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Mastering Right Triangle Trigonometry: Ratios, Solving Techniques, and Special Ratios

Understand the basics of right triangle trigonometry, including ratios, reciprocal ratios, and special trigonometric ratios. Learn how to solve for side lengths and angle measures in right triangles using trigonometric functions and inverse trigonometry. Practice with examples and handy tips for solving right triangles efficiently.

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Mastering Right Triangle Trigonometry: Ratios, Solving Techniques, and Special Ratios

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  1. Right Triangle Trigonometry 23 March 2011

  2. Degree Mode v. Radian Mode

  3. Symbols Theda – Represents the angle measure Adjacent Side Hypotenuse Opposite Side

  4. Six Trigonometric Ratios • 3 Basic Ratios + 3 Reciprocal Ratios • What is a reciprocal?

  5. Basic Trig. Ratio Sine Cosine Tangent Reciprocal Trig. Ratio Cosecant Secant Cotangent Six Trigonometric Ratios, cont. It’s a sin to have two c’s.

  6. Three Basic Trig. Ratios SOH-CAH-TOA

  7. Sine (SOH) 24 25 7

  8. Cosine (CAH) 24 25 7

  9. Tangent (TOA) 24 25 7

  10. Cosecant – Reciprocal of Sine (“It’s a sin to have two C’s.”) 24 25 7

  11. Secant – Reciprocal of Cosine 24 25 7

  12. Cotangent – Reciprocal of Tangent 24 25 7

  13. Your Turn: • Pg. 419: 9 – 14, 27 – 32

  14. Solving for Side Lengths • If given one side and one angle measure, then we can solve for any other side of the triangle. 8 x

  15. Solving Right Triangles, cont. • Ask yourself what types of sides do you have: opposite, adjacent, and/or hypotenuse? • Pick the appropriate trig function to solve for x. • Solve for x. 8 x

  16. Solving for Side Lengths, cont. 8 x

  17. Solving Side Lengths, cont. x 14

  18. Special Trigonometric RatiosMemorize These!!!

  19. Your Turn:

  20. Inverse Trigonometric Ratios • We can “undo” trig ratios • Gives us the angle measurement (theda) • Represented by a small –1 in the upper right hand corner • Ex. • 2nd button → correct trig ratio

  21. Inverse Trigonometric Ratios, cont.

  22. Your Turn: Solve for thedaRound to nearest hundredth

  23. Solving For Angle Measures • If given two sides of a triangle, then we can solve for any of the angles of the triangle. 4 5

  24. Solving for Angle Measures, cont. • Ask yourself what types of sides do you have: opposite, adjacent, and/or hypotenuse? • Pick the appropriate trig function to solve for • Solve for using the inverse trigonometric function 4 5

  25. Solving for Angle Measures, cont. 4 5

  26. Your Turn: • Complete problems 11 – 16 on the Solving Right Triangles Practice handout

  27. Solving Right Triangles • We can use two properties of triangles to solve for all the angles and the side lengths of a right triangle.

  28. Pythagorean Theorem For a righttriangle, a2 + b2 = c2 Triangle Sum Theorem When you add up all the angles in a triangle, they equal 180° Properties of Triangles

  29. Given Two Sides Use Pythagorean Theorem to solve for remaining side. Solve for 1 of the angles using trig ratios Solve for the other angle using Triangle Sum Theorem Given an Angle & a Side Use the Triangle Sum Theorem to solve for the other angle Use trig ratios to solve for 1 of the sides Use the Pythagorean Theorem to solve for the other side Tricks for Solving Right Triangles

  30. Beta – Another symbol for an unknown angle measure

  31. Solving Right Triangles – Examples: Given Two Sides 4 5

  32. Solving Right Triangles – Examples: Given an Angle and a Side 2 30°

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