1. Engineering Fundamentals Session 6 (1.5 hours)

2. Trigonometry Triangle: –Geometric figure with 3 straight sides and 3 angles. Sides Angles

3. Right-Angled Triangle One angle is 90 degrees Pythagorean Theorem ( 畢氏定理 ) ”. Give lengths of any 2 sides -> find the length of 3rd side c 2 = a 2 +b 2 C a b

4. What is a function? An example function: f(x) = 2x – 10 f(x) is a function of x; its value depends on x. Another function example: f 2 (x) = x 2 + 1 Example function of time: v(t) = 2 t 2 + t f(0) = ________ f 2 (5)= ________ v(1) = __________

5. Plotting a Function xf(x) 2 01 12 25 310 f(x) = x 2 + 1 f(x) x

6. Trigonometric Functions Trigonometric functions: sin(θ), cos(θ), tan(θ), … If you know a length and an angle in a right-angled triangle, how do you find the unknown length ? 30 degrees 10

7. Trigonometric functions Consider the triangle with sides of length a, b and hypotenuse c: opposite hypothenuse adjacent Note tan θ = sin θ / cos θ

8. Degrees, Seconds, and Minutes Each degree is further subdivided into 60 minutes each minute may be subdivided into another 60 seconds: –1 degree = 60 minute = 60 ’ –60 minute = 60 second = 60 ”

9. Example: Express the angle 265.46  in Degree- Minute-Second (DMS) notation. Solution:

10. Radian and Second 360  = 2Πradians 180  =Πradians 30  = ___________radians 60  = ___________radians 180  = __________radians 1 radian = ______  Π radian = _____  Π/2 radian = ______  Π/4 radian = ______  Π/6 radian = ______ 

11. Special triangle (1) the right-angled isosceles triangle. According to Pythagoras ’ theorem, the side length ratio is 1:1:  2. Special angles: 45  sin 45  or sin Π/4 = 1/  2 cos 45  or sin Π/4 = 1/  2 tan 45  or sin Π/4 = 1

12. Special triangle (2) the half equilateral triangle side lengths in ratio 1:  3:2 Special angles: 30  and 60  sin 30  or sin Π/6 = 1/2 cos 30  or cos Π/6 =  3/2 tan 30  or tan Π/6 =  3 sin 60  or sin Π/3 =  3/2 cos 60  or sin Π/3 = 1/2 tan 60  or tan Π/3 = 1 /  3

13. Trigonometric Equation (1) Example: 2 sinθ-1 = 0 2 sinθ= 1 sinθ= ½ θ= 30° Is it only this answer? Try to calculate the value of sin 150 ° Why the result is same?

14. Unit Circle and Sin/Cos

15. Angle Convention Start measuring angle from positive x-axis + direction = anticlockwise - direction = clockwise θ

16. Angle Example Θ= _______ 45 ° 30 ° 60° Θ Θ Θ

17. Effect of Quandrant on +/- Quadrant II Sine function is +ve Quadrant I All function are +ve Quadrant IV Cos function is +ve Quadrant III Tan function is + ve You can remember by CAST C A S T

18. Special Angles Find WITHOUT calculator: Sin(30 °) = _________ Cos(45 °)= _________ Tan(315 °)=_______ Sin(60 °)=________ cos(180 °)=_______ Tan(135 °)=_______ Sin(240 °)=______ Cos(-45 °)=_________