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Mastering Special Case Triangles: Coordinate Geometry Exploration

Learn how to determine terminal arm length using Pythagorean Theorem and Distance Formula. Find precise trigonometric ratios for special case triangles and solve equations for exact values. Dive into quadrant I assignments for practical application.

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Mastering Special Case Triangles: Coordinate Geometry Exploration

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  1. Terminal Arm Length and Special Case Triangles DAY 2

  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!”

  3. 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.

  4. Solution 2. Determine the sides of the triangle. Use the Theorem of Pythagoras. • c2= a2 + b2 • c2 = 32 + 42 • c2 = 25 • c = 5

  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.

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

  7. Using the Distance Formula Thedistance formula: d = √[(x2 – x1)2 + (y2 – y1)2] • Example: Given point P (-2, -6), determine the length of the terminal arm.

  8. Review of SOH CAH TOA • Example: Solve for x. • Example: Solve for x.

  9. Example: Determine the ratios for the following:

  10. 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

  11. 45° • By the Pythagorean Theorem, r =

  12. 30° and 60° Draw an equilateral triangle with a perpendicular line from the top straight down • 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 =

  13. Finding Exact Values • Sketch the special case triangles and label • Sketch the given angle • Find the reference angle

  14. Example: cos 45°

  15. Example: sin 60°

  16. Example: Tan 30° • Example: Tan 30° • Example: Cos 30°

  17. Solving Equations using Exact Values, Quadrant I ONLY

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

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