Trigonometric equation and graphing (Test)

• angles in standard position
  • the arrows in initial and terminal arms are optional
  • rotation angles (greater than 360$\mathrm{°}$) would need extra swirls when sketching
• coterminal, reference (positive acute angle), quadrantal angles: $\frac{n\pi }{2}$, $n\in Z$
  • Coterminal angles - angles in standard positions (if the vertex of the angle is at the origin of the x-y plane and its initial arm lies along the positive x-axis) with the same terminal arm
  • Positive angle sketched counterclockwise, negative sketched clockwise
  • Principal angle (0, $360\mathrm{°}$) - smallest positive coterminal angle
  • Reference angle (0, $90\mathrm{°}$) - positive acute angle formed with the terminal arm of $\theta$ and the x-axis
  • Quadrantal angle - any angle whose terminal arm is on either the x- or y-axis
• evaluating trig ratios and finding unknown angles
  • Primary trigonometric ratio
    • SOH: $sin\theta$ = opposite/hypotenuse
      • $csc\theta$ = 1/$sin\theta$
    • CAH: $cos\theta$ = adjacent/hypotenuse
      • $sec\theta$ = 1/$cos\theta$
    • TOA: $tan\theta$ = opposite/adjacent
      • $cot\theta$ = 1/$tan\theta$
  • when calculating the ratios
    • put the negative signs in front of the entire fraction
    • keep the x, y, r, all positive when calculating the reference angles, and then later add the negative sign in brackets if needed
    • if using the scientific calculator, make sure the calculator is in the degree mode and align the manual calculation
  • beware of the range of $\theta$ in the question or if it's a general solution, in radian or degree etc.
  • beware of the domain of $\theta$ it's not necessarily [0, $360\mathrm{°}$)
  • ex. $sin\theta$ = 0.642 - $si{n}^{-1}(0.642)$
  • note radian/degree units in the final answer
  • CAST system: acronym denoting which quadrants the ratios among sin, cos, tan are are (is) positive, in the order of Quadrant 4,1,2,3
• arc length (a=r$\theta$ in radians) sector area (A=$\frac{r^2\theta }{2}$ in radians)
• graphing
  • all roots on the graph: the x-coordinates of all x-int points, note to only add the ones on the graph
  • Desmos graphing online can be used to check the shape
  • Amplitude cannot be negative = $\frac{max-min}{2}$ = |a|
  • Vertical displacement = $\frac{max+min}{2}$ - special name for the vertical translation of a sinusoidal function
  • Phase shift - special name for the horizontal translation of a sinusoidal function; + right, - left
  • Shift & displacement both needed to be conducted at the last as in function translation when graphing
• problem solving
  • if not marked specifically, 1 unit on the square means 1
  • Sinusoidal function graphing, don't come up too steep
  • read the question clearly, like number of periods etc. requirement for sketching
  • include a conclusion statement
  • only round in the final answer
  • when converting, don't forget to add the $\mathrm{°}$
  • beware of the Calculator mode: degree - radian
  • $\theta$ in radian = $\theta \times \frac{\pi }{180\mathrm{°}}$ in degree

Identities (Test)

• Test: Monday 04.17
  • Proofs (2)
  • Simplifying (1)
  • Solving (2)
  • Find an exact value of a trig ratio (2)
• reciprocal trig ratios:
  • $cot\theta =\frac{1}{tan\theta }$
  • $sec\theta =\frac{1}{cos\theta }$
  • $csc\theta =\frac{1}{sin\theta }$
• quotient identities
  • $tan\theta =\frac{sin\theta }{cos\theta }$
• Pythagorean identities
  • $si{n}^2\theta +co{s}^2\theta =1$
  • $1+ta{n}^2\theta =se{c}^2\theta$
  • $co{t}^2\theta +1=cs{c}^2\theta$
• addition identities
  • $cos(A+B)=cosAcosB-sinAsinB$
  • $sin(A+B)=sinAcosB+cosAsinB$
  • $tan(A+B)=\frac{tanA+tanB}{1-tanAtanB}$
• Strategies to simplify a trig expression
  • get a common denominator
  • factor
  • change terms to sine or cosine
  • multiply the numeral and denominator both with a conjugate binomial
    • $5+\sqrt{2}$ conjugate binominal would be $5-\sqrt{2}$
• double angle
  • $sin(2x)+1=(cos(x)+sin(y){)}^2$
  • $cos(2x)=co{s}^2(x)-si{n}^2(x)=2co{s}^2(x)-1=1-2co{s}^2(x)$
• when combining restrictions, beware not to miss any sub-restrictions by testing with a full rotation
• when proving identities, start simplifying with the side identified to be more complicated
• LCD = lowest common denominator
• QED = Latin, meaning which was to be demonstrated
• double cross for fractions equations

Others

• Angles in a Circle
  • Central angle is twice any inscribed angle subtended by the same arc
  • Inscribed angles subtended by the same arc are equal - angles in the same segment theorem: angles in the same segment are equal
  • Angles subtended by the diameter is $90\mathrm{°}$
  • Cyclic quadrilateral: the opposite angles in a cycle
  • 
• Use height of the triangle to prove
• Law of sines: $\frac{a}{\sin (\alpha )}=\frac{b}{\sin (\beta )}=\frac{c}{\sin (\gamma )}=2R$
  • R is the radius of the triangle's circumcircle
• Law of cosines: $c^2=a^2+b^2-2ab\cos (\gamma )$
• Others
  • $si{n}^2\theta +co{s}^2\theta =1$
  • $sin(\alpha +\beta )=sin\alpha cos\beta +cos\alpha sin\beta$
  • $cos(\alpha +\beta )=cos\alpha cos\beta -sin\alpha sin\beta$
  • $tan(\alpha +\beta )=\frac{tan\alpha +tan\beta }{1-tan\alpha tan\beta }$
• angular measure
  • trigonometric ratio
  • solving trigonometric equations
  • algebraic method using calculator
  • ambiguous case (SSA)