CALC - Lesson 3.0: Trig. Basics Unit 3 – Trig and Optimization
Big Picture
Context of this
Lessons
To Do!
-
-
The big picture for this Unit will still revolve around Derivatives. We will
now add the Trig functions into the mix. Our final big idea for Unit 3 will
be working with optimization and the all important creation of
equations. After we have solidified this we will then move into graphs…
but optimization for now!
We just worked with: Derivatives laws and tangent/normal lines. In this
lesson we will continue to solidify the trig ideas with conversations about
trig. This will then transition to some work with Optimization.
Task 1: Warm Up Problems
Task 2: A deeper Look into to Trig…
o Soh-Cah-Toa
o The Unit Circle
o Radians vs. Degrees
o Graphs
Task 4: Optimization
o The Major Concept
o Some Practice
o Challenge Problem Discussion
Task 1: Warm Up Problems
1. Find the equation of the tangent line to the curve y = (3x+4)/(4x – 3) at the point (1,7).
2. The line y = 4x + 3 is tangent to the curve y = x2 + c, then c is?
a. 2
b. 4
c. 7
d. 11
e. 15
CALC - Lesson 3.0: Trig. Basics Unit 3 – Trig and Optimization
3. If f(x) = x2 + 3x and let g(x) and it’s derivative g’(x) have the values shown in the table
above. If h(x) = f(g(x)), what is h’(2)?
x
1
2
3
4
5
6
g(x)
0
1
3
7
2
5
g‘(x)
4
3
1
-1
-2
-1
Task 2: A deeper Look into to Trig…
1. What does Soh-Cah-Toa Stand for.
2. Given the triangle below, find the values given values. Please leave your answers as
fractions… don’t round.
Cos() =
Cos() =
4
Sin() =
Sin() =
Tan() =
Tan() =
7
3. Do your best, with pictures, words, and numbers to explain what a Radian is.
4. Convert the following to from Radians to Degrees, or Degrees to Radians. Show your
work.
120 Degrees  Radians
5/6 Radians  Degrees
225 Degrees  Radians
/6 Radians  Degrees
CALC - Lesson 3.0: Trig. Basics Unit 3 – Trig and Optimization
5. Given the following picture of a part of the unit Circle, Find the values below.
Cos( 240o)=
Sin(330 o)=
Cos( 90o)=
Sin( 60 o)=
Cos( 180o)=
Sin(90 o)=
Cos(5/3)=
Sin(4/3)=
Cos(3/4)=
Sin(/4)=
CALC - Lesson 3.0: Trig. Basics Unit 3 – Trig and Optimization
6. Please match the graph and the function. Please explain your reasoning.
a. f(x) = Sin(x)
b.f(x) = Cos(x).
c. f(x) = Tan(x)
d. f(x) = Sec(x)
e. f(x) = Csc(x)
d. f(x) = Cot(x)
CALC - Lesson 3.0: Trig. Basics Unit 3 – Trig and Optimization
Task 3: Steps to working with Optimization
Step 1: Read the problem and understand what you are trying to “Optimize” in other words
finding the max or min of… UNDERSTAND THE PROBLEM FIRST! Or write down and look at all
your information
Step 2: Create a Picture if possible. Or write down and look at all your information
Step 3: Create a General case for the picture and try and develop a mathematical Model
to fit the situation
CALC - Lesson 3.0: Trig. Basics Unit 3 – Trig and Optimization
Step 4: You will now work with your equations to create a function that you are trying to
optimize. This will require some algebraic manipulation but you should expect that at this
stage
Step 5: We now use derivatives to find the local max or min of the function we are trying to
optimize and set that derivative equal to zero to find our desired input value.
Try these two on your own!
a. What is the area of the largest rectangle with the lower base on the x-axis and the upper
vertices on the curve y = 12 – x2. Draw a picture to help.
b. What is the smallest possible sum of squares of two numbers, if their product is -16?