A List of Definitions and Theorems

Transcription

1 Metropolitan Community College

2 Definition 1. Two angles are called complements if the sum of their measures is 90. Two angles are called supplements if the sum of their measures is 180.

3 Definition 2. One revolution is 360. Degrees can be subdivided into minutes ( ) and seconds ( ), where 60 = 1 and 60 = 1.

4 Definiton 3. An angle in standard position has an initial side on the positive x-axis and a terminal side rotating in a counter-clockwise direction.

5 Definition 4. Two angles in standard position are coterminal if they have the same terminal side.

6 Definition 5. A transversal is a line that intersects two parallel lines, as shown below. Angles 3 and 4 are called alternate interior angles. Angles 1 and 8 are called alternate exterior angles. Angles 3 and 5 are called interior angles on the same side of the transversal. Angles 1 and 5 are called corresponding angles.

7 Theorem 6. When two lines intersect, angles opposite each other are called vertical angles. Vertical angles have the same measure. When two lines intersect, adjacent angles are supplementary.

8 Definition 7. The sum of the angles of a triangle is 180.

9 Definition 8. An acute triangle has three acute angles. A right triangle has a right angle and two acute angles. An obtuse triangle has one obtuse angle and two acute angles. A scalene triangle has three sides of different lengths. An isosceles triangle has two sides of the same length and one side of different legnth. An equilateral triangle has three sides of the same length.

10 Definition 9. Two triangles are similar if their angles have their corresponding angles are congruent. The lengths of their sides are proportional.

11 Definition 10. For an angle θ in standard position with a terminal side that goes through the point (x, y) and r = x 2 + y 2, the six trigonometric functions are defined as follows: sin θ = y r, cos θ = x r, tan θ = y x, csc θ = r y, sec θ = r x, cot θ = x y.

12 Theorem 11. Reciprocal Identities. For an angle θ, csc θ = 1 sin θ, sec θ = 1 cos θ, cot θ = 1 tan θ.

13 Theorem 12. For an angle θ in standard position with terminal lying in the first quadrant, sin θ > 0, cos θ > 0, tan θ > 0. For an angle θ in standard position with terminal lying in the second quadrant, sin θ > 0, cos θ < 0, tan θ < 0. For an angle θ in standard position with terminal lying in the third quadrant, sin θ < 0, cos θ < 0, tan θ > 0. For an angle θ in standard position with terminal lying in the fourth quadrant, sin θ < 0, cos θ > 0, tan θ < 0.

14 Theorem 13. Quotient Identities. For any angle θ, tan θ = sin θ cos θ, cot θ = cos θ sin θ.

15 Theorem 14. Pythagorean Identities. For any angle θ, sin 2 θ + cos 2 θ = 1, tan 2 θ + 1 = sec 2 θ, 1 + cot 2 θ = csc 2 θ.

16 Theorem 15. The range of each trigonometric function is given below. Trigonometric Function Range sin θ [ 1, 1] cos θ [ 1, 1] tan θ (, ) csc θ (, 1] [1, ) sec θ (, 1] [1, ) cot θ (, )

17 Definition 16. For the acute angle θ shown below, sin θ = opp hyp, cos θ = adj hyp, tan θ = opp adj.

18 Theorem 17. Cofunction Identities. For an acute θ, sin θ = cos(90 θ), cos θ = sin(90 θ), tan θ = cot(90 θ), csc θ = sec(90 θ), sec θ = csc(90 θ), cot θ = tan(90 θ).

19 Theorem Triangle. There is a triangle that has angles of 45, 45, and 90, and sides lengths of 1, 1, and 2.

20 Theorem Triangle. There is a triangle that has angles of 30, 60, and 90, and sides lengths of 1, 3, and 2.

21 Definition 20. For an angle in standard position, the reference angle is the acute angle formed with the terminal side of the angle and the x-axis.

22 Definition 21. An angle θ has a measure of 1 radian if it intercepts an arc of a circle that has the same length as the radius of the circle.

