Review Notes for the Calculus I/Precalculus Placement Test

Transcription

1 Review Notes for the Calculus I/Precalculus Placement Test Part 9 -. Degree and radian angle measures a. Relationship between degrees and radians degree 80 radian radian 80 degree Example Convert each angle in radians to degrees. i. 6 radians ii. radians iii. radians 80 5 iv. 7 radians Example Convert each angle in degrees to radians. i radians ii radians iii radians iv radians 80 9 b. Arc length For a circle of radius r, a central angle of in radians subtends an arc whose length s is s r. Example Find the length of the arc of a circle of radius meters subtended by a central angle of 5. Solution To compute s, we need to convert to radians first. s m Right triangle trigonometry: sin, cos, tan, cot, sec, csc Consider a right triangle. Let be one of two acute angles. Let the length of the side that is opposite be b and the side that is adjacent to be a and the the hypotenuse be c. See below.

2 By the Pythagorean Theorem, c a b. Six trigonometric functions of acute angles are defined as sin b c tan b a cos sin csc b c sin cos a c cot a b tan cos sin sec a c cos Since a, b and c are positive, values of these trigonometric functions of are positive when is acute. Example Find the value of each 6 trigonometric functions of the angle if we know c and a. Solution Find b first: b c a 6 9 7, b sin 7. Then 7, cos, tan 7, cot 7 csc 7, sec. Example Let be an acute angle of a right triangle. Suppose we know sin and 8 cos. Find the value of each of the four remaining trigonometric functions of. Solution By the definitions: tan cos sin 8, cot tan 8, csc sin, sec cos 8. Identities: sin cos Since and we have the identities sin cos b c a c b a c sec tan c a b a c b a sin cos and sec tan. c c a a Example Let be an acute angle of a right triangle. Suppose we know sin. Find the value of

3 each of the five remaining trigonometric functions of. Solution By the identity: sin cos cos sin 5 6 6, cos 5 (since is acute) Then tan 5, cot 5, sin, cos 5. Example Let be an acute angle of a right triangle. Suppose we know tan. Find the value of each of the five remaining trigonometric functions of. Solution Since tan b a, let b, a. Then c a b 5 and sin 5, cos 5, cot, csc 5, sec 5.. Values of trigonometric functions at special angles Let 5,anda b. Then c. Values of six trigonometric functions are sin cos tan cot csc sec Let 6 0, b, and c. Then a. Values of six trigonometric functions are sin 6 tan 6 cos 6 cot 6 csc 6 sec 6 Let 60, a, and c. Then b. Values of six trigonometric functions are

4 sin tan csc cos cot sec When 0, the right triangle is a horizontal line segment. So, b 0anda c. Let a c. Then values of six trigonometric functions are sin0 0 0 cos0 tan0 0 0 cot0 does not exist csc0 does not exist sec0 Similarly, when, the right triangle is a vertical line segment. So, a 0andb c. Let b c. Then values of six trigonometric functions are sin cos 0 0 tan does not exist cot 0 0 csc sec does not exist In summary, sin cos tan cot csc sec DNE DNE DNE 0 DNE Example Find the exact value of each expression. a. sin tan b. sec cot c. sin0 cos0 d. csc sec0

5 5. Values of trigonometric functions at general angles To extend the definitions of the trigonometric functions to general angles, we consider an angle in a rectangular coordinate system. An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Let be an angle in standard position and let a, b denote the coordinates of a point, except the origin 0, 0, on the terminal side of. Let r a b. Then sin b r tan b a csc r b cos a r cot a b sec r a Both a and b can positive and negative. The signs of the trigonometric functions are: Quadrant where lies sin, csc cos, sec tan, cot I a 0, b 0 II a 0, b 0 III a 0, b 0 IV a 0, b 0 Example Find the exact value of each of the six trigonometric functions of a positive angle if, is a point on its terminal side. Solution First find r : a, b, r 7. The point, is in the th quadrant. By definition: sin 7, cos 7, tan, cot csc 7, sec 7 Example Find the exact value of each of the six trigonometric functions when and. Solution Let. Then the point on the terminal side is a, 0 where a 0 and r a 0 a. So, 5

6 sin 0 a 0, cos a a, tan 0 0, cot does not exist csc does not exist, sec. Let. Then the point on the terminal side is 0, b where b 0 and r 0 b b. So, sin b b, cos 0 b 0, tan does not exist, cot b 0 0 csc, sec does not exist. Example Name the quadrant in which the angle lies. a. sin 0, and cos 0 is in the second quadrant. b. cos 0, and cot 0 is in the fourth quadrant. 6