1. Preparing for the SAT II Triangle Trigonometry

2. ©Carolyn C. Wheater, 20002 TopicsTopics u Basis of Trigonometry u The Six Ratios u Solving Right Triangles u Special Right Triangles u Law of Sines u Law of Cosines

3. ©Carolyn C. Wheater, 20003 Basis of Trigonometry uTrigonometry, or "triangle measurement," developed as a means to calculate the lengths of sides of right triangles. uIt is based upon similar triangle relationships.

4. ©Carolyn C. Wheater, 20004 Right Triangle Trigonometry uYou can quickly prove that the two right triangles with an acute angle of 25°are similar uAll right triangles containing an angle of 25° are similar 25  You could think of this as the family of 25° right triangles. Every triangle in the family is similar. We could imagine such a family of triangles for any acute angle. You could think of this as the family of 25° right triangles. Every triangle in the family is similar. We could imagine such a family of triangles for any acute angle.

5. ©Carolyn C. Wheater, 20005 Right Triangle Trigonometry uIn any right triangle in the family, the ratio of the side opposite the acute angle to the hypotenuse will always be the same, and the ratios of other pairs of sides will remain constant.

6. ©Carolyn C. Wheater, 20006 The Six Ratios uIf the three sides of the right angle are labeled as n the hypotenuse, n the side opposite a particular acute angle, A, and n the side adjacent to the acute angle A, usix different ratios are possible. A hypotenuse adjacent opposite

7. ©Carolyn C. Wheater, 20007 The Six Ratios

8. ©Carolyn C. Wheater, 20008 Solving Right Triangles uWith these six ratios, it is possible n to solve for any unknown side of the right triangle, if another side and an acute angle are known, or n to find the angle if two sides are known. Once upon a time, students had to rely on tables to look up these values. Now the sine, cosine, and tangent of an angle can be found on your calculator.

9. ©Carolyn C. Wheater, 20009 Sample Problem uIn right triangle ABC, hypotenuse is 6 cm long, and  A measures 32 . Find the length of the shorter leg. n Make a sketch n If one angle is 32 , the other is 58  n The shorter leg is opposite the smaller angle, so you need to find the side opposite the 32  angle. 6 32  58 

10. ©Carolyn C. Wheater, 200010 Choosing the Ratio u... Find the length of the shorter leg. n You need a ratio that talks about opposite and hypotenuse n Can use sine (sin) or cosecant (csc), but since your calculator has a key for sin, sine is more convenient. 6 32  58 

11. ©Carolyn C. Wheater, 200011 Solving the Triangle From your calculator, you can find that sin(32  )  0.53, so 6 32  58 

12. ©Carolyn C. Wheater, 200012 Special Right Triangles  45 – 45 – 90 Triangle n The legs are of equal length n The length of the hypotenuse is times the leg s s 45°

13. ©Carolyn C. Wheater, 200013 Special Right Triangles  30 – 60 – 90 Triangle n The side opposite the 30  angle is half the hypotenuse n The side opposite the 60  angle is half the hypotenuse times 30°

14. ©Carolyn C. Wheater, 200014 Special Right Triangles u30 – 60 – 90 Triangle n The side opposite the 30  angle is half the hypotenuse n The side opposite the 60  angle is half the hypotenuse times 60°

15. ©Carolyn C. Wheater, 200015 Memory Work uKnow these values as well as you know your own name. 30° 45° 60° 1 sincostan

16. ©Carolyn C. Wheater, 200016 Non-Right Triangles uAll these relationships are based on the assumption that the triangle is a right triangle. uIt is possible, however, to use trigonometry to solve for unknown sides or angles in non-right triangles.

17. ©Carolyn C. Wheater, 200017 Law of Sines uIn geometry, you learned that the largest angle of a triangle was opposite the longest side, and the smallest angle opposite the shortest side. uThe Law of Sines says that the ratio of a side to the sine of the opposite angle is constant throughout the triangle.

18. ©Carolyn C. Wheater, 200018 Sample Problem uIn  ABC, m  A = 38 , m  B = 42 , and BC = 12 cm. Find the length of side AC. n Draw a diagram to see the position of the given angles and side. n BC is opposite  A n You must find AC, the side opposite  B. A B C

19. ©Carolyn C. Wheater, 200019 Sample Problem u.... Find the length of side AC. n Use the Law of Sines with m  A = 38 , m  B = 42 , and BC = 12

20. ©Carolyn C. Wheater, 200020 WarningWarning uThe Law of Sines is useful when you know n the sizes of two sides and one angle or n two angles and one side. uHowever, the results can be ambiguous if the given information is two sides and an angle other than the included angle (ssa).

21. ©Carolyn C. Wheater, 200021 WarningWarning uThe Law of Sines gives a unique solution when the given information is n sas n asa n aas uThe ambiguous case is ssa, which is not a way of proving triangles congruent. Remember that these are all sufficient conditions for congruent triangles.

22. ©Carolyn C. Wheater, 200022 Law of Cosines uIf you apply the Law of Cosines to a right triangle, that extra term becomes zero, leaving just the Pythagorean Theorem. uThe Law of Cosines is most useful n when you know the lengths of all three sides and need to find an angle, or n when you two sides and the included angle.

23. ©Carolyn C. Wheater, 200023 Sample Problem uTriangle XYZ has sides of lengths 15, 22, and 35. Find the measure of the largest angle of the triangle. 15 22 35 Largest angle opposite longest side

24. ©Carolyn C. Wheater, 200024 Sample Problem u... Find the measure of the largest angle of the triangle. 15 22 35

25. ©Carolyn C. Wheater, 200025 CautionCaution uMany people, when they reach the line will mistakenly subtract 660 from 709. uDon’t be one of them. uThe multiplication should be done before any addition or subtraction.