terminal arm length and special case triangles
DESCRIPTION
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 . - PowerPoint PPT PresentationTRANSCRIPT
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