A chemical reaction is endothermic if it absorbs heat from its surroundings. It is exothermic if it gives off heat to its surroundings. In her chemistry class, Lily is given a salt (ammonium nitrate) and a beaker of water at 60 ∘F. She dissolves the salt in the water and sees that the temperature of the solution is now 50 ∘F. Did the temperature of the solution go up or go down? By how many degrees? What kind of reaction has occurred?

Prove (a) cosh2(x) − sinh2(x) = 1 and (b) 1 − tanh 2(x) = sech 2(x). solution (a) cosh2(x) − sinh2(x) = ex + e−x 2 2 − 2 = e2x + 2 + e−2x 4 − = 4 = . (b) we start with the identity proved in part (a): cosh2(x) − sinh2(x) = 1. if we divide both sides by cosh2(x), we get 1 − sinh2(x) cosh2(x) = 1 or 1 − tanh 2(x) = .

What is the value of the expression i 0 × i 1 × i 2 × i 3 × i 4? 1 –1 i –i

Use the following statements to write a compound statement for the conjunction or disjunction. then find its truth value. p: an isosceles triangle has two congruent sides. q: a right angle measures 90° r: four points are always coplanar. s: a decagon has 12 sides. a) a right angle measures 90° or an isosceles triangle has two congruent sides; true. b) a right angle measures 90° and an isosceles triangle has two congruent sides; true. c) a right angle measures 90° and an isosceles triangle does not have two congruent sides; false. d) a right angle measures 90° or an isosceles triangle does not have two congruent sides; true.