Pre-Calc Review
SOHCAHTOA
A helpful reminder for the Trig ratios. Sine is the ratio of Opposite to Hypotenuse values, Cosine is the ratio of adjacent to Hypotenuse values, and Tangent is the ratio of Opposite to Adjacent values.
Coterminal angles
Angles with the same initial and terminal sides. Examples are 30 degrees and 390 degrees, π radians and 3π radians, and 100 degrees and 820 degrees. To find the simplest form, subtract 360 degrees/ 2π radians from the angle until the angle is between 0 and 360 degrees/ 0 and 2π radians.
Trig Form of a complex number and shortcuts for * and /
The trig form or polar form of a complex number where a= r* cosθ and b=r*sinθ is (rCosθ ) + (rSinθ )i or r(Cosθ +Sinθi ). Tanθ = b/a T find θ simply do inverse Tan of b/a. R is the radius. The equation to multiply complex numbers is if Z₁= R₁ (Cosθ ₁ +iSinθ ₁ ) and Z₂= R₂( Cosθ ₂ +iSinθ ₂) then: Z₁*Z₂=R₁*R₂ (Cos(θ ₁+θ ₂) +iSin(θ ₁+θ ₂) To divide Z₁ by Z₂, multiply R₁/R₂ by (Cos(θ ₁-θ ₂) + iSin(θ ₁-θ ₂)
Unit Circle
The unit circle is a circle with a radius of one unit. An angle placed on the unit circle now has a location: the coordinate plane. Commonly known angles can be used to determine the corresponding ordered pairs.
Examples of solving equations using identities
2sinx-1=0 2sinx=1 sinx= 1/2 x= π/6 and 5π/6 +2πn Sinx + √2= -sinx -2sinx= √2 sinx= -√2/2 x=5π/4 and 7π/4
Pythagorean Identities
(Sinθ)^2 + (cosθ)^2=1 1 + (tanθ)^2= (secθ)^2 1+ (cotθ)^2= (secθ)^2 Proof: If x is an angle of a triangle with its adjacent side being "a", opposite side being "b", and Hypotenuse being "c", (Sinθ)^2 + (Cosθ)^2= (a²+b²)/c²
Examples of verifying
(Tanθ )^2 /(Cosθ )^2= (sinθ )^2/(Cosθ)^2 *(Cosθ )^2= sinθ ^2 (Secθ )^2-1/(Secθ)^2= (Sinθ)^2 = (Tanθ )^2 / (Secθ )^2 = (Sinθ )^2/(Cosθ)^2 *1/(Cosθ )^2 = (Sinθ )^2
Degrees and Radians
A degree is one 360th of a revolution around a circle. A radian is equal to the length of the radius of a circle, and is 1/2π of a revolution. Since 360 degrees equals 2π radians, π radians equals 180 degrees. To convert from degrees to radians, multiply the number of degrees by π/180. To convert from radians to degrees, multiply the number of radians by 180/π. For example: 90 degrees= π/2 radians, 180 degrees= π radians, 270 degrees= 3π/2 radians, and 360 degrees= π radians
Sum and Difference Formulas
U and V are angle values Sin(u+v)= SinuCosv + SinvCosu Sin(U-V)= SinuCosv -SinvCosu Cos(U+V)= CosUCosV-SinUSinV Cos(U-v)= CosuCosv+ SinuSinv Tan(u+v)= (Tanu+tanv)/1-tanutanv Tan(U-V)= (Tanu-Tanv)/ 1+ TanutanV
Law of Sines
a/Sina=b/Sinb=c/Sinc Sina/a=Sinb/b=Sinc/c
Rules for Verifying
Do not touch the right side of the equation. Look for factoring, adding fractions, square binomials, and common denominators. Look for ways to substitute identities. Last resort, convert everything to sines and cosines.
How to use a calculator to evaluate functions.
First, put the calculator into either radians or degrees depending on the desired result. For degrees/ radians, press the trig function and then enter the degree/radian value. If given a trig ratio, use the 2nd key to put the function into inverse mode and then enter the trig ratio.
Double Angle Formulas
Sin2u= 2SinuCosu Sin420=2sin210cos210 Cos2u= (Cosu)^2 -(Sinu)^2 or 1-2(sinu)^2 or 2(Cosu)^2-1 Tan2u= 2tanu/1-(tanu)^2
Six trig functions of a point not on the unit circle
If (X,Y) is a point not on the circle, then the following is true, Cosθ =x/r, sinθ = Y/r, Tanθ = Y/X, Secθ = r/X, CSCθ = r/Y, and Cotθ = X/Y.
