Exam #2 Review concepts
#1 Use a calculator to evaluate the inverse trig function of a coordinate in decimal form
- check whether the answer should be in degrees or radians - set your calculator to degree or radian mode respectively - input the decimal or fraction value coordinate into the inverse trig function requested
#5 if x = 6sinθ , write the expression in terms of x tan 2θ
- divide by 6 to get the value of what sinθ is equal to 6sin theta = x sinθ= x/6 write tan 2θ in terms of x - use a triangle to find any other needed values or x, y, r - use the double angle formula, or a different formula if a sum and difference, or half-angle - input the x , y, r ratio values into the formula and complete the algebra - make sure to multiply properly if finding the LCD - Not finding the LCD to cancel out the original denominator right away
#2 Evaluate without a calculator ( the inverse trig function of a trig function of an angle measure in radians or degrees) ex: sin^-1(sin 240 degrees)
- sketch the angle in standard form (counterclockwise) - find the reference angle, or theta hat - find the corresponding angle measure within the range of the inverse trig function sin^-1(sin 240 degrees) sin^-1( coodrinate - rad3/2) range of sin^-1 is [-90 degrees, 90 degrees] Matching angle is -60 degrees (the angle with the same sin coordinate of 240 degrees, within the inverse sine Range) Domain of sin^-1 [-1,1] Range of sin^-1 [ -90 degrees, 90 degrees] Domain of cos^-1 [-1,1] Range of cos^-1 [ 0 , pi ] Domain of tan^-1 (-infinity, infinity) Range of tan^-1 ( - 90 degrees, 90 degrees )
#4 Find the exact value of a given non-standard angle (not directly on the unit circle)
- use a sum and difference, double angle, or half angle identity to expand with standard unit circle measure angles or radian measure angles Example: cos 7pi/12 = cos (3pi/12 + 4pi/12) = cos ( pi/4 + pi/3) - matches the sum identity and is within standard unit circle measures Next: Use the coordinate value inputted into the identity formula Example: cos pi/4 = rad2/2 or 1/rad2 it is the x-coordinate for pi/4
#6 Given sec B = -7/3 ; find cos2B; or another sum, difference, double angle, or half angle
- use sec B as the r/x value - if the y value is needed, use a triangle for x^2 + y^2 = r^2 with r always positive - to find cos2B find the reciprocal of sec B = -7/3 to find the x/r value x/r = -3/7 Use a double angle formula of cos2B - easiest is the one with only cos in it -input (-3/7)^2 as the cos^2 value and evaluate the rest of the sequence
#9-13 Odd and Even functions part 1
Cosine is an even function Cos(x) = cos (-x) And there is a reflection when we have: -cos(x) or -cos(-x) [ not when we have cos(-x)] And I am almost certain that is correct
#14 -16 Proofs: use Chapter 5 Identities plus Pythagorean, reciprocal, and ratio identities as needed
Less obvious Pythagorean identities to remember: cot² +1 = csc² tan² + 1 = sec² Sin = +- rad 1-cos^2 Cos = +- rad 1-sin^2 Can be deduced from: Sin^2 + cos^ = 1 ALSO REMEMBER: You can use the double angle identity divided by 2, or the half angle identity multiplied by 2 to deduce another proof. Assign proper factors to each; continuation on next notecard
#14-16 Proofs and deducing and using other proofs from the chapter 5 identities
Sin2A = 2cos sin Can be divided by 2 Cos2A and Tan2A can be similarly divided by 2 to create a different proof of needed SinA/2 and the other half angle identities can be multiplied by 2 to get other proofs (similar to the processes Professor used on the Exam Review and handouts in some instances)
#9-13 Odd and Even functions part 2
Sine is an odd function: -sin(x) = sin (-x) And there is a reflection when we have: -sin(x) or sin(-x)
#2 sin^-1, cos^-1, tan^-1 of an Exact Value
applied to an Exact Value to find the angle or radian measure
#2 Sin, cos, tan of an Angle or Radian measure
applied to an angle or radian measure to find the Exact Value or coordinate
#7b Graph y = sec x
parabola at each critical point of the graph of y = cos x - vertex connected to peak and valley parabolas open with respect to the peaks and valleys of the graph at the peaks, the parabola opens up; at the valleys the parabola opens down Asymptotes of y = sec x are at the zeroes of y = cos x -starts at pi/2 units positive and negative - then 3pi/2 units positive and negative
#7c Graph y = csc x
parabola at each critical point of the graph of y = sin x - vertex connected to peak and valley parabolas open with respect to the peaks and valleys of the graph at the peaks, the parabola opens up; at the valleys the parabola opens down Asymptotes of y = csc x are at the zeroes of y = sin x - Starts at pi units positive and negative - then 2pi units positive and negative
#7a Graph y= tan x
use table of exact values + asymptotes Coordinates are (1,1) , (-1,-1) ,(0,0) and repeat the pattern in between asymptotes Asymptotes start at pi/2 units positive and negative direction Then 3pi/2 positive and negative direction
#3 Write the expression as a single trigonometric expression, and simplify
use the sum and difference, double angle, or half angle formula to combine the expression into one statement Example: cos3x^2 cos8x^2 - sin8x^2 sin3x^2 same as: cosA cosB - sinA sin B same as: cos(A + B) = cos (3x^2 + 8x^2) = cos(11x^2) Pay attention to order
#8 Period of trig functions
y = sin x: Period: 2π y = cos x Period: 2π y = tan x Period: π
