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# Trigonometry MATH 1060

Utah State University

GPA 3.77

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This 26 page Class Notes was uploaded by Darren Schulist on Wednesday October 28, 2015. The Class Notes belongs to MATH 1060 at Utah State University taught by Staff in Fall. Since its upload, it has received 10 views. For similar materials see /class/230421/math-1060-utah-state-university in Mathematics (M) at Utah State University.

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Date Created: 10/28/15

Evaluating Trigonometric Functions If we have any angle7 9 in standard position with a point my on the terminal side oft9 and r xz2 y2 gt 07 then use the following de nitions to evaluate the six trigonometric functions I sm 0 g cos B 7 7 7 tanHg7 10 cotH 5 y9 0 I y secH 10 cscl97 y9 0 I y The following gure shows us the quadrants and will also help us to evaluate the functions Quadrant II Quadrant I sins 209 7 2059 tans 7 cans Quadrant III Quadrant IV in sins 7 209 7 2059 tans tans 7 Problem 5 Determine the value of the six trigonometric functions of the angle7 t9 2 taut g 6 lies in quadrant l Solution Step 1 First7 nd the value of r Since taut a a T 2 yz 132 22 9 4 13 Solution Step 2 Since 6 lies in quadrant 17 all of the trigonometric functions are positive Using this fact and the information above7 we can determine the value of the six trigonometric functions of 0 sint9 y cost9 i r r sint9 7 cost9 7 3 3 i V13 V13 s1n0 7 cost9 7 13 13 tan6 g7 7 0 cott9 7 y7 0 9 2 3 tan6 g cott9 i sect9 7 7 0 csct9 277 y7 0 V13 V13 sect9 T csc 6 T Evaluating Trigonometric Functions If we have any angle7 9 in standard position with a point my on the terminal side oft9 and r 412 y2 gt 07 then use the following de nitions to evaluate the six trigonometric functions y 1 sm 0 7 cos B 7 7 7 10 cotH y9 0 tan 0 a I I y secH 10 cscl97 y9 0 I y The following gure shows us the quadrants and will also help us to evaluate the functions Quadrant II Quadrant I sins sins 2059 7 2059 tans 7 cans Quadrant III Quadrant IV sin 7 sin 7 209 7 2059 tans tans 7 Problem 4 Give the quadrant in which 6 lies csc l 039 I tam Solution Step 1 Are cscd and tam positive or negative Each will be true in two quad rants Find the quadrant that is true for both You will get quadrant ll Evaluating Trigonometric Functions Given an angle 9 in standard position Where 9 gt 90 or 9 lt 0 the referece angle 9 is the acute angle formed by the terminal side of 9 and the horizontal axis Use the following to nd the reference angle 9 o If 9 lies in quadrant ll 9 7r 7 9 9 180 7 9 o If 9 lies in quadrant lll 9 9 7 7r 9 9 7 180 o If 9 lies in quadrant lV 9 27f 7 9 9 360 7 9 The following is a table of trigonometric values of common angles Which Will be usefull 9degrees 0 30 45 60 90 180 270 9radians 0 g E g g 7r 3 sin 9 0 g g g 1 0 1 cos 9 l g g g 0 1 0 tan 9 0 g 1 V3 Undefl 0 Undefl Problem 5 Evaluate the sine7 cosine7 and tangent of the given 0 t9 7630 Solution Step 1 Find an angle with which 6 is coterrninal 7630 360 7270 7270 360 90 Thus7 90 is coterrninal with 7630 Solution Step 2 Since 6 is the y axis between quadrants l and ll7 the sine is positive Solution Step 3 Use the table above or the unit circle to evaluate the sine7 cosine and tangent of 0 Make sure to give them the appropriate sign sint9 sin 7630 cos0 cos 7630 tant9 tan7630 sin 90 cos 90 tan 90 1 0 undefined Tangent and Cotangent Graphs Tangent Cotangent Function tanz Function cotz Period 7r Period 7r Domainzza gE Damainzzfnw Range 700700 N Range 70000 VerticalAsymptote z g Z VerticalAsymptote z n7r Problem 4 Sketch the graph of the following function rltxgtcotltgt Solution Step 1 Begin by nding two consecutive asymptotes The length between the two asymptotes will be the period for the new function Set the part inside the tangent function equal to consecutive asymptotes for the regular tanx Two consecutive vertical asymptotes occur at 0 and 27139 Therefore7 the period of this function is 27139 Evaluating Trigonometric Functions If we have any angle7 9 in standard position with a point my on the terminal side oft9 and r 412 y2 gt 07 then use the following de nitions to evaluate the six trigonometric functions I sin 0 g cos B 7 tanH a 10 C0tl9E I y 7 ya secH 10 CSCH y9 0 I y 7 The following gure shows us the quadrants and will also help us to evaluate the functions Quadrant II Quadrant I i si 209 7 2059 tans 7 cans Quadrant III Quadrant IV sing 7 sing 7 209 7 2059 tans tans 7 Problem 1 Given the following