SECTION 3.3 DERIVATIVES OF TRIG FUNCTIONS
1. Fill in the table below.
Derivatives of Trigonometric Functions:
• ddx(sinx) =
• ddx(cosx) =
• ddx(tanx) =
• ddx(cscx) =
• ddx(secx) =
• ddx(cotx) =
2. Find the derivative of y =secx
1� x tanx.
3. If f(✓) = e✓ sin(✓), find f 00(✓). Simplify your answers.
4. Findd
dt[t sin t cos t].
UAF Calculus I 1 3-3 Derivatives of Trig Functions
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f G echos O sin O e e4 cos Ot Sino
f O eOf sin cos 0 t SCO since e
eosin0 eocosO teocosO eosinO2eocosO
dfzfftsintlcostl ftsinltdf sintltcosttlddtlts.int
tlsinltvtws.lt tcoftI sinLtI
ts inLt 5ttkosLtD2tsinLtIwsCtJCjustfo fun
5. An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the massis pulled down 2 cm past its rest position and then released, it vibrates vertically. The equation ofmotion is
s = 2 cos t+ 3 sin t, for t � 0,
where s is measured in centimeters and t is measured in seconds. (We are taking the positivedirection to be downward.)
(a) Why might you expect to use sines and cosines to model this particular problem?
(b) Find s(0), s0(0), and s00(0) including units.
(c) What do the numbers from part (a) indicate about the mass in the context of the problem?
6. A 12 foot ladder rests against a wall. Let ✓ be the angle between the ladder and the wall and let xbe the distance from the base of the ladder and the wall.
(a) Compute x as a function of ✓. (Drawing a picture will help.)
(b) How fast does x change with respect to ✓ when ✓ = ⇡/6? (Get an exact answer and a decimalapproximation.)
(c) Interpret your answer from part (b) in the context of the problem. (Units will help you here.)
(d) Determine how far the ladder is from the wall when ✓ = ⇡/6.
UAF Calculus I 2 3-3 Derivatives of Trig Functions
Thebouncing is periodic as are sines cosines 9o A a massat
S o 2 em I g12cmrest
S Ith LL Sint 3cosLt 2sin 1 3cost t
S t 2cost 3 sint S lo 3 curlsS o 2curlsIs 2cms2
Thedisplacement at time t o is 2cmThe initial velocity is 3curlsThe initial acceleration is 2cms2
TY Fp sin0 x 12 since µ12cos0 so d o
12 Ws 1275 6538 10.39
Whenthe angle is The the endofthe ladder is sliding away fromthe wall at a rateof GT3 ttfradians 210.39 fttradians
When 0 476 Xe 12 Sir 6 12 12 6The ladder is 6ft from the wall Check
f z