Preparing for University Calculus - Math and Computer Science

3.12 Geometry and basic trigonometryCalculus does not use very advanced geometry, but you should be thoroughly familiarwith similar triangles, Pythagoras’ theorem, and parallel lines; and, fromanalytic geometry, the midpoint and distance formulae, and the “negative reciprocal”rule. Trigonometry is important in various branches of science, but especiallyin mathematics, physics, and engineering.You should be able to convert between degrees and radians (180 ◦ = π radians).You should know the definitions of the trig functions, and be able to use them tofind sides and angles of triangles. You should know and be able to use the sine andcosine laws for triangles.Most angles do not have trig functions that are easy to give as exact expressionsrather than decimal approximations (and you will not be expected to do so), butyou should know the trig functions of a few common angles, such as 0 ◦ ,30 ◦ ,45 ◦ ,60 ◦ ,and 90 ◦ . You should also know how to find the trig functions of angles outsidethe range [0 ◦ , 90 ◦ ] in terms of trig functions of angles in that range.You should also be familiar with the inverse trigonometric functions. Note thatalthough (for instance) sin 2 (x) means (sin(x)) 2 , and sin(x) −1 means 1/ sin(x), whichis csc(x), sin −1 (x) means arcsin(x).Examples:1. If the hypotenuse of a right triangle is 10, and one leg has length 8, how longis the other leg?Solution: Let the length of the unknown leg be x. By Pythagoras’ theorem,x 2 +8 2 =10 2 ,sox 2 = 100 − 64 = 36 and x =6.2. Find the midpoint of the segment from the point (2, 3) to (8, −3).Solution: ( By the midpoint formula, the coordinates of the midpoint are2+82 , 3+−3 )which simplifies to (5, 0).23. If two sides of a triangle have length 1 and 2, and the angle between them is45 ◦ , what is the length x of the remaining side?Solution: By the cosine law, x 2 =1 2 +2 2 −2(1)(2) cos(45 ◦ )=5−4( √ 2/2) =5 − 2 √ 2; and therefore x = √ 5 − 2 √ 2.We cannot simplify this further so we leave it in this form.32