14 Pull It All Together

BIG idea Equivalence

You can use symbols to represent an expression in an unlimited number of ways, where all representations have the same value when each variable is replaced with its assigned number.

Task 1

Prove the identity cos x + cos y = 2 cos ( x + y 2 ) cos ( x − y 2 ) . cosine x plus cosine y equals 2 cosine . open . fraction x plus y , over 2 end fraction . close cosine . open . fraction x minus y , over 2 end fraction . close . .

1. Show that x + y 2 + x − y 2 = x . fraction x plus y , over 2 end fraction . plus . fraction x minus y , over 2 end fraction . equals x . .
2. Find a similar expression using x + y 2 fraction x plus y , over 2 end fraction and x − y 2 fraction x minus y , over 2 end fraction that equals y.
3. Use parts (a) and (b) to prove the identity.

BIG idea Function

Function You can often represent a relationship between variables as a function, in which each value of the input variable is associated with a unique value of the output variable.

Task 2

The graphs below are of the representative parts of the sine and cosine functions for which their inverse functions are defined.

1. Use the two graphs to help you sketch the graph of the distance between x and cos − 1 ( sin x ) cosine super negative 1 end super . open sine x close for − π 2 ≤ x ≤ π 2 . negative , pi over 2 , less than or equal to x less than or equal to , pi over 2 . .
2. Sketch the graph of the distance between x and sin − 1 ( cos x ) sine super negative 1 end super . open cosine x close for 0 ≤ x ≤ π . 0 less than or equal to x less than or equal to pi .
3. How do the two graphs appear to be related?

Task 3

A regular n-gon is inscribed in the unit circle. What is the perimeter for each n?

1. 3
2. 5
3. 6
4. 10
5. 57
6. 542
7. n
8. The perimeter in part (f) should be close to what number? How close is it? (Hint: Use the Law of Cosines. Also, you need not begin with part (a).)

BIG idea Equivalence

The facts about a quantity may be expressed by many different equations.

Task 4

For a 30 ° − 60 ° − 90 ° 30 degrees , minus 60 degrees negative 90 degrees triangle, how does the distance from the incenter to each vertex compare to the radius of the incircle?

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