Instructor: Dr. R.A.G. Seel Trigonometr Algebr & Functions (Mths 0 0) 0th Prctice Assignment hpotenuse hpotenuse side opposite side sin = opposite hpotenuse tn = opposite. Find sin, cos nd tn in 9 sin = hpotenuse = opposite hpotenuse = tn = opposite = 4. Find sin β, cos β nd tn β in b c β cos β = tn β = sin β =. Find sin ν, cos ν nd tn ν in ν. Find sin ν, cos ν nd tn ν in ν sin ν = tn ν = cos ν = sin ν = tn ν = cos ν =. Find sin, cos nd tn in sin = tn =. Find sin, cos nd tn in the following tringle: 4 sin = tn =
We define the following three trigonometric functions: sec = hpotenuse ; csc = sin = hpotenuse opposite ; cot = tn = opposite Emple: Referring to eercise #, sec = hpotenuse = 9 =, csc = sin = 4 nd cot = 4. 7. Referring to eercise #, find the vlues of the other three trigonometric functions. 8. Referring to eercise #, find the vlues of the other three trigonometric functions. 9. Referring to eercise #4, find the vlues of the other three trigonometric functions. 0. Referring to eercise #, find the vlues of the other three trigonometric functions.. Referring to eercise #, find the vlues of the other three trigonometric functions. 0. Pthgoren Theorem c c = + b b Emple: Find the si trigonometric functions of the ngle ν in 7 ν Solution: From the informtion given, we cn clculte the vlues of cos ν nd sec ν: cos ν = 7, nd sec ν = cos ν = 7. Let us denote the side opposite of ν s. Then, b the Pthgoren Theorem, 7 = +, or 7 =. Once we hve the vlue of =, we cn finish the problem: sin ν = 7, tn ν = 7, csc ν = 7 nd cot ν =.
Find the si trigonometric functions of θ nd ω, in the following tringles:. ω 4.. ω. + β Emple: If cos θ = /, find the vlues of the other five trigonometric functions. Solution: It is given tht cos θ = hpotenuse =, so we hve two sides of the right-ngle tringle. Let s sketch such tringle, denoting the missing side s. Using the Pthgoren Theorem, we cn esil find the length of : = =. Once we know the three sides of the tringle it is ver es to finish the nswer: sin = ; tn = ; cot = = ; csc = ; sec =.. Suppose sin β = /7. Find the vlues of csc β, tn β nd cos β. 7. Suppose cos θ = /7. Find the vlues of sin θ, tn θ nd csc θ. 8. Suppose tn ν = /. Find the vlues of sin ν, cos ν nd cot ν. 9. Suppose csc =. Find the vlues of sin, cos nd cot. 0. Suppose sec = /. Find the vlues of sin, tn nd cos in terms of.. Suppose sin = /. Find the vlue of cos nd the vlue of the epression sin + cos.
