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 = a 2 + b 2 c 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 – x 1 ) 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: 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: Cos 30°

Solving Equations using Exact Values, Quadrant I ONLY

ASSIGNMENT: Text pg 83 #8; 84 #10, 11, 12, 13