Trigonometry Concepts: Secant, Cosecant, Cotangent and Inverse Functions

Trigonometry Trigonometry

Introduction In this chapter you will learn about secant, cosecant and cotangent, based on cosine, sine and tan We will also look at the inverse functions of sine, cosine and tan, known as arcsin, arccos and arctan We will build on the Trigonometric Equation solving from C2

Teachings for Exercise 6A Teachings for Exercise 6A

Trigonometry You need to know the functions secant , cosecant and cotangent 1 x = You should remember the index law: 1 x 1 1 = 1 It is NOT written like this in Trigonometry sec cos cos cos All 3 are undefined if cos , sin or tan = 0 1 1 cos sec = cosec sin 1 = Something which will be VERY useful later in the chapter cot tan sin cos cos sin = = tan cot so 6A

Trigonometry You need to know the functions secant , cosecant and cotangent Example Questions Will cosec200 be positive or negative? 1 = cosec200 sin200 1 = sec cos y = Sin 90 180 270 360 1 = cosec sin As sin200 is negative, cosec200 will be as well! 1 = cot tan 6A

Trigonometry You need to know the functions secant , cosecant and cotangent Example Questions Find the value of: sec280 to 2dp 1 = sec cos 1 = sec280 cos280 1 = Just use your calculator! cosec sin = sec280 5.76 1 = cot tan 6A

Trigonometry You need to know the functions secant , cosecant and cotangent Example Questions Find the value of: cot115 to 2dp 1 = sec cos 1 = cot115 tan115 1 = Just use your calculator! cosec sin = cot115 0.47 1 = cot tan 6A

1 1 = = cot sec tan Trigonometry cos 1 = cosec sin 1 You need to know the functions secant , cosecant and cotangent = sec210 cos210 30 -60 -60 y = Cos 90 180 270 360 Example Questions Work out the exact value of: sec210 210 By symmetry, we will get the same value for cos210 at cos30 (but with the reversed sign) 1 (you may need to use surds ) = sec210 cos30 Cos30 = 3/2 1 = sec210 3 2 Flip the denominator 2 2 3 3 = sec210 or 3 6A

1 1 = = cot sec tan Trigonometry cos 1 = cosec sin 3 1 = cosec You need to know the functions secant , cosecant and cotangent 3 4 sin 4 /4 3 /4 y = Sin /2 3 /2 2 Example Questions Work out the exact value of: Sin(3 /4) = Sin( /4) 3 3 1 cosec = cosec 4 4 sin4 Sin( /4) = Sin45 1/ 2 (you may need to use surds ) = 3 1 1 cosec 4 2 Flip the denominator = 3 cosec 2 4 6A

Teachings for Exercise Teachings for Exercise 6B 6B

1 1 = = cot sec tan Trigonometry cos 1 = cosec sin You need to know the graphs of sec , cosec and cot At 90 , Sin = 1 Cosec = 1 At 180 , Sin = 0 Cosec = undefined We get an asymptote wherever Sin = 0 1 1 = cosec y = Sin 0 sin 90 180 270 360 -1 y = Cosec 6B

1 1 = = cot sec tan Trigonometry cos 1 = cosec sin You need to know the graphs of sec , cosec and cot At 0 , Cos = 1 Sec = 1 At 90 , Cos = 0 Sec = undefined We get asymptotes wherever Cos = 0 1 1 y = Cos = sec cos 0 90 180 270 360 -1 y = Sec 6B

1 1 = = cot sec tan Trigonometry cos 1 = cosec sin At 45 , tan = 1 Cot = 1 You need to know the graphs of sec , cosec and cot At 90 , tan = undefined Cot = 0 y = Tan 1 = cot 180 270 360 90 tan y = Cot At 180 , tan = 0 Cot = undefined 6B

1 1 = = cot sec tan Trigonometry cos 1 = cosec sin You need to know the graphs of sec , cosec and cot 1 y = Sin 0 90 180 270 360 -1 Maxima/Minima at (90,1) and (270,-1) (and every 180 from then) 1 0 90 180 270 360 -1 Asymptotes at 0, 180, 360 (and every 180 from then) y = Cosec 6B

1 1 = = cot sec tan Trigonometry cos 1 = cosec sin You need to know the graphs of sec , cosec and cot 1 y = Cos 0 90 180 270 360 -1 Maxima/Minima at (0,1) (180,-1) and (360,1) (and every 180 from then) 1 0 90 180 270 360 -1 Asymptotes at 90 and 270 (and every 180 from then) y = Sec 6B

1 1 = = cot sec tan Trigonometry cos 1 = cosec sin You need to know the graphs of sec , cosec and cot y = Tan 360 270 180 90 Asymptotes at 0, 180 and 360 (and every 180 from then) 360 180 270 90 y = Cot 6B