. Referring to the picture below, how much is sin + cos? Completel simplif our nswer. (Hint: c = + b ). c b. Referring to the picture in eercise, how much is sec tn? 0. Specil Tringles Consider squre with side of length. Divide the squre long the digonl (clculte the length of the digonl). You will get two equl tringles. Let us concentrte on one of these. This tringle is n isosceles right-ngle tringle with hpothenuse nd legs with length. The cute ngles re 4. This is one of the two specil tringles tht we hve to lern. B now it should be n es eercise to clculte the vlues of: sin 4 = cos 4 =, tn 4 =, csc 4 =, sec 4 = nd cot 4 =. =, The second specil tringle is creted b dividing the equilterl tringle with side long one of the heights. 0 0 0 0 From the picture we cn find the vlues of sin 0 =, cos 0 = csc 0 =, sec 0 = nd cot 0 =. You cn esil find the trigonometric vlues of 0., tn 0 =, 4
You hve to know these vlues. This tsk is much esier if ou just remember the two specil tringles nd derive the vlues of the trigonometric functions of 0, 4 nd 0. Emple: Find the vlues of nd in 0 From wht hs been covered so fr we know tht, cos 0 =. On the other hnd, we know tht cos 0 =. So, cos 0 = = 0. Solving for we get = = 0 Once we hve the vlue of the hpothenuse in the right ngle tringle we cn simpl use the Pthgoren Theorem, (or use n other suitble trigonometric function involving ) ( in order to find the vlue of : = 0 ) = = 00 =. So, =. Note tht there re mn different ws to pproch this problem, but ll of them involve knowing nd understnding the specil tringles. In the right-ngled tringles below, find the vlues of the unknowns: 4.. 8 0 7.. 0 0 Evlute: 8. sin 0 + cos 0 ; 9. sin 0 cos 0 ; 0. sin 4 + sin 0 tn 4 ;. csc 0 cos 4 + cot 0 ;. sin 0 + cos 0 (Hint: eercise #8);. tn 0 cot 0 ; 4. csc 0 cot 0 ;. ( tn 0 cot 4 ) sec 4 ;
Emple: Find the cute ngle θ given sec θ =. Solution: In order to solve these kind of problems, gin we hve to rel on our ver good friends, the specil tringles. On one hnd we know tht sec θ = hpothennuse nd on the other hnd it is given tht sec θ =. It follows tht hpothennuse =, which mens tht we re tlking bout the right-ngle tringle with hpothenuse with length nd. This mens tht we re tlking bout the 0 0 90 -tringle. The lst thing we hve to nswer is which one of the two ngles it is? 0 or 0? Once we sketch the tringle the nswer is in front of our ees. The ngle to the side is 0, so θ = 0. 90 0 0 Find the cute ngle θ given:. sin θ = 40. cot θ = 44. csc θ = 48. csc θ = 7. cos θ = 4. cot θ = 4. sec θ = 49. tn θ = 8. cos θ = 4. tn θ = 4. sin θ = 0. sec θ = 9. sec θ = 4. tn θ = 47. cos θ =. sin θ =
Answers:. 4/ sin = / tn = /4. sin ν = / tn ν = / cos ν = /. sin ν = / tn ν = / cos ν = / 4. sin β = b/c tn β = b/ cos β = /c. sin = / tn = /. sin = 4/ tn = 4/ / 7. cot ν = / csc ν = / sec ν = / 8. cot ν = csc ν = sec ν = / 9. cot β = /b csc β = c/b sec β = c/ 0. cot = / sec = csc = /. cot = /4 sec = / csc = /4. sin = 4/4 4/4 tn = / cot = / csc = 4/ sec = 4/ sin ω = 4/4 cos ω = 4/4 tn ω = / cot ω = / csc ω = 4/ sec ω = 4/. sin = / / tn = / cot = csc = sec = / sin ω = / cos ω = / tn ω = cot ω = / sec ω = / csc ω = 4. sin = / / tn = cot = csc = sec =. sin = + /( + ) + /( + ) tn = cot = / csc = + / sec = + sin β = + /( + ) cos β = + /( + ) tn β = / cot β = csc β = + sec β = + /. csc β = 7/ tn β = /8 8/7 7. sin θ = 0/7 csc θ = 0/ sec θ = 7 0/0 8. sin ν = / cos ν = / cot ν = / 9. sin = / / cot = 0. sin = tn = /. 4/ sin + cos =. sin + cos =. sec tn = 4. = /, = 4 /;. = /, = /, = 0 ;. = 4 ;, = 4 ; 7. =, = ;, = 0 ; 8. + ; 9. 0; 0. 0;. + ;. ;. ; 4. ;. ;. θ = 4 ; 7. θ = 4 ; 8. θ = 0 ; 9. θ = 0 ; 40. θ = 0 ; 4. θ = 4 ; 4. θ = 4 ; 4. θ = 0 ; 44. θ = 4 ; 4. θ = 0 ; 4. θ = 0 ; 47. θ = 0 ; 48. θ = 0 ; 49. θ = 0 ; 0. θ = 4 ;. θ = 0 ; 7