Appendix D. Trigonometry.Angles can be measured in degrees or radians. The angle given by a complete revolutioncontains 360o, or 2π radians.360o= 2π rad1 rad =180πo1o=π180radExample 1.(a.) Convert 9oto radians.(b.) Convert5π12to degrees.The standard position of an angle occurs when we place its vertex at the origin of acoordinate system and its initial side on the positive x-axis. A positive angle is obtained byrotating the initial side counterclo ckwise until it coincides with the terminal side.Negative angles are obtained by clockwise rotation1The trigonometric functions.sin θ =opphypcos θ =adjhyptan θ =oppadjcot θ =adjoppsec θ =hypadjcsc θ =hypoppSpecial triangles:sinπ4=√22sinπ6=12sinπ3=√32cosπ4=√22cosπ6=√32cosπ3=12tanπ4= 1 tanπ6=1√3sinπ3=√32Signs of the trigonometric functions:Example 2.(a.) If sin θ =35(0 < θ < π/2), find the r emaining trigonometric r ations.(b.) If cos θ = −13(π < θ < 3π/2 ), find the remaining trigonometric rations.3Trigonometric Identities:csc θ =1sin θsec θ =1cos θcot θ =1tan θtan θ =sin θcos θsin2θ + cos2θ = 1tan2θ + 1 = sec2θcot2θ + 1 = csc2θsin(−θ) = − sin θ cos(−θ) = cos θsin(θ + 2π) = sin θ cos(θ + 2π) = cos θtan(θ + π) = tan θ cot(θ + π) = cot θsin(x + y) = sin x cos y + cos x sin ysin(x − y) = sin x cos y − cos x sin ycos(x + y) = cos x cos y − sin x sin ycos(x − y) = cos x cos y + sin x sin ytan(x + y) =tan x + tan y1 − tan x tan ytan(x − y) =tan x − tan y1 + tan x tan ysin 2x = 2 sin x cos xcos 2x = cos2x − sin2x = 1 − 2 sin2x = 2 cos2x − 1cos2x =1 + cos 2x2sin2x =1 − cos 2x2sin x cos y =12[sin(x + y) + sin(x − y)]cos x cos y =12[cos(x + y) + cos(x − y) ]sin x sin y =12[cos(x − y) − cos(x + y)]Example 4. If sin x =13and sec y =54, where x and y lie between 0 and π/2, evaluatethe expression(a.) cos(x − y)4(b.) sin 2yExample 5. Find all values of x in the interval [0, 2π] that satisfy the equation:(a.) 2 cos x − 1 = 0(b.) 2 cos x + sin 2x = 05Graphs of the trigonometric functions.y = sin xy = cos xy = tan x y = cot x6y = csc x y = sec xExample 6. Sketch the graph of f(x) = 1 − sin
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