1. Learning Objective: To be able to describe the sides of right-angled triangle for use in trigonometry. Setting up ratios Trig in the Calculator

2. Trigonometry is concerned with the connection between the sides and angles in any right angled triangle. Angle

3. A A The sides of a right -angled triangle are given special names: The hypotenuse, the opposite and the adjacent. The hypotenuse is the longest side and is always opposite the right angle. The opposite and adjacent sides refer to another angle (given to us), other than the 90 o.

4. There are three formulae involved in trigonometry: sin A= cos A= tan A = S O H C A H T O A

5. Finding the ratios The simplest form of question is finding the decimal value of the ratio of a given angle. Find: 1)sin 32= sin 32 = 2)cos 23 = 3)tan 78= 4)tan 27= 5)sin 68=

6. Using ratios to find angles It can also be used in reverse, finding an angle from a ratio. To do this we use the sin -1, cos -1 and tan -1 function keys. (hitting the 2 nd key first)

7. Example: 1.sin x = 0.1115 find angle x. x = sin -1 (0.1115) x = 6.4 o 2. cos x = 0.8988 find angle x x = cos -1 (0.8988) x = 26 o sin -1 0.1115 = shiftsin () cos -1 0.8988 = shiftcos ()

8. Ex. 1: Finding Trig Ratios Compare the sine, the cosine, and the tangent ratios for  A in each triangle beside.

9. Ex. 1: Finding Trig Ratios LargeSmall sin A = opposi te hypotenu se cosA = adjace nt hypotenu se tanA = opposi te adjacent 8 1717 ≈ 0.470 6 1515 1717 ≈ 0.882 4 8 1515 ≈ 0.533 3 4 8. 5 ≈ 0.470 6 7. 5 8. 5 ≈ 0.882 4 4 7. 5 ≈ 0.533 3 Trig ratios are often expressed as decimal approximations.

10. Ex. 2: Finding Trig Ratios SS sin S = opposi te hypotenu se cosS = adjace nt hypotenu se tanS = opposi te adjacent 5 1313 ≈ 0.384 6 1212 1313 ≈ 0.923 1 5 1212 ≈ 0.416 7

11. Ex. 2: Finding Trig Ratios—Find the sine, the cosine, and the tangent of the indicated angle. RR sin S = opposi te hypotenu se cosS = adjace nt hypotenu se tanS = opposi te adjacent 1212 1313 ≈ 0.923 1 5 1313 ≈ 0.384 6 1212 5 ≈ 2.4

12. Examples of Trig Ratios 12 20 16 Q P Opposite

13. Similar Triangles and Trig Ratios 3 5 4 A B 12 20 16 Q P R C