Introduction to Trigonometry

1. Introduction to Trigonometry This section presents the 3 basic trigonometric ratios sine, cosine, and tangent. The concept of similar triangles and the Pythagorean Theorem can be used to develop the trigonometry of right triangles.

2. Engineers and scientists have found it convenient to formalize the relationships by naming the ratios of the sides.You will memorize these 3 basic ratios.

3. The Trigonometric Functions SINE COSINE TANGENT

4. SINE Pronounced like “sign” COSINE Pronounced like “co-sign” TANGENT Pronounced “tan-gent”

5. B With Respect to angle A, label the three sides Hypotenuse Opposite A C Adjacent

6. We need a way to remember all of these ratios…

7. Sin SOHCAHTOA Opp Hyp Cos Adj Hyp Tan Opp Adj

8. Finding sin, cos, and tan.(Just writing a ratio or decimal.)

9. Find the sine, the cosine, and the tangent of M. Give a fraction and decimal answer (round to 4 places). N 10.8 9 P 6 M

10. B Find the sine, cosine, and the tangent of angle A 24.5 8.2 A C 23.1 Give a fraction and decimal answer. Round to 4 decimal places

11. Finding a side.(Figuring out which ratio to use and getting to use a trig button.)

12. 20 m 20 tan x ) 55 Ex: 1 Find x. Round to the nearest tenth. Figure out which ratio to use. What we’re looking for… What we know… adj opp We can find the tangent of 55 using a calculator

13. 283 m x Ex: 2 Find the missing side.Round to the nearest tenth.

14. x Ex: 3 Find the missing side.Round to the nearest tenth. 20 m

15. 80  72 = ) tan Ex: 4 Find the missing side.Round to the nearest tenth. 80 m Note: When the variable is in the denominator, you end up dividing x

16. 5 Sometimes the right triangle is hiding ABC is an isosceles triangle as marked. Find sin C. Answer as a fraction. A 13 13 12 C B 10

17. Ex. 5 A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge? cos 60° x (cos 60°) = 200 200 60° x x X = 400 yards

18. Ex: 6 A surveyor is standing 50 metres from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree? tan 71.5° tan 71.5° ? 50 (tan 71.5°) = y 71.5° 50 m y 149.4 m

19. For some applications of trig, we need to know these meanings:angle of elevation andangle of depression.

20. Angle of Elevation Angle of Elevation If an observer looks UPWARD toward an object, the angle the line of sight makes with the horizontal. Angle of elevation

21. Angle of Depression If an observer looks DOWNWARD toward an object, the angle the line of sight makes with the horizontal. Angle of depression

22. Finding an angle.(Figuring out which ratio to use and getting to use the 2nd button and one of the trig buttons. These are the inverse functions.)

23. 17.2 ) 9 2nd tan Ex. 1: Find . Round to four decimal places. 17.2 9 Make sure you are in degree mode (not radians).

24. 7 23 ) 2nd cos Ex. 2: Find . Round to three decimal places. 7 23 Make sure you are in degree mode (not radians).

25. 200 400 ) 2nd sin Ex. 3: Find . Round to three decimal places. 200 400 Make sure you are in degree mode (not radians).

26. When we are trying to find a sidewe use sin, cos, or tan. When we need to find an angle we use sin-1, cos-1,ortan-1.