1 1 = = cot sec tan Trigonometry cos 1 = cosec sin You need to know the graphs of sec , cosec and cot y = Sec 1 0 Sketch, in the interval 0 360, the graph of: 90 180 270 360 -1 1 sec2 = + y y = 1 + Sec2 = sec y 2 y = Sec2 Horizontal stretch, scale factor 1/2 1 = sec2 y 0 90 180 270 360 Vertical translation, 1 unit up 1 sec2 = + -1 y 6B

Teachings for Exercise Teachings for Exercise 6C 6C

1 1 sin cos cos sin = = = = cot sec tan cot tan Trigonometry cos 1 = cosec sin Example Questions Simplify You need to be able to simplify expressions, prove identities and solve equations involving sec , cosec and cot sin cot sec Remember how we can rewrite cot from earlier? cos sin 1 sin cos This is similar to the work covered in C2, but there are now more possibilities Group up as a single fraction sin cos sin cos = As in C2, you must practice as much as possible in order to get a feel for what to do and when Numerator and denominator are equal = 1 6C

1 1 sin cos cos sin = = = = cot sec tan cot tan Trigonometry cos 1 = cosec sin Example Questions Simplify ( sin cos sec You need to be able to simplify expressions, prove identities and solve equations involving sec , cosec and cot ) + cosec Rewrite the part in brackets 1 1 = sin cos + cos sin Multiply each fraction by the opposite s denominator This is similar to the work covered in C2, but there are now more possibilities sin cos = sin cos + sin cos sin cos Group up since the denominators are now the same + sin sin cos cos = sin cos As in C2, you must practice as much as possible in order to get a feel for what to do and when Multiply the part on top by the part outside the bracket ( ) sin cos + sin cos = sin cos Cancel the common factor to the top and bottom = + sin cos 6C

1 1 sin cos cos sin = = = = cot sec tan cot tan Trigonometry cos 1 = cosec sin cot cosec sec Putting them together 3 cos Show that: cot cosec sec + 2 2 cosec + 2 2 cosec Replace numerator and denominator Left side cos sin cot cosec sec 2 = + 2 2 cosec 1 2 2 cos sin Numerator Denominator sec + This is just a division cot cosec 2 2 cosec cos sin 1 Rewrite both Rewrite both Multiply by the opposite s denominator = cos cos sin 1 2 2 2 sin 1 1 sin + Change to a multiplication 2 2 cos sin Group up cos sin cos sin 2 2 cos sin 1 = = 2 2 sin cos 2 2 + Group up 2 2 2 2 cos sin cos sin Group up 3 2 cos sin = + 2 2 sin cos cos sin 2 = sin 2 2 From C2 Simplify sin2 + cos2 = 1 1 = cos 3 = 6C 2 2 cos sin

1 1 sin cos cos sin = = = = cot sec tan cot tan Trigonometry cos 1 = cosec sin = sec 2.5 You need to be able to simplify expressions, prove identities and solve equations involving sec , cosec and cot Rewrite using cos 1 cos = 2.5 Rearrange 1 2.5 = cos Work out the fraction You can solve equations by rearranging them in terms of sin, cos or tan, then using their respective graphs = cos 0.4 Inverse cos ( ) = 0.4 1 cos Work out the first answer. Add 360 if not in the range we want Subtract from 360 (to find the equivalent value in the range = 113.6 Example Question Solve the equation: sec In the range: = 246.4 = 2.5 0 360 1 y = Cos 0 90 180 270 360 -1 6C

1 1 sin cos cos sin = = = = cot sec tan cot tan Trigonometry cos 1 = cosec sin = cot2 0.6 You need to be able to simplify expressions, prove identities and solve equations involving sec , cosec and cot Rewrite using tan 1 = tan2 0.6 Inverse tan 1 1 0.6 = 2 tan Work out the first value, and others in the original range (0-360) You can add 180 to these as the period of tan is 180 You can solve equations by rearranging them in terms of sin, cos or tan, then using their respective graphs = , 239.04 2 59.04 = , 599.04 2 419.04 Divide all by 2 (answers to 3sf) = 29.5, 120, 210, 300 Example Question Solve the equation: cot2 In the range: = 360 0.6 0 y = Tan 90 180 270 360 0 360 Remember to adjust the acceptable range for 2 0 2 720 6C

1 1 sin cos cos sin = = = = cot sec tan cot tan Trigonometry cos 1 = cosec sin You need to be able to simplify expressions, prove identities and solve equations involving sec , cosec and cot Rewrite each side Cross multiply Divide by Cos You can solve equations by rearranging them in terms of sin, cos or tan, then using their respective graphs Divide by 2 Rewrite the right-hand side Example Question Solve the equation: In the range: 0 360 6C

