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M130 Course Notes Exam 3 to Exam 4 4.1 Exponentials 1. The exponential function with base a is f (x) = ax for a > 0, a 6= 1. 2. Graphs of exponential when 0 < a < 1 and a > 1. (a) Domain (b) Range (c) Passes through (0, 1) (d) Always increasing/decreasing (e) Horizontal asymptote (f) Transformations 3. Base e 4.2 Logarithms 1. Note f (x) = ax is injective. Hence f −1 exists. 2. Defn: Let a 6= 1 be a positive number. The logarithmic function with base a is denoted by loga and is defined by loga x = y ⇐⇒ ay = x 3. Evaluating logs (a) log1 = 0 (b) loga a = 1 (c) loga ax = x (d) aloga x = x 4. Graphing (mono-inc.), passes through (1, 0) 5. Natural log E.g., 1. Simplify log4 (1/2) 2. Solve for x: log4 (x) = 2 3. Convert to log form 811/2 = 9 4. Convert to exp form log3 81 = 4 4.3 Logarithm Laws 1. log xy = log x + log y 2. log(x/y) = log x − log y 3. log xy = y log x 4. logb x = logB x logB b 5. Expand/Contract 4.4 Exponential and Logarithmic Equations E.g., 1. 2x = 8 2. 23x = 34 3. x2 2x − x2x = 2 · 2x 4. e2x − ex − 6 = 0 5. ln(2 + x) = 1 6. log2 (x2 − x − 2) = 2 7. ln(x − 1) + ln(x + 2) = 1 5.1 Unit Circle 1. x2 + y 2 = 1 2. is P on S 1 ? 3. Locating a point on S 1 given x/y value and Quadrant ... 4. Defn: The terminal point P (x, y) is the value obtained by moving t units from (1, 0) on S 1 counterclockwise if t > 0 and clockwise for t < 0. (a) t = π (b) t = 0 (c) t = π/2 5. Defn: The reference number t corresponding to the terminal point for t is the shortest distance on S 1 between T P (t) and the x-axis 5.2 Trigonometric Functions 1. On S 1 , sin θ = y, csc θ = 1/y, etc. 2. Consider rt. 4 3. Evaluating trig. function 4. Where each is positive/negative 5. Domain (a) sin, cos: R (b) tan, sec: R\ {π/2 + nπ}n∈Z (c) cot, csc: R\ {nπ}n∈Z 6. csc θ = 1/ sin θ, etc. 7. Even/Odd 8. Pythagorean Identities (a) sin2 θ + cos2 θ = 1 (b) tan2 θ + 1 = sec2 θ (c) 1 + cot2 θ = csc2 θ E.g., 1. sin, cos, tan, cot, tan, sec, csc @0, π, π/ 2. (−3/5, 4/5) is terminal point evaluate trig functions. 3. Sign of tan t sec t in Q IV 4. Write first in terms of second when in Q III tan t, cos t 5. Write first in terms of second when in Q IV sin t, sec t 5.3 Trigonometric Graphs 1. Periodicity Identities: (a) sin(t + 2πn) = sin t for n ∈ Z (b) cos(t + 2πn) = cos t for n ∈ Z 2. Graphs of Sine, Cosine 3. Ψ(x) = A sin k(x − γ) and χ(x) = A cos k(x − γ) for positive k have: (a) Amplitude |A| (b) Period 2π/k (c) Phase Shift γ (d) One period appears in the interval [γ, γ + 2π/k] E.g., 1. Find the amplitude, period, phase shift, and graph one period. (a) y = 2 sin(x − π/3) (b) y = cos[1/2(x + π/4)] (c) y = 2 sin(2/3x − π/6) 2. The graph of one period is given. Determine the period, phase shift, and amplitude; and give an equation. 5.4 More Trigonometric Graphs 1. Periodicity Identities: (a) tan(t + π) = tan t (b) cot(t + π) = cot t (c) csc(t + 2π) = csc t (d) sec(t + 2π) = sec t 2. Graphs of sec x, csc x, tan x, cot x (a) On (−π/2, π/2), y = tan t is %. (b) On (0, π), y = cot t is &. (c) y = csc t: (0, π) ∼ ∪ (π, 2π) ∼ ∩. (d) y = sec t: (0, π/2) ∼ RHS(∪) (π, 3π/2) ∼ ∩ (3π/2, 2π) ∼ LHS(∪). 