Terminal Arm Length and Special Case Triangles - PowerPoint Presentation

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Terminal Arm Length and Special Case Triangles

DAY 2

Using Coordinates to Determine Length of the Terminal Arm

There

are two methods which can be used:

Pythagorean

Theorem

Distance

Formula

Tip: “Always Sketch First!”

Using the Theorem of Pythagoras

Given

the point (3, 4), draw the terminal arm.

1. Complete

the right triangle by joining the terminal point to the x-axis.

Solution

2. Determine

the sides of the triangle. Use the Theorem of Pythagoras.

c

2

= a2

+ b

2

c

2

= 3

2

+ 4

2

c

2

= 25

c = 5

Solution continued

3. Since

we are using angles rotated from the origin, we label the sides as being x, y and r for the radius of the circle that the terminal arm would make.

Example: Draw the following angle in standard position given any point (x, y) and determine the value of r.

Using the Distance Formula

The

distance

formula: d = √[(x

2 – x1

)

2

+ (y

2

– y

1

)

2

]

Example: Given point P (-2, -6), determine the length of the terminal arm.

Review of SOH CAH TOA

Example: Solve for

x.

Example: Solve for x.

Example: Determine the ratios for the following:

Special Case Triangles – Exact Trigonometric Ratios

We

can use squares or equilateral triangles to calculate exact trigonometric ratios for 30°, 45° and 60°.

Solution

Draw a square with a diagonal.

A

square with a diagonal will have angles of 45°.

All

sides are equal.

Let

the sides equal 1

45°

By the Pythagorean Theorem,

r =

30° and 60°

All

angles are equal in an equilateral triangle (60°)

After

drawing the perpendicular line, we know the small angle is 30°

Let

each side equal 2

By

the Theorem of Pythagoras, y =

Draw an equilateral triangle with a perpendicular line from the top straight down

Finding Exact Values

Sketch the special case triangles and label

Sketch

the given angle

Find

the reference angle

Example: cos 45°

Example: sin 60°

Example: Tan 30°

Example: Tan 30°

Example: Cos 30°

Solving Equations using Exact Values, Quadrant I ONLY

ASSIGNMENT:

Text

pg 83 #8; 84 #10, 11, 12, 13