Teachings for Exercise Teachings for Exercise 6D 6D

1 1 cos sin = = cot sec = cot tan cos Trigonometry 1 sin cos = = cosec tan sin Adj Hyp Example Question = cos Given that: Replace A and H from the triangle 12 13 5 = cos A = tan 12 and A is obtuse, find the exact value of secA 1 y = Cos A is obtuse (in the 2nd quadrant) Cos is negative in this range 0 90 180 270 360 Opp Adj -1 = tan 12 13 = cos 13 5 Flip the fraction to get Sec 13 12 = sec 12 Ignore the negative, and use Pythagoras to work out the missing side 6D

1 1 cos sin = = cot sec = cot tan cos Trigonometry 1 sin cos = = cosec tan sin Opp Hyp Example Question = sin Given that: Replace A and H from the triangle 5 5 = sin A = tan 13 12 and A is obtuse, find the exact value of cosecA 1 A is obtuse (in the 2nd quadrant) Sin is positive in this range y = Sin 0 90 180 270 360 Opp Adj -1 = tan 5 = sin 13 13 5 Flip the fraction to get Sec 13 5 = cosec 12 Ignore the negative, and use Pythagoras to work out the missing side 6D

1 1 sin cos cos sin = = = = cot sec tan cot tan Trigonometry cos 1 = cosec sin You need to know and be able to use the following identities + 2 2 sin cos 1 Divide all by cos2 2 2 sin cos cos cos 1 1 tan + + 2 2 sec + 2 2 2 cos Simplify each part 2 2 1 cot cosec tan + sec 2 2 1 You might be asked to show where these come from 6D

1 1 sin cos cos sin = = = = cot sec tan cot tan Trigonometry cos 1 = cosec sin You need to know and be able to use the following identities + 2 2 sin cos 1 Divide all by sin2 2 2 sin sin cos sin 1 1 tan + + 2 2 sec + 2 2 2 sin Simplify each part 2 2 1 cot cosec + cot cosec 2 2 1 You might be asked to show where these come from 6D

1 1 cos sin 1 tan + 2 2 1 cot + 2 2 sec cosec = = cot sec = cot tan cos Trigonometry 2 cos 1 1 sin cos = = cosec tan sin + 2 sin Left hand side Example Question 4 4 cosec cot Factorise into a double bracket Prove that: ( )( ) = + 2 2 2 2 cosec cot cosec cot + 2 1 cos 1 cos 4 4 cosec cot Replace cosec2 2 ( )( ) = + 1 cot + 1 2 2 2 2 cosec cot cot The second bracket = 1 ( ) = + 2 2 cosec cot Rewrite 2 1 cos sin = + sin 2 2 Group up into 1 fraction + 2 1 cos sin 1 cos 1 cos = 2 2 Rearrange the bottom (as in C2) + = 2 6D

1 1 cos sin 1 tan + 2 2 1 cot + 2 2 sec cosec = = cot sec = cot tan cos Trigonometry 2 cos 1 1 sin cos = = cosec tan sin + 2 sin Right hand side Example Question (1 sec + 2 2 sin ) Prove that: sin Multiply out the bracket = + (1 sec + 2 2 2 2 2 2 2 sin sin sec sec cos ) Replace sec2 1 = + 2 2 sin sin 2 cos Rewrite the second term 2 sin cos = + 2 sin 2 This requires a lot of practice and will be slow to begin with. The more questions you do, the faster you will get! Replace the fraction = + 2 2 sin tan Rewrite both terms based on the inequalities ( ) ( ) = 1 cos + 2 2 sec 1 The 1s cancel out = 2 2 sec cos 6D

1 1 cos sin 1 tan + 2 2 1 cot + 2 2 sec cosec = = cot sec = cot tan cos Trigonometry 2 cos 1 1 sin cos = = cosec tan sin + 2 sin = 2 4cosec 9 cot Example Question Replace cosec2 360 = 2 4cosec 9 cot ( ) Solve the Equation: in the interval: 0 4 1 cot + = 2 9 cot Multiply out the bracket 4 4cot + = 2 9 cot A general strategy is to replace terms until they are all of the same type (eg cos , cot etc ) Group terms on the left side = 2 4cot cot 5 0 Factorise + = (4cot 5)(cot 1) 0 4/5 y = Tan 360 90 180 270 Solve -1 5 4 = = cot 1 cot or Invert so we can use the tan graph 4 5 = = tan 1 tan or Use a calculator for the first answer Be sure to check for others in the given range = = 38.7, 135, 315 219 6D