Demoivre's Theorm
If a complex number if in trig form, use this equation to raise it to a power: Z^N = R^N * (Cos(Nθ) +iSin(Nθ )
Inverse Trig Functions
If you have the Trig identity value and want the corresponding angle value you should use the necessary inverse trig function. These are found by clicking "2nd" and then the Trig identity button of the inverse trig identity which is desired. For example, if sinθ is 1, sin-1= π/2.
Examples
Sinθ *Cotθ = Sinθ *(Cosθ /sinθ) =Cosθ Tanθ *Cosθ **Cotθ = (Sinθ /Cosθ )* Cosθ * (Cosθ /Sinθ)= Cosθ (Tanθ )^2 *Cotθ = Tanθ
Fundamental Trig Indentities
Sinθ = 1/cscθ Cosθ = 1/secθ Tanθ = 1/cotθ and sinθ /cosθ Cscθ = 1/sinθ secθ = 1/cosθ Cotθ = 1/Tanθ and Cosθ /Sinθ
Identities
Tangent is the ratio of the opposite side to the adjacent side and of Sinθ/Cosθ. Cotangent is the ratio of 1/tanθ. Cosecant is the ratio of 1/Sinθ. Secant is the ratio of 1/cosθ.
Six Trig functions
The Six Trig Functions are Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent.
Absolute Value of a complex Number
The absolute value of a complex number a +bi is equal to the square root of the quantity a²+b²
Linear Speed
The distance traveled over the time that passed. Expressed as s/t or arc length/time or Radius*θ/time.
Arc Length
The length of an arc of a circle. It is equal to the radius of a circle times the central angle in radians. Denoted by "s".
Angular Speed
The measurement of how fast an angle changes over time. Expressed as θ/t. Can be converted to linear speed by multiplying by the radius of the angle.
Law of Cosines
a²=b²+c²-2bcCos(A) b²= a²+c²-2acCos(B) c²=a²+b²-2abCos(C)
Half-Angle Formulas
refer to image
Graphing Angles
Angles are graphed with the initial side on the x-axis and the terminal side graphed counter-clockwise the amount of degrees/radians of the angle from the initial side. An angle can be measured as the amount of rotation around the unit circle. The x axis heading right is "0" degrees/ radians. The y axis heading up is 90/.5π. The X axis heading left is 180/π. The y axis heading down is 270/1.5π.
Graphing
The domain, or input values, are angle values. The range, or output values, are sin and cos values. The domain restrictions are - ∞ to ∞. The range restrictions are -1 to 1. The first quadrant is +,+, the second is -,+, the third is -,-, and the fourth is +,-. For graphing Sine and cosine, y= aSin(bx+c) +d y=aCos(bx+c)+d a alters the amplitude or height of the wave, a negative a reflects the graph across the x axis, the frequency of the wave is b, a negative b reflects the graph across the y axis, c causes a horizontal translation, d causes a vertical translation. Steps to graph: 1. Determine amplitude (a). 2. bx+c=0 is start of first wave. bx+c=2π is end point. Solve for x 3. Determine vertical translation 4. Graph Function a. Wave b. label starting point and end of first wave c.Determine the midpoints d. Draw Axes For tan and cot, 1. determine a 2. Determine asymptotes. For tan, set bx+c= -π/2 and bx+c= π/2. For cot, set bx+c=0 and bx+c=π. 3. 1.Axes 2. asymptotes. 3. 2 full waves. 4. 6 Key points. For Sec and Csc, 1. Graph the corresponding sin/cos function. 2. Mark undefined values with asymptotes. 3 Sketch Sec/csc portion as parabolas.
vectors
The symbol that denotes a vector is a ray above the two points that make up the vector, such as PQ, where P would be the initial point and Q would be the terminal point. To put a vector in component form (<v₁, v₂>),subtract the initial coordinates from the terminal coordinates. This anchors the vector at the origin. For example, if the inital point is (4,3) and the terminal point is (10,5), the component form would be <6,2>. To numerically add vectors, add the x components together and the y components together. To pictorially add vectors, add them head to toe so that the initial side of the added vector is placed at the terminal side of another. The Unit Vector is a vector of length 1. To put a vector in this form, divide both components by the magnitude of the vector. <v₁,v₂> /IIVII. If u is a unit vector such that θ is the angle from the x axis to u, the terminal point of u will lie on the unit circle and U= IIVII<Cosθ , Sinθ > will be true. θ is the direction angle of the vector U. Example: Find the unit vector in the direction of v where v= <-2,5> Unit vector= <-2,5>/√ 29= -2/√29, 5/√ 29 = -2√29/29, 5√29/29 The dot product of two vectors yields a scalar quantity. The equation for a dot product is U•V=U₁V₁+U₂V₂ Properties of dot product: V•V=IIVII The equation for the angle between two vectors is Cosθ = U•V/IIUII IIVII which can be rewritten as U•V= IIUII IIVII Cosθ