"The only way to be totally free is through education" - Jose Marti
Study Journal Notes
Trigonometry:
Lesson 1:
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Know these definitions: Standard position, coterminal angle, principal angle, reference angle, terminal arm
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Positive angles rotate counter clockwise from the positive x-axis, negative angles rotate clockwise from the positive x-axis
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In order to find coterminal angles, you can use this formula: principal angle + n(360), where n is any integer
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The primary trig ratios are sin, cos, and tan (sine, cosine, and tangent)
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The reciprocal trig ratios are csc, sec, and cot (cosecant, secant, and cotangent)
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You can make a triangle from any point on a terminal arm to the x-axis, with the x-axis as the base of the triangle, then use the trig ratios to solve for the reference angle.
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EACH TRIG RATIO IS A RATIO - that means you can write your answer in EXACT terms as a fraction.
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The CAST system tells you which trig ratios will be positive or negative based on the quadrant that the terminal arm is in.
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When solving a problem, follow these steps:
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draw a picture of the rotation and terminal arm;
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turn your picture into a triangle with the x-axis as the base of the triangle;
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use the Pythagorean Theorem to fill in the values for any missing sides (use exact numbers, no decimals allowed);
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use SOH CAH TOA to guide you in solving the trig ratios for the given angle;
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use the CAST system to determine which trig ratios are positive and negative.
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Lesson 2:
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Know how to convert from radians to degrees and vice-versa
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Know how the four quadrants of the cartesian plane relate to the coordinates of any point
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Know the CAST system for positive/negative trig values in a given coordinate
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Know the right-triangle ratios for the trig identities:
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sin(a) = y/r
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cos(a) = x/r
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tan(a) = y/x
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csc(a) =r/y
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sec(a) = r/x
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cot(a) = x/y
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Know how to find the arc length given an angle of rotation and the radius of the circle
Lesson 3:
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Know how the Pythagorean Theorem relates to the unit circle
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Know the right triangle diagrams for the 3 basic angles (30 degrees, 45 degrees, 60 degrees)
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Know how to write the six trigonometric identities in terms of x, y, and r (see lesson 2)
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Know how to find the quadrant given an angle or coordinates of a point and how to draw the right and gle triangle diagram for any point or given angle
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Know how to identify the coordinates on the exterior of the unit circle as symmetric reflections of the first quadrant (you can use your three basic angles/triangle diagrams to calculate all of the standard angles in the unit circle and find their coordinates).
Lesson 4:
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Steps to solving for an angle for a given equation
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check the domain: are you working in radians or degrees?
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Solve the equation algebraically so that you have one of the six trig identities isolated (sine, cos, tan, csc, sec, cot)
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Put your equation into the form sin(a) = b, cos(a) = b, or tan(a) = b; where a is the mystery angle you are solving for and b is everything else outside of the trig identity (b will be some number, maybe a fraction)
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Use CAST to identify which quadrant you can be in
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Solve for the reference angle (if it is a known identity then use one of the three special triangles, if it is an unknown identity then find the inverse trig ratio of b when b is positive)
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Use the reference angle to solve for the angle in the given quadrants across the given domain.
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Lesson 5:
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Know the general shape of the three primary trig functions
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sin(x) "starts" or has a y-intercept at 0 (since the y-value at 0 on the unit circle is 0)
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cos(x) "starts" or has a y-intercept at 1 (since the x-value at 0 on the unit circle is 1)
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tan(x) "starts" or has a y-intercept at 0 (since sin(x)/cos(x) = 0/1 = 0 for the location 0 on the unit circle)
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Know the descriptive terms for sinusoidal graphs:
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period - horizontal length that it takes for a periodic function to repeat itself
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amplitude - the distance from the median line to the minimum or maximum value of a periodic function
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median line - the average of the maximum and minimum values
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phase shift -the horizontal translation away from the y-axis
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Lesson 6:
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Know the amplitude, intercepts, domain, and range for the three primary trigonometric functions
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Make a connection between the transformations unit and the formula asin(b(x-c))+d
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a = amplitude and reflections
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b = factor of period (2pi/b = period for sin and cos; pi/b = period for tan)
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c = horizontal shift (phase shift)
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d = vertical shift
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know that cosine and sine follow the same transformation rules, but have a different phase shift
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know that the period for tan(x) is different than the period of cos(x) or sin(x)
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graph maximums and minimums (the range of the graph) is affected by the vertical stretch (amplitude) and the vertical shift
Lesson 7:
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know the information in the boxes on page 20 of your notepack:
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for sin/cos:
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amplitude = (max-min)/2
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period = 2pi/b or 360/b
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phase shift is right for c>0, left for c<0
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vertical shift = up for d>0, down for d<0, where d = (max+min)/2
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for tan:
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amplitude = not applicable
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period = pi/b or 180/b
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phase shift is right for c>0, left for c<0
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vertical shift = up for d>0, down for d<0, where d = (max+min)/2
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Know where the standard graphs "start" to calculate the phase shift (see Lesson 5).
Lesson 8:
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Know the numeric values of the standard radian measures in case you are given a graph where the x-axis is numeric rather than pi-based, for example:
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pi/2 = ~1.57
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pi = ~3.14
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2pi = ~6.28
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recognize the context of a question, and use it to guide your substitutions
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for example, if an equation is given as d(t) = ... and you are given a value for t, substitute t in the equation. If you are given a value for distance, substitute d(t) in the equation.
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Translate the circular or cyclic nature of a situation to a cosine or sine graph.
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if the situation is circular, the radius is the amplitude, and one full revolution relates to the period
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if the situation is cyclical, one full cycle relates to the period
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