Transcript Lecture 2
Basic Trigonometry Review for Engineers PLUS! Solving systems of equations 26-Mar-16 Instant Trig • Trigonometry is math, so many people find it scary • It’s usually taught in a one-semester highschool course • However, 95% of all the “trig” needed to solve cool problems can be covered in 15 minutes Laws of Trig you will Master • Sum of internal angles = 180 (all triangles) • Pythagorean theorem: a2+b2=c2 (right triangles) • Definition of sin, cos, tan, arcsin, arccos, arctan (right triangles) • Law of Cosines (all triangles) • Law of Sines (all triangles) Angles add to 180° • The angles of a triangle always add up to 180° 20° 44° 68° 44° 68° + 68° 180° 68° 30° 120° 20° 30° + 130° 180° Right triangles • Right triangles are a special case in trig opposite – A right triangle is one in which one of the angles is 90° – Here’s a right triangle: Here’s the angle we are looking at Here’s the right angle adjacent • We call the longest side the hypotenuse • We pick one of the other angles--not the right angle • We name the other two sides relative to that angle The Pythagorean Theorem • If you square the length of the two shorter sides and add them, you get the square of the length of the hypotenuse • adj2 + opp2 = hyp2 • 32 + 42 = 52, or 9 + 16 = 25 • hyp = sqrt(adj2 + opp2) • 5 = sqrt(9 + 16) Defining sin, cos, tan • also, a = c cos(q) , b = c sin(q) q = arccos(a/ c) = arcsin(b/c) = arctan(b/a) Example The length of the shadow of a tree 22.02 m tall is 28.34 m. Find the angle of elevation of the sun. Draw a sketch. 22.02 m B The angle of elevation of the sun is 37.85. Rev.S08 8 28.34 m http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Example 2 The length of the shadow of a tree is 100 ft, and the angle of elevation of the sun is 60 degrees. Find the height of the tree Draw a sketch. The angle of elevation of the sun is 37.85. Rev.S08 9 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Law of Cosines • Use when you know an angle and two adacent sides to find the 3rd side • Use when you know three sides and want to find an angle Find x Law of Sines • Use when you know an angle and the opposite side and want to find another angle or side: Find b and unknown side Example of a narrative statement of a system of the equations: The air-mail rate for letters to Europe is 45 cents per half-ounce and to Africa is 65 cents per half-ounce. If Shirley paid $18.55 to send 35 halfounce letters abroad, how many did she send to Africa? Example of an algebraic statement of the same system of the equations: A system of linear equations can be solved four different ways Substitution Gaussian Elimination Matrices Graphing and, #5 ... on the computer with Matlab Example 1: A total of $12,000 is invested in two funds paying 9% and 11% simple interest. If the yearly interest is $1,180, how much of the $12,000 is invested at each rate? Before you work this problem, you must know the definition of simple interest. Simple interest can be calculated by multiplying the amount invested at the interest rate.