. 2°=𝑎

6 . 25 𝑎=6 .25 cos32 . 2° 𝑎=5 .29 feet

or   use sec 𝛽=𝑐𝑎

or   cos𝛼=𝑏𝑐

or   cos𝛼=𝑎𝑐

6.2. Solving Right Triangles

Matched Problem 1:

Solutions:

Solve the right triangle with meters. and

The inverse of sine.

If 0.4196

and ” are same

𝐨𝐫

θ=24 . 81  °To the nearest hundredth degree

𝐨𝐫θ=24  ° 49′To the nearest minute

does not mean

We use the same process with the other 5 trigonometric functions

6.2. Solving Right Triangles

Solve the right triangle with cm and b= .

Solutions:

Example 3: Right Triangle Solution

First, draw a figure and label the parts .

Solve For : tan 𝛽=2 . 624 .32

or

Solve For : sin 𝛽=2 . 62𝑐

𝑐=2 .62

sin 31 . 2° 𝑐=5 . 06 𝑐𝑚

or   use csc 𝛽=𝑐

2. 62

Pythagorean Theorem𝑐=√4 . 322+2 . 622=5 . 05

Solve For : =

use to solve

58° 5 0 ′

α ∧𝛽 are complementary

0.2 = [

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6.2. Solving Right Triangles

Matched Problem 3:

Solutions:

Solve the right triangle with km and b = .

Trigonometric Functions: A Unit Circle Approach

Section 6.3

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The Wrapping Function.

6.3. Trigonometric Functions: A Unit Circle Approach

Definitions of the Trigonometric Functions.

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6.3. Trigonometric Functions: A Unit Circle Approach

The Wrapping Function:

in standard position

The point is called a circular point

is the point of intersection of the terminal side of with unit circle + =1

denote the length of the arc opposite

radians

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= =1=

6.3. Trigonometric Functions: A Unit Circle Approach

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(1,0)

(0,1)

(-1,0)

(0,-1)

(1,0)

1

1

1

𝑎=± √32

𝑊 (𝜋6 )=(√32

,12)

Solutions:

Example 1: Coordinates of Circular Points

)A () =0,-1(

Find the coordinates of the following circular points .(A) (B) (C) (D) )E (

)B () = 0, 1(

)C = (,) (

))D ( -)=

)E ( ( ) =

6.3. Trigonometric Functions: A Unit Circle Approach

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Matched Problem 1:

Solutions:

6.3. Trigonometric Functions: A Unit Circle Approach

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Coordinates of the key circular point

Defining the Trigonometric Functions:

Definition1: Trigonometric Functions

6.3. Trigonometric Functions: A Unit Circle Approach

6.3. Trigonometric Functions: A Unit Circle Approach

2) If is the point on the terminal side that lies on the circle with radius .

1) Note that the point on the unit circle

Remarks:

sin 𝑥cos 𝑥

= =1 ) , lies on the unit circle

sin 𝑥=𝑏𝑟

cos𝑥=𝑎𝑟

tan𝑥=𝑏𝑎

csc 𝑥=𝑟𝑏

,𝑏≠ 0

sec𝑥=𝑟𝑎

,𝑎≠ 0

cot 𝑥=𝑎𝑏𝑏≠ 0

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Solutions:

Example 2: Evaluating Trigonometric Functions

)A (Identify that in unit circle or not:

Find the values of all six trigonometric functions of the angle if .(A) =( ,- )

(B) The terminal side of contains the point (- 60,- 11).

in unit circle Thus:

6.3. Trigonometric Functions: A Unit Circle Approach

𝑟=1

-

=

-

sec𝑥=1𝑎

=53

= -

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6.3. Trigonometric Functions: A Unit Circle Approach

)B (Identify that in unit circle or not: 𝑟=√(− 60)2+  (− 11)2=61

r = 61 = = =

csc 𝑥=𝑟𝑏

=−6111

sec𝑥=𝑟𝑎

=−6160

cot 𝑥=𝑎𝑏

=6 011

Matched Problem 2:

Find the values of all six trigonometric functions of the angle if .(A) =( - , )

(B) The terminal side of contains the point (13, 84).

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Solutions:

Example 3: Calculator Evaluation

)A ( =14.10

)B(

6.3. Trigonometric Functions: A Unit Circle Approach

Evaluate to four significant digits.(A) (B) (C) )(D) The coordinates of .

= =

)C ( = =

)D (

Matched Problem 1:

Solutions:

6.3. Trigonometric Functions: A Unit Circle Approach

Evaluate to four significant digits.(A) (B) (C)

)D (The coordinates of.

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Sample questions( Test Your Self )

Choose the correct answer for the following questions:

1. The degree measure of the angle formed by rotation is…………

2. The value of angle in decimal degree is ……………..

3. The exact value of in radian…………….

4. For the given triangle, the trigonometric function that corresponds the

ratio is………

Choose the correct answer for the following questions :

a ) b ) c ) d )

a ) b ) c ) d)

a ) b ) c) d)

a ) b- ) ) c) d)

Sample Questions

5. The coordinates of the circular point is…………

6. In which quadrants must lie such that and

7. The value of )) is…………….

4. The value of is …………………

a ) b) c) d)

a) Quadrant III b) Quadrant II c) Quadrant I d) Quadrant IV

a ) b ) c) d)

a ) b ) c ) 0 d) Undefined

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Home Work

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• P. 8, #7, 14, 19,23,25,37,41,55.• P.18 , # 10,18,21,29,35,34-42.• P.30, #9,11,29,33,39, 47, 53-56,75,81.Give all the same details as we did before .