2. 1-1 Using Trigonometry to Find Lengths

3. You have been hired to refurbish the Weslyville Tower… (copy the diagram, 10 lines high, the width of your page.) In order to bring enough gear, you need to know the height of the tower…… How would you determine the tower’s height?

4. When it is too difficult to obtain the measurements directly, we can operate on a model instead. A model is a larger or smaller version of the original object.

5. A model must have similar proportions as the initial object to be useful. Trigonometry uses TRIANGLES for models. We construct a similar triangle to represent the situation being examined.

6. Imagine the sun casting a shadow on the ground. Turn this situation into a right angled triangle

7. The length of the shadow can be measured directly The primary angle can also be measured directly The Height? 200 m 40 O X Sooo…

8. Make a model!! Draw a right angled triangle with a base of 20 cm and a primary angle of 40 O, then just measure the height!

9. We can generate an equation using equivalent fractions to determine the actual height! Height Base = 17 cm 20 cm = 20 000 cm X cm General Model Real 0.85 = 20 000 cm X cm 20 000 (0.85) = X 170 m = X

10. In the interest of efficiency.. Drawing triangles every time is too time consuming. Someone has already done it for us, taken all the measurements, and loaded them into your calculator Examine the following diagram

11. O O OO As the angle changes, so shall all the sides of the triangle. Recall the Trig names for different sides of a triangle…

12. Geometry O “theta” adjacent opposite hypotenuse Trigonometry base height hypotenuse

13. Trig was first studied by Hipparchus (Greek), in 140 BC. Aryabhata (Hindu) began to study specific ratios. For the ratio OPP/HYP, the word “Jya” was used

14. Brahmagupta, in 628, continued studying the same relationship and “Jya” became “Jiba” “Jiba became Jaib” which means “fold” in arabic

15. European Mathmeticians translated “jaib” into latin: SINUS (later compressed to SIN by Edmund gunter in 1624)

16. Given a right triangle, the 2 remaining angles must total 90 O. A = 10 O, then B = 80 O A = 30 O, then B = 60 O A B C A “compliments” B

17. The ratio ADJ/HYP compliments the ratio OPP/HYP in the similar mathematical way. Therefore, ADJ/HYP is called “Complimentary Sinus” COSINE

18. The 3 Primary Trig Ratios O SINO = opp opp adj hyp COSO = adj hyp TANO = opp adj

19. soh cah toa FIND A: 25 O A 17m COS25 O = A 17 X 17 17 X 1 1 A = 17 X cos25 O A = 15.4 m

20. soh cah toa FIND A: 32 O A 12 m SIN32 O = A 12 X 12 12 X 1 1 A = 12 X SIN32 O A = 6.4 m

21. soh cah toa FIND A: 63 O A 10 m TAN63 O = A 10 X 10 10 X 1 1 A = 10 X TAN63 O A = 19.6 m

22. Tan 40 O = X 200 m 40 O X 200 200 (Tan40 O ) = X 168 m = X

23. Remember: Equivalent fractions can be inverted 2 4 5 10 = 4 2 5 =

24. Page 8 [1,2] a,c 3-7

25. Find the height of the building 150 m 50 O OPP ADJ HYP H TAN 50 = H 150 (150) TAN 50 = H X 150 150 X 1 1