point7 which lies on the terminal side of an angle in standard position7 determine the value of the six trigonometric functions of the angle7 0 727 76 Solution Step 1 First7 nd the value of r You will get r 2V10 Solution Step 2 Use the above information and nd the value of each trigonometric func tion You will get the following 3x10 V10 s1n0 77 cos 6 if 10 10 1 tan 6 3 cot 6 g sec 19 7V10 csc t9 7 O Evaluating Trigonometric Functions If we have any angle7 9 in standard position with a point my on the terminal side oft9 and 7 12 y2 gt 07 then use the following de nitions to evaluate the six trigonometric functions y 1 sm 0 7 cos B 7 7 7 tanH 7 10 cotl9 7 y9 0 I y 7 secH 10 CSCH y9 0 I y The following gure shows us the quadrants and will also help us to evaluate the functions lt0 i E s N 2 lt0 i E s N N sm 9 2059 7 2059 tans 7 cans Quadmnt III Quadrant IV sin 7 Sin 2059 7 2059 tan a tan a 7 Problem 4 Give the quadrant in which 6 lies 2V5 5 7 sin0 sec 6 Evaluating Trigonometric Functions If we have any angle7 9 in standard position with a point my on the terminal side oft9 and r xz2 y2 gt 07 then use the following de nitions to evaluate the six trigonometric functions y 1 sm 0 7 cos B 7 7 7 tanH 7 10 cotH y9 0 I I y secH 10 cscl97 y9 0 I y The following gure shows us the quadrants and will also help us to evaluate the functions Quadrant II Quadrant I sin sin 209 7 2059 tans 7 cans Quadrant III Quadrant IV sin 7 sin 7 209 7 2059 tans tans 7 Problem 6 Determine the value of the six trigonometric functions of the angle7 t9 1 sun 77 cot gt 0 2 Solution Step 1 With values for y and r7 we can nd the value for x Remember that r is a distance and can never be negative Thus T2 2y2 T2 7 yz 2 22 71V 2 471 952 Since sin lt 0 and cot0 gt 0 in quadrant HL we know that 0030 lt 0 Thus pm Evaluating Trigonometric Functions If we have any angle7 9 in standard position with a point my on the terminal side oft9 and r 12 y2 gt 07 then use the following de nitions to evaluate the six trigonometric functions y 1 sm 0 7 cos B 7 7 7 tanH 10 cotl9 7 y9 0 y secH 10 cscl97 y9 0 I y The following gure shows us the quadrants and will also help us to evaluate the functions Quadrant II Quadrant I sin sin 209 7 2059 tans 7 cans Quadrant III Quadrant IV sin 7 sin 7 209 7 2059 tans tans 7 Problem 5 Determine the value of the six trigonometric functions of 0 cot0 7 6 lies in quadrant lV Solution Step 1 Find the value of r Since cot0 77 you will get 9 r5 Solution Step 2 Use the information above to determine the value of the six trigonometric functions of 0 Remember that 6 lies in quadrant IV You will get 2V5 xB sin 6 77 cos 6 7 5 5 1 tan6 72 cott9 ii sec 6 V5 csc 6 ii Evaluating Trigonometric Functions If we have any angle7 9 in standard position with a point my on the terminal side oft9 and T 12 y2 gt 07 then use the following de nitions to evaluate the six trigonometric functions I sin 0 g cos B 7 T T I tanH 7 10 cotl977 y9 0 I y secH 10 cscl97 y9 0 I y The following gure shows us the quadrants and will also help us to evaluate the functions QuadTant II QuadTant I sin sin 209 7 2059 tans 7 cans Quadmnt III QuadTant IV sin sin 7 209 7 2059 tans tans 7 Problem 8 Determine the value of the six trigonometric functions of 0 y 2x 5 6 lies m quadrant l Solution Step 1 First7 nd a point on the line Choose a value for x that will be in quadrant ll and solve for y You will get 177 You may get a different point This is okay7 as the nal answers Will be equivalent It is just easier to use is Whenever we cant Solution Step 2 Next7 solve for 7 You will get r5 Solution Step 3 With values for 7y7T7 we can use the information above to determine the value of the six trigonometric functions of 0 7V V sin 6 7 cos 6 7 10 10 1 tan 6 7 cot t9 7 7 sec 6 5V csc 6 57 Evaluating Trigonometric Functions Given an angle 9 in standard position Where 9 gt 90 or 9 lt 0 the referece angle 9 is the acute angle formed by the terminal side of 9 and the horizontal axis Use the following to nd the reference angle 9 o If 9 lies in quadrant ll 9 7r 7 9 9 180 7 9 o If 9 lies in quadrant lll 9 9 7 7r 9 9 7 180 o If 9 lies in quadrant lV 9 27f 7 9 9 360 7 9 The following is a table of trigonometric values of common angles Which Will be usefull 9degrees 0 30 45 60 90 180 270 9radians 0 g E g g 7r 3 sin 9 0 g g g 1 0 1 cos 9 l g g g 0 1 0 tan 9 0 g 1 V3 Undefl 0 Undefl Problem 3 Given 67 nd the reference angle7 6 6 778 Solution Step 1 Since 6 lt 07 nd the angle with which it is coterrninal 778 360 282 Solution Step 2 Determine the quadrant in which 