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Derivatives of Trig Functions
© G. Battaly 2019 1
October 20, 2019
Calc Web Pages
Study 3.5 # 175197, 201, 203
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
GOALS: 1. Recognize the derivatives of: f(x)= sin(x), g(x)= cos(x), h(x)= tan(x)2. Recognize the derivatives of reciprocal functions: f(x)= csc(x), g(x)= sec(x), h(x)= cot(x)3. Find the derivatives of all 6 functions.4. Notice that the derivatives of all the cofunctions are negative. cosine(x) cosecant(x) cotangent(c)
Calc Web Pages
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
Reference Graphs of Trig Functions
y= sin x
y= csc x
y= cos x
y= sec x y= cot x
y= tan x
Derivatives of Trig Functions
© G. Battaly 2019 2
October 20, 2019
Calc Web Pages
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
Investigate: d [ sin(x) ] dx
1. Go to: https://www.geogebra.org/classic This opens an online software called geogebra.
2. Click upper right bars and select the folder option.3. In the window for the file name, type:http://www.battaly.com/calc/geogebra/trig/derivative_sinx.ggb
4. Click the X under the previous bars to clear the graphing window. 5. Then click the circles to the left of f: y= sin(x) and A=(c,sin(c))6. Click and drag either the point A or the c bar and watch the point move along the curve of y=sin(x)
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
Investigate: d [ sin(x) ] dx
7. Click the circle to the left of T, the tangent line at point A.8. Notice the slope of the tangent line. What is its value? What is it, in words? How does it change as the A is moved?When is it positive? negative? zero?
9. Find point B on the left side, and click the circle to the left of B. B has the same xcoordinate as A, but its ycoordinate is the slope of the tangent line T or the derivative of y=sin(x) at that x value. 10. Move point A to see what happens to point B. 10. Return to point B on the left. Right click on it and Turn ON SHOW TRACE. Then move point A again.11. What do you see? What does the resulting curve represent?
Derivatives of Trig Functions
© G. Battaly 2019 3
October 20, 2019
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework
Investigate: d [ sin(x) ] dx
Just finished a very visual representation of the derivative of sin(x) How do we do that analytically?
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
Investigate: d [ sin(x) ] dx
p.773
sin x csc x = 1
cos x sec x = 1
tan x cot x = 1
Derivatives of Trig Functions
© G. Battaly 2019 4
October 20, 2019
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
Derivatives of Trig FunctionsCofunctions: derivatives are neg
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
F: tangent line
Derivatives of Trig Functions
© G. Battaly 2019 5
October 20, 2019
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
Derivatives of Trig Functions
© G. Battaly 2019 6
October 20, 2019
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
Derivatives of Trig Functions
© G. Battaly 2019 7
October 20, 2019
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
Derivatives of Trig Functions
© G. Battaly 2019 8
October 20, 2019
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
F: eq of tangent line at x = π/4
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
y y1 = mT(x x1)
need (x1 , y1) and mT
F: eq of tangent line at x = π/4 45
45(x1,y
1)= (π/4, 2 ); need mT
Derivatives of Trig Functions
© G. Battaly 2019 9
October 20, 2019
45
4530
60
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
Derivative Practice
Derivatives of Trig Functions
© G. Battaly 2019 10
October 20, 2019
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
v(t)=0 at t=0,π, 2πs(t)=5cost
a(t)=v '(t)=5cost
v(t)=s '(t)=5(sint)=5sint
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
F: tangent line
Derivatives of Trig Functions
© G. Battaly 2019 11
October 20, 2019
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
Derivative Practice
3.5 Derivatives of Trig Functions
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NYHomework
Derivatives of Trig Functions
© G. Battaly 2019 12
October 20, 2019
HW: 183 answer can look different and still be correct.