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Chapter 8: Trigonometric Equations and Applications

Chapter 8: Trigonometric Equations and Applications. L8.1 Solving Simple Trigonometric Equations and Angles of Inclination . This equation has many solutions (where these 2 curves intersect) :. y = ½ . π. 3 π. −2 π. 2 π. − π. −3 π. y = sin x.

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Chapter 8: Trigonometric Equations and Applications

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  1. Chapter 8: Trigonometric Equations and Applications L8.1 Solving Simple Trigonometric Equations and Angles of Inclination

  2. This equation has many solutions (where these 2 curves intersect): y = ½ π 3π −2π 2π −π −3π y = sin x Solving Simple Trigonometric Equation e.g., sin x = ½ A trigonometric equation is an equation that involves at least one trigonometric function. To solve, find value(s) of the variable that make the equation true. default period • You may be asked to find • solutions on a specific interval • for ex, for 0 ≤ x < 2π, the solns are x = π/6, x = 5π/6. • general solutions (i.e., all solutions) • in general, x = π/6 + 2πn, x = 5π/6 + 2πn, n ϵ Z.

  3. Solving Trigonometric Equations use the unit circle (for special trig values) OR use your calculator’s inverse trig functions (sin-1, cos-1, tan-1). For ex, cos x = occurs at 5π/6 & 7π/6 Some observations: • ▪ sin(x) = value or cos(x) = value occurs twice in ea 2π period* • ▪ sin(x) > 0; x ϵ Q1 & Q2sin(x) < 0; x ϵ Q3 & Q4 • ▪ cos(x) > 0; x ϵ Q1 & Q4cos(x) < 0; x ϵ Q2 & Q3 • ▪ tan(x) = value occurs once on each π period • ▪ tan(x) > 0; x ϵ Q1 & Q3tan(x) < 0; x ϵ Q2 & Q4 * Exception: the extreme values, 1 & −1, occur only once in each period or rotation To find the angles that have a specific value, you can

  4. Solving Trigonometric Equation: General Steps θref θref θref θref QII QI QIII QIV • Isolate trig function using gen’l algebra techniques, if nec • Convert to primary trig function, if nec • Identify the quadrants of solutions. • Find reference angle (in Q1), then use angleref to find solns*. • If general solutions requested, add multiple periods to each solution. * To find θ given θref : [Prior to this, we went the other way, θ → θref ].

  5. Solving Trigonometric Equation: Examples Ref angle: Solve for 0 ≤ x < 360°. Do not use calculator unless required. Solns in Q3 & Q4. Q3 x = 180 + 45 = 225° and Q4 x = 360 – 45 = 315°. 2. Ref angle: cos-1(⅝) ≈ 51.3° Solns in Q1 & Q4. → cos x = ⅝ Q1 x = 51.3° and Q4 x = 360 – 51.3 = 308.7°. Solns in Q2 & Q4. 3. 4tan x = −3 Ref angle: tan-1(¾) = 36.9° Q2 x = 180 – 36.9 = 143.1° and Q4 x = 360 – 36.9 = 323.1°. Whenever you solve any equation, you can check your answers by plugging them in and confirming that they make the original equation true.

  6. Solving Trigonometric Equation: Examples Solve for 0 ≤ x < 2π. Do not use calculator unless required. • csc x = −2 Solve (this means general solution or all solutions). x in radian. • 8cos(x) + 7 = 0 • tan(x) = 2

  7. Angles of Inclination The inclination of a line is an angle α, where 0 ≤ α < 180°. The slope of a nonvertical line is rise / run or Δy / Δx [= tangent!] y y l2 l1 α α x x m = tan(α) The slope of l1 = tan α tan α > 0 (m > 0) The slope of l2 = tan α tan α < 0 (m < 0) • Examples: • Write the equation of a line with inclination 158° which goes thru (−3,5). • To the nearest degree, find the inclination of the line 7x + 8y = 16. 5 = −.4(−3) + b → b = 3.8 m = tan(158°) ≈ −.4 So y = −.4x + 3.8 αref = tan-1(⅞) ≈ 41° → y = −⅞x + 2, so m = −⅞ m < 0, soα ≈ 180 − 41 = 139°

  8. Chapter 8: Trigonometric Equations and Applications L8.2 Solving Trigonometric Equations of the form trigfcn(ax) = value

  9. Warm up 1. Solve for 0 ≤ x < 2π to the nearest 100th of a radian. • 2sec x + 9 = 6 sec x = −3/2 cos x = −⅔ → x is in Q2 & Q3 xref = cos-1(⅔) ≈ 0.84 → x ≈ π − 0.84 ≈ 2.30 or x ≈ π + 0.84 ≈ 3.98 Check answer: 2sec(2.30) ≈ 5.998 √ 2sec(3.98) ≈ 6.009 √ 2. To the nearest 10th degree, find the inclination of the line 3x + 5y = 15. y = (−3/5)x + 5 → m = (−3/5)x m = tan(θ) and θ is in Q2 θref = tan-1(3/5) ≈ 31.0º → θ = 180 − 31.0 = 149.0º

  10. Solving Trigonometric Equation: Examples Solve for 0 ≤ x < 2π. Do not use calculator unless required. • log2(sec(x)) = 1 • log3(sin(x)) = −1 Solve (this means general solution or all solutions). x in radian. 1. |sec(x)| = 2

  11. Solving trigfcn(ax) = value For example: sin(3x) = ½ or tan(x/4) = −1 IMPORTANT NOTE: sin(3x) ≠ 3·sin(x) b/c for any function, f, f(ax) ≠ af(x) To solve these trig equations of the form trigfcn(ax) = value: • Readjust the interval of interest for the variable • Solve for ax, as you would for x • At the very end, solve for x Example: Solve sin(3x) = ½ for 0 ≤ x < 2π • 0 ≤ x < 2π → 0 ≤ 3x < 6π – need angles on 3 full periods! • sin(3x) = ½ → 3x = π/6, 5π/6, 13π/6, 17π/6, 25π/6, 29π/6 • So x = π/18, 5π/18, 13π/18, 17π/18, 25π/18, 29π/18 Note: These values for x are within [0, 2π)

  12. Solving Simple Trigonometric Equations Steps to solve trigfcn(ax) = value • If a ≠ 1, readjust the interval of interest • Isolate trig function using gen’l algebra techniques, if nec • Convert to primary trig function, if nec • Identify the quadrants of solutions. • Find reference angle (in Q1), then use angleref to find solns*. • If a ≠ 1, solve for x • If general solutions requested, add multiple periods to each solution.

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