Teachings for Exercise Teachings for Exercise 6E 6E

Trigonometry Copy and complete, using surds where appropriate 0 30 45 60 90 Sin 0 0.5 1/ 2 or 2/2 3/2 1 Cos 1 3/2 1/ 2 or 2/2 0.5 0 Tan 0 1/ 3 or 3/3 1 3 Undefined 6E

Trigonometry The same values apply in radians as well 0 /6 /4 /3 /2 Sin 0 0.5 1/ 2 or 2/2 3/2 1 Cos 1 3/2 1/ 2 or 2/2 0.5 0 Tan 0 1/ 3 or 3/3 1 3 Undefined 6E

Trigonometry You need to be able to use the inverse trigonometric functions, arcsinx, arccosx and arctanx y = x y = arcsinx /2 These are the inverse functions of sin, cos and tan respectively 1 y = sinx However, an inverse function can only be drawn for a one-to-one function - /2 -1 1 /2 -1 (when reflected in y = x, a many-to- one function would become one-to many, hence not a function) - /2 y = sinx y = arcsinx Domain: -1 x 1 Range: - /2 arcsinx /2 Domain: - /2 x /2 Range: -1 sinx 1 Remember that from a function to its inverse, the domain and range swap round (as do all co-ordinates) 6E

We cant use /2 x /2 as the domain for cos, since it is many-to-one Trigonometry y = arccosx You need to be able to use the inverse trigonometric functions, arcsinx, arccosx and arctanx y = x /2 These are the inverse functions of sin, cos and tan respectively 1 However, an inverse function can only be drawn for a one-to-one function -1 1 /2 -1 y = cosx (when reflected in y = x, a many-to- one function would become one-to many, hence not a function) y = cosx y = arccosx Domain: -1 x 1 Range: 0 arccosx Domain: 0 x Range: -1 cosx 1 Remember that from a function to its inverse, the domain and range swap round (as do all co-ordinates) 6E

Trigonometry You need to be able to use the inverse trigonometric functions, arcsinx, arccosx and arctanx y = tanx /2 y = arctanx These are the inverse functions of sin, cos and tan respectively - /2 /2 However, an inverse function can only be drawn for a one-to-one function - /2 (when reflected in y = x, a many-to- one function would become one-to many, hence not a function) y = tanx y = arctanx Domain: x R Range: - /2 < arctanx < /2 Subtle differences The domain for tanx cannot equal /2 or /2 The range can be any real number! Domain: - /2 < x < /2 Range: x R 6E

Trigonometry y = arccosx /2 -1 1 /2 y = arcsinx /2 y = arctanx -1 1 - /2 - /2 6E

Trigonometry You need to be able to use the inverse trigonometric functions, arcsinx, arccosx and arctanx arcsin(0.5) Arctan just means inverse sin = 1 sin (0.5) Remember the exact values from earlier ( ) 30 = Work out, in radians, the value of: arcsin(0.5) 6 6E

Trigonometry You need to be able to use the inverse trigonometric functions, arcsinx, arccosx and arctanx arctan( 3) Arctan just means inverse tan = 1 tan ( 3) Remember the exact values from earlier ( ) 60 = Work out, in radians, the value of: arctan( 3) 3 6E

Trigonometry 2 arcsin You need to be able to use the inverse trigonometric functions, arcsinx, arccosx and arctanx arcsinx, arccosx and arctanx You need to be able to use the inverse trigonometric functions, 2 Arcsin just means inverse sin 2 1 sin Work out, in radians, the value of: Work out, in radians, the value of: 2 Ignore the negative for now, and remember the values from earlier 2 arcsin 2 = 1 sin 2 4 2 Sin(- ) = -Sin (or imagine the Sine graph ) 2 = 1 sin 1 4 2 y = sinx 2/2 ( ) - /4 45 /4 - /2 /2 - 2/2 -1 6E

Trigonometry ( ) cos arcsin 1 You need to be able to use the inverse trigonometric functions, arcsinx, arccosx and arctanx arcsinx, arccosx and arctanx You need to be able to use the inverse trigonometric functions, Arcsin just means inverse sin ( ) cos sin 1 1 Work out, in radians, the value of: Work out, in radians, the value of: Think about what value you need for x to get Sin x = 1 ( ) cos arcsin 1 cos 2 Cos(- ) = Cos( ) cos 2 Remember it, or read from the graph y = sinx 1 1 y = cosx = 0 - /2 /2 - /2 /2 -1 -1 6E

Summary We have learnt about 3 new functions, based on sin, cos and tan We have seen some new identities we can use in solving equations and proof We have also looked at the inverse functions, arc sin/cos/tanx 6E