Introduction to Trigonometry

1. Introduction to Trigonometry Right Triangle Trigonometry

2. Introduction • What special theorem do you already know that applies to a right triangle? • Pythagorean Theorem:a2 + b2 = c2 c a b

3. Introduction • Trigonometry is a branch of mathematics that uses right triangles to help you solve problems. • Trig is useful to surveyors, engineers, navigators, and machinists (and others too.)

5. Finding Trigonometric Ratios A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and means measurement of triangles. The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan, respectively.

6. Topic 1 Before we can understand the trigonometric ratios we need to know how to label Right Triangles.

7. Labeling Right Triangles • The most important skill you need right now is the ability to correctly label the sides of a right triangle. • The names of the sides are: • the hypotenuse • the opposite side • the adjacent side

8. Labeling Right Triangles • The hypotenuse is easy to locate because it is always found across from the right angle. Since this side is across from the right angle, this must be the hypotenuse. Here is the right angle...

9. B C A Labeling Right Triangles • Before you label the other two sides you must have a reference angle selected. • It can be either of the two acute angles. • In the triangle below, let’s pick angle B as the reference angle. This will be our reference angle...

10. B (ref. angle) C A Labeling Right Triangles • Remember, angle B is our reference angle. • The hypotenuse is side BC because it is across from the right angle. hypotenuse

11. B (ref. angle) C A Labeling Right Triangles • Side AC is across from our reference angle B. So it is labeled: opposite. hypotenuse opposite

12. B (ref. angle) C A Labeling Right Triangles Adjacent means beside or next to • The only side unnamed is side AB. This must be the adjacent side. adjacent hypotenuse opposite

13. B (ref. angle) C A Labeling Right Triangles • Let’s put it all together. • Given that angle B is the reference angle, here is how you must label the triangle: hypotenuse adjacent opposite

14. Labeling Right Triangles • Given the same triangle, how would the sides be labeled if angle C were the reference angle? • Will there be any difference?

15. Labeling Right Triangles • Angle C is now the reference angle. • Side BC is still the hypotenuse since it is across from the right angle. B hypotenuse C (ref. angle) A

16. Labeling Right Triangles • However, side AB is now the side opposite since it is across from angle C. B opposite hypotenuse C (ref. angle) A

17. Labeling Right Triangles • That leaves side AC to be labeled as the adjacent side. B hypotenuse opposite C (ref. angle) A adjacent

18. B hypotenuse opposite C (ref. angle) A adjacent Labeling Right Triangles • Let’s put it all together. • Given that angle C is the reference angle, here is how you must label the triangle:

19. W X Y Labeling Practice • Given that angle X is the reference angle, label all three sides of triangle WXY. • Do this on your own. Click to see the answers when you are ready.

20. W X Y Labeling Practice • How did you do? • Click to try another one... adjacent opposite hypotenuse

21. R T S Labeling Practice • Given that angle R is the reference angle, label the triangle’s sides. • Click to see the correct answers.

22. R T S Labeling Practice • The answers are shown below: hypotenuse adjacent opposite

23. Finding Trigonometric Ratios Let be a right triangle. The sine, the cosine, and the tangent of the acute angle A are defined as follows. o h side opposite A hypotenuse = ABC a h side adjacent A hypotenuse = side opposite A side adjacent to A o a = A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. TRIGONOMETRIC RATIOS B sinA= hypotenuse side opposite A h o cosA= A C a tanA= side adjacent to A The value of the trigonometric ratio depends only on the measure of the acute angle, not on the particular right triangle that is used to compute the value.

24. Finding Trigonometric Ratios B 8.5 4 C A 7.5 B 17 8 A C 15 opposite hypotenuse sinA = 7.5 8.5 4 8.5 4 7.5 15 17 8 17 8 15  0.4706  0.4706 adjacent hypotenuse cosA =  0.8824  0.8824 opposite adjacent tanA =  0.5333  0.5333 Compare the sine, the cosine, and the tangent ratios for A in each triangle below. SOLUTION These triangles were created so that A has the same measurement in both triangles. Trigonometric ratios are frequently expressed as decimal approximations. The ratios of the sides for a certain angle size stays constant. We can use this to help us find missing sides and missing angles.

25. How do I remember these?

26. Finding Trigonometric Ratios R 13 5 T S 12 opp. adj. opp. hyp. adj. hyp. = = R 5 12 12 13 5 13 13 5 = = S T 12 = = Find the sine, the cosine, and the tangent of the indicated angle. S SOLUTION The length of the hypotenuse is 13. For S, the length of the opposite side is 5, and the length of the adjacent side is 12.  0.3846 sinS hyp.  0.9231 cosS opp. adj.  0.4167 tanS

27. Finding Trigonometric Ratios R 13 5 T S 12 opp. hyp. opp. adj. adj. hyp. = = R 12 5 5 13 12 13 13 5 = = S T 12 = = Find the sine, the cosine, and the tangent of the indicated angle. R SOLUTION The length of the hypotenuse is 13. For R, the length of the opposite side is 12, and the length of the adjacent side is 5. sinR  0.9231 hyp. cosR  0.3846 adj. opp. tanR = 2.4

28. Homework Pg. 469 #3,4,13 Pg. 477 # 3,4,6,7,8