Trig Study Sheet

Trigonometry test tomorrow! I've promised to get a study sheet up before 7PM, at which time I have to be back at school for the "Grade 8 IB Info Night". More on that tomorrow, here's the study sheet. I am following the order in our textbook, Mathematics for the international student: Mathematics SL, not necessarily the order we did in class. Anything discussed in class that isn't in chapters 10-13 of the book, I've added at the end of the sheet.

Chapter 10: Right-Angle Trigonometry

Pythagoras

• a²+b²=c² (where c is the hypotenuse)
• this deals with sides, not angles
• often used twice in 3D problems (eg. diagonal of a box)

"soh-cah-toa"

• sin A = Opposite/Hypotenuse (soh)
• cos A = Adjacent/Hypotenuse (cah)
• tan A = Opposite/Adjacent (toa)

Solving (sides, angles, chords, and tangents)

• Isosceles Triangles: drop an altitude from the base so that you have two identical right-angled triangles
• Chords (right) can be made into isosceles triangles by subtending it from the centre (connect radii to both ends for the chord)
• "A chord subtends an angle of 112 degrees at it's centre" means that angle AOB is 112 degrees
• A radius can be drawn from the point a tangent touches the circle to form a right angle
• Remeber that you can find right angles in other figures too, usually by drawing altitudes or diagonals
• Angle of elevation is the angle formed between the horizontal and the up-ward line, angle of depression is the horizontal and the down-ward line.

Slope of a line (the dotted lines are the axes)

• slope x = tan a
• slope x = rise/run
• tan a = opposite/adjacent
• (drop a perpendicular from line x to the axis, this forms a right-angled triangle)

The two most important triangles ever

Chapter 11: The Unit Circle

You only really need to know the first quarter of the Unit Circle, and the transformations associated with the other quadrants.

• for any point on the circle: P(cos a, sin a) when angle a is measured from the positive x-axis
• the equation of this circle is x² + y² = 1 or
  cos² a + sin² a = 1

CAST Rule
S A All trig ratios are positive in
TC quadrant A, only sin is positive in quadrant
S, etc.
We know they are negative because of the coordinates for each angle.

Corresponding angles
quad A ~ quad S ~ quad T ~ quad C
sin x = sin (180-x) = - sin (180+x) = - sin (360-x)
cos x = - cos (180-x) = - cos (180+x) = cos (360-x)
tan x = - tan (180-x) = tan (180+x) = - tan (360-x)

Chapter 12: Non-Right Angled Triangle Trigonometry

• Area of Triangle = (1/2)ab sin C [where cappital letters are angles and lowercase letters are their opposite sides]
• Area of Triangle = root of [s(s-a)(s-b)(s-c)] where s= (a+b+c)/2

• length of an arc = (θ/360)2πr
• area of a sector = (θ/360)πr²

• Cosine Rule: a²=b²+c²-2bc cos A
• also: cos A = (b² + c² -a²)/(2bc)
• Sine Rule:
  sin A = sin B = sin C
  a b c

Chapter 13: Periodic Phenomena

• Periodic function: f(x) = f(x+p)
• "over some interval of x, the y is repeating"
• p = period
• A = amplitude (half dist between max and min
• principal axis = the line between max and min

Radians

• (θ degrees)(π/180) = radian measure
  (θ radians)(180/π) = degree measure
• length of arc = rθ
  Area=(1/2)r²θ
• A radian is the length of an arc that is equal to the radius of the circle.
• since the formula for the circumferance of a circle is 2πr, when we use the unit circle (radius of 1) we get a circumferance of 2π.
• 2π is the distance once around the circle (360 degrees)
• π is the distance travelled around half the circle (180 degrees)
• π/2 is a quarter distance around the circle (90 degrees)
• π/3 is the length traveled when the angle that subtends that arc is 60 degrees
• π/4 is the length traveled when the angle that subtends that arc is 45 degrees
• π/6 is the length traveled when the angle that subtends that arc is 30 degrees

General Sinusodal Functions

• f(x) = A sin [B(x-C)]+D (works for both sin and cos)
• period= (2π)/B
• Transformations
  • A is the amplitude or vertical stretch, when negative it is a reflection in the x-axis
  • B helps to find the period (see above), it is the horizontal compression, if negative it is a reflection in the y-axis
  • C in the horizontal shift or "phase shift"
  • D is the principal axis or vertical shift
• Graphing a wavelength
  sin: o-max-0-min-0
  cos: max-0-min-0-max

Trig Modeling

• Examples: tides, monthly temperature, Feris wheels
• identify the max and min
• find the principal axis (D) and Amplitude (A)
• find the period and B value
• decide whether to use a sine of cosine graph
• make the appropriate phase shift (eg. if high tide is at noon and you have to use a sine function, shift so that noon is on your y-axis to make things easy to work with)

Simplifying/Proving Expresssions

• make the left side equal the right side
• do NOT "multiply both sides by4" or move anything over
• use trig identities to simplify to one variable (all sin x instead of both sin x and cos x)

Identities

Quotient identities

• tan θ = (sin θ)/(cos θ)
• cot θ = (co θ)/(sin θ)

Reciprocal identities

• csc θ = 1/(sin θ)
• sec θ = 1/(cos θ)
• cot θ = 1/(tan θ)

Pythagorian identities

• sin² θ + cos² θ =1
• if you divide the above by sin²θ, you get: 1 + cot² θ= csc² θ
• if you divide the above by cos² θ, you get: tan² θ + 1 = sec² θ

All of the identities are things we have learned before. We learned what tan was in Grade 9 (or 10). The reciprocal identities are just reciprocals, the only new thing is the names (secant is the reciprocal of cosine, cosecant is the reciprocal of sine, and cotan is the reciprocal of tan). We learned the basic pythagorian identity ages ago, and the others are just different versions of it by division and application of the other identity names. When solving the identity questions, just

Addition Formulas

• sin (a+b) = sin(a)cos(b) + sin(b)cos(a)
  sin (a-b) = sin(a)cos(b) - sin(b)cos(a)
  sin (2a) = 2sin(a)cos(a) = sin (a+a)
• cos (a+b) = cos(a)cos(b) - sin(a)sin(b)
  cos (a-b) = cos(a)cos(b) + sin(a)sin(b)
  cos (2a) = cos²(a) - sin²(a) = cos (a+a) =1-2sin²(a) =2cos²(a)-1
• the final two in the line above used the pythagorean identity to replace one of the values so as to simplify

There are only really two formulas, just different instances of them. The first is sin(a+b) and the second is cos(a+b). The italicised ones are just the simplified form of sin or cos of (a+a). Remember that to fully simplify and solve you should be using the Trig Identities to replace things you notice. for example, if you see cos²(a), recall the pythagorian identity of cos²(a) + sin²(a) = 1. You can rearrange this to cos²(a) = 1 - sin²(a) and then replace cos²(a) with it. This may help you to reduce the equation since you only need to solve for sin(a) now!

I hope this has helped! I've already received some positive feedback, and thank you very much for stopping by.

Good luck,
Ace