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Trigonometric Ratios

Trigonometric Ratios. Please view this tutorial and answer the follow-up questions on loose leaf to turn in to your teacher. Identifying Parts of a Right Triangle. A. Hypotenuse – always across from the 90° angle Side Opposite – always across from the angle being referenced

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Trigonometric Ratios

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  1. Trigonometric Ratios Please view this tutorial and answer the follow-up questions on loose leaf to turn in to your teacher.

  2. Identifying Parts of a Right Triangle A • Hypotenuse – always across from the 90° angle • Side Opposite – always across from the angle being referenced • Side Adjacent- always touching the angle being referenced • *Note that all angles are marked with capitol letters and sides are marked with lower case letters C B Angle C measures 90°

  3. Identifying Parts of a Right Triangle A • What side is opposite of angle A? • Side BC • What side is opposite of angle B? • Side AC • What side is adjacent to angle A? • Side AC • What side is adjacent to angle B? • Side BC • What side is the hypotenuse? • Side AB C B

  4. Trigonometric Ratios (only apply to right triangles) • Sine (abbreviated sin) • Sin x° = • Example: A C B Sin A =

  5. Trigonometric Ratios (only apply to right triangles) • Cosine (abbreviated cos) • Cos x° = • Example: A C B Cos A =

  6. Trigonometric Ratios (only apply to right triangles) • Tangent (abbreviated tan) • Tan x° = • Example: A C B Tan A =

  7. Helpful Hint to Remember the Trig Ratios • SOH (sine = opposite / hypotenuse) • CAH (cosine = adjacent / hypotenuse) • TOA (tangent = opposite / adjacent) • Remember SOH CAH TOA

  8. Time to Practice • Identify the following trig ratio values C 3 B Sin A = Sin B= Cos A = Cos B= Tan A = Tan B= 4 5 A

  9. Time to Practice • Identify the following trig ratio values C 3 Sin A = Sin B= Cos A = Cos B= Tan A = Tan B= B 4 5 A

  10. More Practice • Identify the following trig ratio values B Sin A = Sin B= Cos A = Cos B= Tan A = Tan B= 13 5 C A 12

  11. More Practice • Identify the following trig ratio values B Sin A = Sin B= Cos A = Cos B= Tan A = Tan B= 13 5 C A 12

  12. How to use the trig ratios to find missing sides • Step 1: Make sure your calculator is in degree mode • Step 2: Label the right triangle with the words opposite, adjacent, and hypotenuse based on the given angle (Note: Do not use the right angle.) • Step 3: From the given information, determine which trig ratio should be used to find the side length • Step 4: Substitute in the given information

  13. How to use the trig ratios to find missing sides (continued) • Step 5: Put a 1 under the trig ratio • Step 6: Cross multiply • Step 7: When x=, put problem into your calculator (Note: you may have to divide first to get x by itself) • (NOTE: The angles of a triangle MUST add up to be 180°)

  14. Example • Given the following triangle, solve for x. 60° 8 cm x

  15. Let’s Talk Through the Steps • Step 1 : Check calculator for degree mode • Press the Mode button and make sure Degree is highlighted as in the picture below

  16. Step 2 • Label the triangle according to the given angle 60° 8 cm- HYPOTENUSE X - OPPOSITE

  17. Step 3 • Identify the trig ratio we should use to solve for x. 60° 8 cm- HYPOTENUSE From the 60° angle, we know the hypotenuse and need to find the opposite. So we need to use SINE. X - OPPOSITE

  18. Step 4 • Substitute in the given information into the equation. 60° 8 cm- HYPOTENUSE Sin x°= Sin 60° = X - OPPOSITE

  19. Step 5 • Put a 1 under the trig ratio 60° 8 cm- HYPOTENUSE Sin x°= Sin 60° = 1 X - OPPOSITE

  20. Step 6 • Cross multiply to solve for x Sin 60° = 1 8 sin (60°) = x

  21. Step 7 • Since x is already by itself, I can enter the information into the calculator. Therefore, we can state that x=6.93.

  22. Let’s Look at Another Example • Suppose that when we set-up the ratio equation, we have the following: • Tan 20° =

  23. What Happens When We Cross Multiply? • Tan 20° = 1 X tan 20° = 4 (How do we get x by itself?) tan 20° tan 20° (Now we have to divide by tan 20° in order to solve for x) X = 4 tan 20° X = 10.99

  24. How to use the trig ratios to find missing angles • Step 1: Make sure your calculator is in degree mode (See slide 15) • Step 2: Label the right triangle with the words opposite, adjacent, and hypotenuse based on the given angle (Note: Do not use the right angle.) • Step 3: From the given information, determine which trig ratio should be used to find the side length • Step 4: Substitute in the given information

  25. How to use the trig ratios to find missing sides (continued) • Step 5: Solve for x by taking the inverse (opposite operation) of the trig ratio. • Step 6: When x=, put problem into your calculator.

  26. Calculator Steps for Finding Angles • To solve for x, remember to take the inverse trig function. • On the calculator, you can find the inverse trig functions by pressing 2nd and then the trig function.

  27. Let’s Look at an Example • Given the following triangle, solve for x. 62 cm 90° x 200 cm

  28. Step 2 Label the sides opposite, adjacent, or hypotenuse from angle x. OPPOSITE 62 cm 90° 200 cm HYPOTENUSE x

  29. Step 3 Since we have the opposite and the hypotenuse, we need to use SINE. OPPOSITE 62 cm 90° 200 cm HYPOTENUSE x

  30. Step 4 Substitute in the given information into the equation. OPPOSITE 62 cm Sin x = 90° 200 cm HYPOTENUSE x

  31. Step 5 To solve for x, we need to take the inverse of sine on both sides. OPPOSITE 62 cm Sin x = 90° Sin-1 (sin x) = Sin-1 200 cm HYPOTENUSE x

  32. Step 6 Now just type in the x= on your calculator. Sin x = Sin-1 (sin x) = Sin-1 X = Sin-1 X = 18°

  33. Now It’s Your Turn! • Use what you’ve just reviewed to help you answer the following questions. • Submit all of your work to your teacher after completing the tutorial. • Don’t be afraid to go back through the slides if you get stuck. • GOOD LUCK!

  34. Problem #1 • Complete the following ratios. Sin A = Sin B = Cos A= Cos B= Tan A= Tan B= 6 cm C A 90° 8 cm 10 cm B

  35. Problem #2 • Solve for x and y. 40 ft 90° y 55° x

  36. Problem #3 • Solve for angles A and B. A 5 in B 90° 7 in C

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