23 Theorem 22. A 360 angle has a measure of 2π radians. A 180 angle has a measure of π radians.

24 Theorem 23. The length of an arc is s = rθ, where θ is measured in radians.

25 Theorem 24. The area of a sector is A = 1 2 θr 2, where θ is measured in radians.

26 Definition 25. The circle of radius 1 centered at the origin is called the unit circle.

27 Theorem 26. Any point on the unit circle is given by (cos θ, sin θ), where θ is an angle in standard position.

28 Theorem 27. Sixteen points on the unit circle are shown below. ( ( 1 ) ) 2 2, 2, π ( ), 1 5π 6 2π 3 (0, 1) π 2 π 3 ( ) 1 2, 3 ) ( 2 2 2, 2 π ( , 1 π 2 6 ) ( 1, 0) π 0 2π (1, 0) ( 3 ( 2, ) 7π 6 ) 5π 4 ) 4π 3 2, 2 ( 2 1 2, 3 2 3π 2 (0, 1) 5π 3 ( 3 11π 6( 2 7π 4 ) 2, 1 ) 2 2, 2 ( 2 1 2, 3 2 )

29 Definition 28. Linear speed on a circular path is given by v = rθ t. Angular speed measures the number of rotations with respect to time and is given by ω = θ t.

30 Definition 29. Sine Curve. One period of y = sin x is shown below. 1 π 2 π 3π 2 2π 1

31 Definition 30. Cosine Curve. One period of y = cos x is shown below. 1 π 2 π 3π 2 2π 1

32 Definition 31. Tangent Curve. One period of y = tan x is shown below. 1 π 2 π 4 π 4 π 2 1

33 Definition 32. Cotangent Curve. One period of y = cot x is shown below. 1 π 4 π 2 3π 4 π 1

34 Definition 33. Secant Curve. One period of y = sec x is shown below. 1 π 2 π 2 π 3π 2 1

35 Definition 34. Cosecant Curve. One period of y = csc x is shown below. 1 π 2 π 3π 2 2π 1

36 Theorem 35. Even-Odd Identities. sin( θ) = sin θ, cos( θ) = cos θ, tan( θ) = tan θ, csc( θ) = csc θ, sec( θ) = sec θ, cot( θ) = cot θ.

37 Theorem 36. Difference Identity for Cosine. cos(a B) = cos A cos B + sin A sin B

38 Theorem 37. Sum Identity for Cosine. cos(a + B) = cos A cos B sin A sin B

39 Theorem 38. ( cos x π ) = sin x 2 ( sin x π ) = cos x 2

40 Theorem 39. Sum Identity for Sine. sin(a + B) = sin A cos B + cos A sin B

41 Theorem 40. Difference Identity for Sine. sin(a B) = sin A cos B cos A sin B

42 Theorem 41. Sum Identity for Tangent. tan(a + B) = tan A + tan B 1 tan A tan B

43 Theorem 42. Difference Identity for Tangent. tan(a B) = tan A tan B 1 + tan A tan B

44 Theorem 43. Double Angle Identities. sin 2θ = 2 sin θ cos θ cos 2θ = cos 2 θ sin 2 θ = 1 2 sin 2 θ = 2 cos 2 θ 1 tan 2θ = 2 tan θ 1 tan 2 θ

45 Definition 44. Inverse Trigonometric Functions. The six inverse trigonometric functions have the following ranges. Inverse Function arcsin x Range [ π 2, π ] 2 arccos x [0, π] arctan x ( π 2, π ) 2 arccsc x arcsec x [ π ) ( 2, 0 0, π ] 2 [ 0, π ) ( π ] 2 2, π arccot x (0, π)

46 Definition 45. An oblique triangle is a triangle that does not have a 90 angle.

47 Theorem 46. Law of Sines. For any triangle ABC, sin A a = sin B b = sin C c.

48 Theorem 47. Law of Cosines. For any triangle ABC, c 2 = a 2 + b 2 ab cos C and C = cos 1 ( a 2 + b 2 c 2 2ab ).

49 Theorem 48. Product Theorem. [r 1 (cos θ 1 +i sin θ 1 )] [r 2 (cos θ 2 +i sin θ 2 )] = r 1 r 2 [cos(θ 1 +θ 2 )+i sin(θ 1 +θ 2 )]

50 Theorem 49. Quotient Theorem. r 1 (cos θ 1 + i sin θ 1 ) r 2 (cos θ 2 + i sin θ 2 ) = r 1 r 2 [cos(θ 1 θ 2 ) + i sin(θ 1 θ 2 )]

51 Theorem 50. DeMoivre s Theorem. [r(cos θ + i sin θ)] n = r n [cos(nθ) + i sin(nθ)]