3. The functions y = a tan(kx) and y = a cot(kx) have period π/k. 4. Then functions y = a csc(kx) and y = a sec(kx) have period 2π/k. E.g., 1. Find the period and graph the function. (a) y = tan(x/2) (b) y = tan(x − π/4) (c) y = 2 csc(x − π/3) (d) y = sec[2(x + π/2)] (e) y = cot(x − π/2) (f) y = 1/2 sec(2πx − π) 6.1 Angle Measure 1. Convert between radians/degrees. 2. Defn: An angle is in standard position if one side of the angle is the x-axis. 3. Defn: Two angles in standard position are coterminal if their (non x-axis) sides coincide 4. How to reduce angles. (Review) 5. Arc length s = rθ 6. Area of a sector A = 21 r2 θ E.g., 1. (Length of arc/area of sector) that subtends central angle of 45◦ in a circle of radius 10. 2. A sector of a circle of radius 24 has an area of 288. Find the central angle. 6.2 Right Triangle Trigonometry 1. Trig. functions in reference to a triangle. 2. Special triangles E.g., 1. From the top of a 200ft building you check out hot c/hicks. If you spy a hottie and the angle of depression to the hottie is 23◦ . How far are you from the hottie? How far is the hottie from the building? 6.3 Trigonometric Functions of Angles 1. Trig. functions in reference to an imbedded triangle in a circle of radius r. E.g., 1. Find the values of the trig functions of θ if cos θ = −7/12 with θ ∈ QIII. 2. Find the values of the trig functions of θ if cot θ = 1/4 with sin θ < 0. 7.1 Trigonometric Identities Pythagorean 1. sin2 θ + cos2 θ = 1 2. tan2 θ + 1 = sec2 θ 3. cot2 θ + 1 = csc2 θ Reciprocal 1. tan θ = sin θ cos θ 2. cot θ = cos θ sin θ 3. sec θ = 1 cos θ 4. csc θ = 1 sin θ = 1 tan θ Cofunction 1. sin( π2 − θ) = cos θ 2. cos( π2 − θ) = sin θ 3. tan( π2 − θ) = cot θ 4. sec( π2 − θ) = csc θ 5. csc( π2 − θ) = sec θ 6. cot( π2 − θ) = tan θ Even/Odd 1. sin(−θ) = − sin θ 2. cos(−θ) = cos θ 3. tan(−θ) = − tan θ E.g., 1. Simplify cos t csc t 2. Simplify cos3 t + sin2 t cos t 3. Simplify 1+cot A csc A 4. Verify cos x/ sec x + sin x/ csc x = 1 5. Verify 1 − sin x = (sec x − tan x)2 1 + sin x 6. Verify sec t csc t(tan t + cot t) = sec2 t + csc2 t 7. Verify tan x + tan y = tan x tan y cot x + cot y 7.2 Addition and Subtraction Formulae 1. sin(α ± β) = sin α cos β ± cos α sin β 2. cos(α ± β) = cos α cos β ∓ sin α sin β 3. tan(α ± β) = tan α ± tan β 1 ∓ tan α tan β E.g., 1. Determine sin 15◦ 2. Determine cos 17π/12 3. Verify cot(π/2 − u) = tan u 4. Verify cos(x − π) = − cos x 5. Verify 1 − tan x tan y = cos(x + y) cos x cos y 6. Verify cos(x + y) cos(x − y) = cos2 x − sin2 y 7. Verify cos(x + y) + cos(x − y) = 2 cos x cos y 7.3 Double-Angle and Half-Angle Formulae Double-Angle 1. sin 2x = 2 sin x cos x 2. cos 2x = cos2 x − sin2 x = 1 − 2 sin2 x = 2 cos2 x − 1 3. tan 2x = 2 tan x 1 − tan2 x Half-Angle r 1 − cos x 2 r 1 + cos x 2. cos(x/2) = ± 2 1. sin(x/2) = ± 3. tan(x/2) = 1 − cos x sin x = sin x 1 + cos x E.g., 1. Find sin 2x if x is in QII and tan x = −4/3 2. Determine tan 15◦ 3. Determine cos 3π/8 4. Verify sin 8x = 2 sin 4x cos 4x 5. Verify tan2 (x/2 + π/4) = 1 + sin x 1 − sin x 7.4 Inverse Trigonometric Functions 1. a 7.5 Trigonometric Equations 1. a