Precalculus and Trigonometry
Spring Semester 2017
Professor: Ryan Reynolds
Notes Taken by Albany Ashton
Study Guide for Exam One:
Converting Radians and Degrees:
∙ To convert a number that is currently in degrees into radians, use the formula: θ(π
∙ To convert a number that is currently in radians into degrees, use the formula: θ(180o
Finding Coterminal Angles:
∙ To find coterminal angles for a number in radians, use the formula: θ+¿−2 π ∙ To find coterminal angles for a number in degrees, use the formula: θ+¿−360o Finding the Arc Length and Area of a Sector:
∙ For finding the arc length of a circle, use the formula: s=θr
o s represents arc length, and r represents the radius of the circle
∙ For finding the area of a sector within a circle, use the formula: Asect=12θr2 o Asect represents area of a sector, and r represents the radius of the circle ∙ For both of these formulas, θ must be in radians We also discuss several other topics like What are the three ways to form magma?
Finding Linear Speed and Angular Speed:
∙ For finding the linear speed, use the formula: υ=st
o s represents arc length, and t represents time
∙ For finding the angular speed, use the formula: ϖ=θt
o θ represents revolutions or radians, and t represents time
∙ The algebraic relationship between linear speed and angular speed is shown through the formula: ϖr=υ∨θtr=rθt If you want to learn more check out What is the punishment for status offense?
∙ The equation for a circle is: x2+y2=1 Don't forget about the age old question of What makes an event independent or dependent?
∙ The equation of the Pythagorean Identity is: cos2+sin2=1
∙ The reference angles for the Unit Circle include:
o 30o∨π6 (√32,12 )
o 45o∨π4 (√22,√22 )
o 60o∨π3 (12,√32 ) Don't forget about the age old question of How do you use certainty in a sentence?
Domain and Range of Cosine and Sine Functions:
∙ The domain of a function is the set of all possible input values.
o The domain for both sine and cosine can be any real numbers.
o Interval notation: (−∞ ,∞)
∙ The range of a function is the set of all the possible output values.
o The largest output that can be obtained for both sine and cosine is 1.
o The smallest output that can be obtained for both sine and cosine is 1.Don't forget about the age old question of What is bowlby’s attachment theory?
We also discuss several other topics like What are the difference between disintegrative shaming and reintegrative shaming?
o Interval notation: [−1,1 ]
Unit Circle Trig Functions:
∙ sin θ=y
∙ tan θ=yx∨sinθ
∙ csc θ=1y∨1
∙ sec θ=1x∨1
∙ cot θ=xy∨cosθ
Secant, Cosecant, and Cotangent of the Reference Angles: ∙π6∨30o
Even and Odd Functions:
∙ Even functions satisfy the equation: f (−θ)= f (θ) . On a line graph, this would be depicted by the line reflecting across the yaxis.
o Which trig functions are even?
Cosine, and secant
∙ Odd functions satisfy the equation: f (−θ)=− f (θ) . On a line graph, this would be depicted by the line reflecting across both the xaxis and yaxis.
o Which trig functions are odd?
Sine, tangent, cosecant, and cotangent
Alternate Forms of the Pythagorean Identity: ∙ 1+cot 2(θ)=csc2(θ)
The Periods for each Unit Circle Function: ∙ sine: 2π
∙ cosine: 2π
∙ tangent: π
∙ cosecant: 2π
∙ secant: 2π
∙ cotangent: π
SOH CAH TOA:
∙ sin (θ)=opp
∙ tan (θ)=opp
Pythagorean Thorium: ∙ a2+b2=c2
The Cosine Rule:
∙ ∙ ∙
a2=b2+c2−2 bc ¿
The Sine Rule:
∙ Cofunction identity relating sine and cosine of different angles in the same right triangle: o sin (α)=cos(β)
o sin (β)=cos(α)
∙ Cofunction identities in terms of one angle, in degrees:
o sin (900−θ)=cos(θ)
o cos( 90o−θ)=sin (θ)
o tan ( 90o−θ)=cot (θ)
o cot ( 90o−θ)=tan(θ)
o sec (90o−θ)=csc (θ)
o csc (90o−θ)=sec (θ)
∙ Cofunction identities in terms of one angle, in radians:
o cos(π2−θ)=sin (θ)
o tan(π2−θ)=cot (θ)
o cot(π2−θ)=tan (θ)
Elevation and Depression Angles:
∙ Elevation angle:
o An angle based on the rotation from our eyeline to an upward position. ∙ Depression angle:
o An angle based on the rotation from our eyeline to a downward position. Graphs of Sine and Cosine:
∙ For any given angle θ from the original point, f (θ) will be the vertical height, or the distance from the xaxis.
∙ For any given angle θ from the original point, g (θ) will be the horizontal length, or the distance from the yaxis.
∙ What are the five points needed to graph either a sine or cosine graph? o The 2 peaks, the 2 midline points, and the trough.
Amplitude, Period, Phase Shift and Frequency
∙ To reiterate, the Period of both sine and cosine is 2 π
∙ The Amplitude is the height from the center line to the peak, or to the trough.
∙ The Phase Shift is how far the function is horizontally to the right of the usual position.
∙ The Vertical Shift is how the function is vertically up from the usual position. ∙ We can have all of them in one equation:
o y=Asin ( Bx+C)+D
amplitude is A
period is 2π/ B
phase shift is –C/ B
vertical shift is D