Trigonometric Functions () 1 / 28
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1 Trigonometric Functions () 1 / 28
2 Trigonometric Moel On a certain ay, ig tie at Pacific Beac was at minigt. Te water level at ig tie was 9.9 feet an later at te following low tie, te tie eigt was 0.1 ft. Te next ig tie was at 12 noon. Te eigt of te water is given by te following function (t) = cos((π/6)t), were t = 0 correspons to minigt. Interpret tis equation. () 2 / 28
3 Trigonometric Moel On a certain ay, ig tie at Pacific Beac was at minigt. Te water level at ig tie was 9.9 feet an later at te following low tie, te tie eigt was 0.1 ft. Te next ig tie was at 12 noon. Te eigt of te water is given by te following function (t) = cos((π/6)t), were t = 0 correspons to minigt. Interpret tis equation. () 2 / 28
4 Trigonometric Moel On a certain ay, ig tie at Pacific Beac was at minigt. Te water level at ig tie was 9.9 feet an later at te following low tie, te tie eigt was 0.1 ft. Te next ig tie was at 12 noon. Te eigt of te water is given by te following function (t) = cos((π/6)t), were t = 0 correspons to minigt. Interpret tis equation. () 2 / 28
5 Derivative of Cosine Will Nee: Ientity: cos(a + b) = cos(a) cos(b) sin(a) sin(b) Teorem 2.4.7: lim x 0 sin(x) x = 1 Half Angle Formula: sin 2 (θ/2) = 1 cos(θ) 2 cos(x + ) cos(x) cos(x) cos() sin(x) sin() cos(x) lim = lim = lim 0 cos(x)(cos() 1) sin(x) sin() (cos() 1) = lim cos(x) sin(x) sin() () 3 / 28
6 Derivative of Cosine Will Nee: Ientity: cos(a + b) = cos(a) cos(b) sin(a) sin(b) Teorem 2.4.7: lim x 0 sin(x) x = 1 Half Angle Formula: sin 2 (θ/2) = 1 cos(θ) 2 cos(x + ) cos(x) cos(x) cos() sin(x) sin() cos(x) lim = lim = lim 0 cos(x)(cos() 1) sin(x) sin() (cos() 1) = lim cos(x) sin(x) sin() () 3 / 28
7 Derivative of Cosine Will Nee: Ientity: cos(a + b) = cos(a) cos(b) sin(a) sin(b) Teorem 2.4.7: lim x 0 sin(x) x = 1 Half Angle Formula: sin 2 (θ/2) = 1 cos(θ) 2 cos(x + ) cos(x) cos(x) cos() sin(x) sin() cos(x) lim = lim = lim 0 cos(x)(cos() 1) sin(x) sin() (cos() 1) = lim cos(x) sin(x) sin() () 3 / 28
8 Derivative of Cosine Will Nee: Ientity: cos(a + b) = cos(a) cos(b) sin(a) sin(b) Teorem 2.4.7: lim x 0 sin(x) x = 1 Half Angle Formula: sin 2 (θ/2) = 1 cos(θ) 2 cos(x + ) cos(x) cos(x) cos() sin(x) sin() cos(x) lim = lim = lim 0 cos(x)(cos() 1) sin(x) sin() (cos() 1) = lim cos(x) sin(x) sin() () 3 / 28
9 ...Derivative of Cosine By Teorem lim 0 sin() = 1. Terefore, lim sin(x)sin() = sin(x). By te alf angle formula, cos() 1 = 2 sin 2 (/2). Combine wit Teorem tis gives lim cos(x)cos() 1 2 sin(/2) = lim cos(x) sin(/2) =cos(x)(0) =0. Tus, cos(x) =0 sin(x) = sin(x). x A similar proof sows x sin(x) =cos(x). () 4 / 28
10 ...Derivative of Cosine By Teorem lim 0 sin() = 1. Terefore, lim sin(x)sin() = sin(x). By te alf angle formula, cos() 1 = 2 sin 2 (/2). Combine wit Teorem tis gives lim cos(x)cos() 1 2 sin(/2) = lim cos(x) sin(/2) =cos(x)(0) =0. Tus, cos(x) =0 sin(x) = sin(x). x A similar proof sows x sin(x) =cos(x). () 4 / 28
11 ...Derivative of Cosine By Teorem lim 0 sin() = 1. Terefore, lim sin(x)sin() = sin(x). By te alf angle formula, cos() 1 = 2 sin 2 (/2). Combine wit Teorem tis gives lim cos(x)cos() 1 2 sin(/2) = lim cos(x) sin(/2) =cos(x)(0) =0. Tus, cos(x) =0 sin(x) = sin(x). x A similar proof sows x sin(x) =cos(x). () 4 / 28
12 ...Derivative of Cosine By Teorem lim 0 sin() = 1. Terefore, lim sin(x)sin() = sin(x). By te alf angle formula, cos() 1 = 2 sin 2 (/2). Combine wit Teorem tis gives lim cos(x)cos() 1 2 sin(/2) = lim cos(x) sin(/2) =cos(x)(0) =0. Tus, cos(x) =0 sin(x) = sin(x). x A similar proof sows x sin(x) =cos(x). () 4 / 28
13 ...Derivative of Cosine By Teorem lim 0 sin() = 1. Terefore, lim sin(x)sin() = sin(x). By te alf angle formula, cos() 1 = 2 sin 2 (/2). Combine wit Teorem tis gives lim cos(x)cos() 1 2 sin(/2) = lim cos(x) sin(/2) =cos(x)(0) =0. Tus, cos(x) =0 sin(x) = sin(x). x A similar proof sows x sin(x) =cos(x). () 4 / 28
14 ...Derivative of Cosine By Teorem lim 0 sin() = 1. Terefore, lim sin(x)sin() = sin(x). By te alf angle formula, cos() 1 = 2 sin 2 (/2). Combine wit Teorem tis gives lim cos(x)cos() 1 2 sin(/2) = lim cos(x) sin(/2) =cos(x)(0) =0. Tus, cos(x) =0 sin(x) = sin(x). x A similar proof sows x sin(x) =cos(x). () 4 / 28
15 Derivatives of Trig Functions x cos(x) = sin(x), an sin(x) =cos(x) x Derivatives of oter trig functions? () 5 / 28
16 3.5- Derivatives of Trig Functions x cos(x) = sin(x) x sin(x) =cos(x) x tan(x) =sec2 (x) =(sec(x)) 2 x sec(x) =sec(x) tan(x) x csc(x) = csc(x) cot(x) x cot(x) = csc2 (x) () 6 / 28
17 Gears ttp:// () 7 / 28
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