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Mathematics LibreTexts

3.5E: Trig Derivatives Exercises

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3.5: Derivatives of Trigonometric Functions

Exercise:

For the following exercises, find dydx for the given functions.

175) y=x2secx+1

Answer:
dydx=2xsecxtanx

176) y=3cscx+5x

177) y=x2cotx

Answer:
dydx=2xcotxx2csc2x

178) y=xx3sinx

179) y=secxx

Answer:
dydx=xsecxtanxsecxx2

180) y=sinxtanx

181) y=(x+cosx)(1sinx)

Answer:
dydx=(1sinx)(1sinx)cosx(x+cosx)

182) y=tanx1secx

183) y=1cotx1+cotx

Answer:
dydx=2csc2x(1+cotx)2

184) y=cosx(1+cscx)

For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of x. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.

185) [T]f(x)=sinx,x=0

Answer:

y=x

CNX_Calc_Figure_03_05_201.jpeg

186) [T]f(x)=cscx,x=π2

187) [T]f(x)=1+cosx,x=3π2

Answer:

y=x+23π2

CNX_Calc_Figure_03_05_203.jpeg

188) [T]f(x)=secx,x=π4

189) [T]f(x)=x2tanx=0

Answer:

y=x

CNX_Calc_Figure_03_05_205.jpeg

190) [T]f(x)=5cotxx=π4

For the following exercises, find d2ydx2 for the given functions.

191) y=xsinxcosx

Answer:
3cosxxsinx

192) y=sinxcosx

193) y=x12sinx

Answer:
12sinx

194) y=1x+tanx

195) y=2cscx

Answer:
csc(x)(3csc2(x)1+cot2(x))

196) y=sec2x

197) Find all x values on the graph of f(x)=3sinxcosx where the tangent line is horizontal.

Answer:
(2n+1)π4,where n is an integer

198) Find all x values on the graph of f(x)=x2cosx for 0<x<2π where the tangent line has slope 2.

199) Let f(x)=cotx. Determine the points on the graph of f for 0<x<2π where the tangent line(s) is (are) parallel to the line y=2x.

Answer:
(π4,1),(3π4,1)

200) [T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function s(t)=6cost where s is measured in inches and t is measured in seconds. Find the rate at which the spring is oscillating at t=5 s.

201) Let the position of a swinging pendulum in simple harmonic motion be given by s(t)=acost+bsint. Find the constants a and b such that when the velocity is 3 cm/s, s=0 and t=0.

Answer:
a=0,b=3

202) After a diver jumps off a diving board, the edge of the board oscillates with position given by s(t)=5cost cm at tseconds after the jump.

a. Sketch one period of the position function for t0.

b. Find the velocity function.

c. Sketch one period of the velocity function for t0.

d. Determine the times when the velocity is 0 over one period.

e. Find the acceleration function.

f. Sketch one period of the acceleration function for t0.

203) The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by y=10+5sinx where y is the number of hamburgers sold and x represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find y and determine the intervals where the number of burgers being sold is increasing.

Answer:
y=5cos(x), increasing on (0,π2),(3π2,5π2), and (7π2,12)

204) [T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by y(t)=0.5+0.3cost, where t is months since January. Find yand use a calculator to determine the intervals where the amount of rain falling is decreasing.

For the following exercises, use the quotient rule to derive the given equations.

205) ddx(cotx)=csc2x

206) ddx(secx)=secxtanx

207) ddx(cscx)=cscxcotx

208) Use the definition of derivative and the identity cos(x+h)=cosxcoshsinxsinh

to prove that d(cosx)dx=sinx.

For the following exercises, find the requested higher-order derivative for the given functions.

209) d3ydx3 of y=3cosx

Answer:
3sinx

210) d2ydx2 of y=3sinx+x2cosx

211) d4ydx4 of y=5cosx

Answer:
5cosx

212) d2ydx2 of y=secx+cotx

213) d3ydx3 of y=x10secx

Answer:
720x75tan(x)sec3(x)tan3(x)sec(x)

ADD IN J214 - J217


3.5E: Trig Derivatives Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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