Use identities to find the exact value.
24) cos(-75°)

( - )

Step-by-step explanation:

Using the addition formula for cosine

cos(x - y) = cosxcosy + sinxsiny

and the exact values

sin45° = cos45° =

cos60° = , sin60° =

Note that

cos(- 75)° = cos(45 - 120)°, thus

cos(45 - 120)°

= cos45° cos120° + sin45° sin120°

= cos45° ( - cos60°) + sin45° sin60°

= × - + ×

= - +

= ×

=

= ( )

or

step-by-step explanation:

we need to solve for 'x' the following equation:  

we have that:

⇒

⇒ ⇒

factorizing, we get that:

  ⇒

then, the solutions are: or

nothing extraneous has been found!

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240