‘ln’ is “natural log” which is the SAME thing as log e x. ‘e’ is a constant with a value 2.718. It has special meaning because the slope of a tangent line of a function of ‘e’ is the function itself. (Dont worry if you don’t understand this bit) DON’T GET CONFUSED BY THE TERM ‘ln’. It just means log base ‘e’. You don’t need to memorize the value of ‘e’. You can just express your answer in terms of ‘e’!!
Circle of Radius 1 Constructed from pythagoream’s theorem Angles correspond to points along the circle: these define our values for cosine and sine
Trigonometric Functions From the unit circle we can deduce: sin(x) = y/r cos(x) = x/r tan(x) = y/x …and of course we all know that x 2 + y 2 = r 2
Example Problems with Right Triangle: Given that cos (θ) = − 3/ 4 and that sin (θ) > 0, ﬁnd tan (θ) and express your answer in simpliﬁed form (without trigonometric functions). For some real number x, it is known that sin(x) = ¼ and cos(x) > 0. The value of cos(x) can be written as √α / 4. What is α?
Common Trig Values to Memorize: Note how sin is ascending and cos is descending If you have these down, then the tan is simple sin/cos! Add npi to go to different quadrants on the coordinate axes All Students (sin is positive in 2nd) Take (tan is positive in 3rd) Calculus (cos is positive in 4 th )
Inverse Trig Functions The expression sin(x) = ½ reads “the sine of x is one half”. The expression sin -1 (1/2) = ? is the opposite of the above expression. It can also be interpreted as: “The sine of some angle is ½, what is the angle?”
Example Problem Evaluate: tan(sin -1 (1/√2)) tan(cot -1 (1/3))