6 lies Since 270 lt 282 lt 360 7 6 lies in quadrant lV Evaluating Trigonometric Functions If we have any angle7 9 in standard position with a point my on the terminal side oft9 and r 12 y2 gt 07 then use the following de nitions to evaluate the six trigonometric functions sinHg C0sl9E 7 7 y tanH77 10 cotH y9 0 I secH 10 CSCH y9 0 I y The following gure shows us the quadrants and will also help us to evaluate the functions Quadrant II Quadrant I sin a 2059 7 2059 tans 7 cans Quadrant III Quadrant IV sin sin 209 7 2059 tans tans 7 Problem 7 Determine the value of the six trigonometric functions of 0 y x 6 lies in quadrant lll Solution Step 1 First7 nd a point on the line Since y x7 any value will work as long as we use the same value for both z and y Let7s use 717 71 because we are in quadrant lll Tangent and Cotangent Graphs Tangent Cotangent Function tanz Function cotz Period 7r Period 7r Domainzza gE Damainzzfnw Range 700700 N Range 70000 VerticalAsymptote z g Z VerticalAsymptote z n7r Problem 2 Sketch the graph of the following function fx 3tan Solution Step 1 Begin by nding two consecutive asymptotes The length between the two asymptotes will be the period for the new function Set the part inside the tangent function equal to consecutive asymptotes for the regular tanx 5 5 Two consecutive vertical asymptotes occur at ii and Therefore7 the period of this function is 5 Solution Step 2 Between the two asymptotes plot a few points to see the movement of the graph Make sure to include the x intercept Create a table like the following and ll in the blanks to help you get points to plot Tree 3 tan 5 Sketch the points you have and connect them Evaluating Trigonometric Functions If we have any angle7 9 in standard position with a point my on the terminal side oft9 and T 12 y2 gt 07 then use the following de nitions to evaluate the six trigonometric functions I sin 0 g cos B 7 T T I tanH 7 10 cotl977 y9 0 I y secH 10 cscl97 y9 0 I y The following gure shows us the quadrants and will also help us to evaluate the functions QuadTant II QuadTant I sin sin 209 7 2059 tans 7 cans Quadmnt III QuadTant IV sin 7 sin 7 209 7 2059 tans tans 7 Problem 6 Determine the value of the six trigonometric functions of the angle7 t9 7 1 Sim 7 cot gt 0 Evaluating Trigonometric Functions Given an angle 9 in standard position Where 9 gt 90 or 9 lt 0 the referece angle 9 is the acute angle formed by the terminal side of 9 and the horizontal axis Use the following to nd the reference angle 9 o If 9 lies in quadrant ll 9 7r 7 9 9 180 7 9 o If 9 lies in quadrant lll 9 9 7 7r 9 9 7 180 o If 9 lies in quadrant lV 9 27f 7 9 9 360 7 9 The following is a table of trigonometric values of common angles Which Will be usefull 9degrees 0 30 45 60 90 180 270 9radians 0 g E g g 7r 3 sin 9 0 g g g 1 0 1 cos 9 l g g g 0 1 0 tan 9 0 g 1 V3 Undefl 0 Undefl Problem 2 Given 67 nd the reference angle7 6 787139 6 7 Solution Step 1 Determine the quadrant in which 6 lies 8 3 Since 7139 lt 7 lt 67 6 lies in quadrant HI Solution Step 2 Since 6 lies in quadrant HL use the above information to determine 6 6 677T Evaluating Trigonometric Functions lfwe have any angle7 9 in standard position with a point my on the terminal side oft9 and T 12 y2 gt 07 then use the following de nitions to evaluate the six trigonometric functions 1 sin 0 g cos B 7 T T I tan973 mm cot977 mm 1 y T T secl977 zy O cscl977 y7 0 I y The following gure shows us the quadrants and will also help us to evaluate the functions QuadTant II QuadTant I sin Sin 2059 7 2059 tans 7 tans Quadmnt III QuadTant IV sin 7 sin 7 209 7 209 cans cans 7 Problem 8 Determine the value of the six trigonometric functions of 0 1 is 7 y 0 t9 lzes m quadrant 1 Solution Step 1 1 First7 nd a point on the line Since y EL plug any value in for m and solve for y Let7s use z 2 HAD HAD H 82 y 9 Now we have the point 271 which lies in quadrant I Tangent and Cotangent Graphs Tangent Cotangent Function tanz Function cotz Period 7r Period 7r Domainzza gE Damainzzfnw Range 700700 N Range 70000 VerticalAsymptote z g Z VerticalAsymptote z n7r Problem 2 Sketch the graph of the following function fx 73tan2x Solution Step 1 Begin by nding two consecutive asymptotes The length between the two asymptotes will be the period for the new function Set the part inside the tangent function equal to consecutive asymptotes for the regular tanx 2x7 2x Ma m2 7T 7T Two consecutive vertical asymptotes occur at 71 and Therefore7 the 7r period of this function is

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