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.

• c2 = a2 + b2 • c2 = 32 + 42 • c2 = 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 = √[(x2 – x1)2 + (y2 – y1)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 = 2

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 = 3

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°

1cos452

1 22 2

22

Example: sin 60°

3sin602

Example: Tan 30°

• Example: Tan 30° • Example: Cos 30°

1 3tan3033

3cos302

Solving Equations using Exact Values, Quadrant I ONLY

ASSIGNMENT:

• Text pg 83 #8; 84 #10, 11, 12, 13