Chapter 4 (4.1 to 4.7) Flashcards
What are co-terminal angles and how do you find them?
Two or more different angles with the same initial and terminal rays
+/- 360*
+/- 2pie
What are the two ways you can write an angle in degrees?
Decimal 25.5*
Degrees, minutes, seconds 25*30’0”
What is 360* in radians?
2 pie
What is 180* in radians?
Pie
What is 90* in radians?
Pie/2
What is 45* in radians?
Pie/4
What is 30* in radians?
Pie/6
What is 60* in radians?
Pie/3
What are the four quadrants and what is it called when it is on an axis?
1-4 counter clockwise
Quadrantal
What are complementary angles?
2 angles that add to 90*or pie/2
What are supplementary angles?
2 angles that add to 180* or pie
How do you convert between degrees and radians?
D->R multiply by pie/180*
R->D multiply by 180*/pie
What are the three ways to find the area of a sector? (Angle in D and R, arc length)
(A is area)
A = theta/360* times pie times radius^2 (degrees) A = theta times 1/2 times radius^2 (radians) A = 1/2 times radius times arc length
What are the two ways to find arc length? (D and R)
(S is arc length)
S = theta times radius (radians)
S = theta/360* times 2pie times radius (degrees)
What is the unit circle?
A circle centered at the origin with a radius of 1
How do you find the six trig functions withs unit circle?
Cos theta is x Sec theta is 1/x Sin theta is y Csc theta is 1/y Tan theta is y/x Cot theta is x/y
What are the unit circle coordinates for 30, 45, and 60*, and where do you measure from?
30* or pie/6: (square root of 3 over 2, 1/2)
45* or pie/4: (square root of 2 over 2, “”)
60* or pie/3: (1/2, square root of 3 over 2)
X-axis
What are the six trig functions of a right triangle?
Sin= opp/hyp Csc= hyp/opp Cos= adj/hyp Sec= hyp/adj Tan= opp/adj Cot= adj/opp
Soh-cah-toa
When sketching angles, what are the three things you need and what direction do you go?
Two rays, one starting on the positive x-axis and a theta sign
Positive is counter-clockwise
Negative is clockwise
What is Sin(90* - theta)?
Cos theta
What is Cos(90* - theta)?
Sin theta
What is Tan(90* - theta)?
Cot theta
What is Cot(90* - theta)?
Tan theta
What is Sec(90* - theta)?
Csc theta
What is Csc(90* - theta)?
Sec theta
What are the six reciprocal identies?
Cos theta=1/sec theta Sec theta=1/cos theta Sin theta=1/csc theta Csc theta=1/sin theta Tan theta=1/cot theta Cot theta=1/tan theta
What are the two quotient identies?
Tan theta = sin theta/cos theta
Cot theta = cos theta/sin theta
What are the three pythagorean identies?
Cos^2 theta + sin^2 theta = 1
1 + tan^2 theta = sec^2 theta
Cot^2 theta + 1 = csc^2 theta
How do you find the six trig functions with a circle that has a radius other than 1?
Cos theta: x/r, Sec theta: r/x, Sin theta: y/r, Csc theta: r/y, Tan theta: y/x, Cot theta: x/y
What is the sentence to remember where the three trig functions are positive?
All Students Take Calculus, all functions are + in Quad 1, only sin in 2, only tan in 3, only cos in 4
What is the reference angle?
The positive acute angle formed by the terminal side of theta and the x-axis. It is called theta’
What does the graph of sin look like? (Unit circle, with fundamental period and domain and range)
At the origin, up at one at pie/2, 0 at pie, -1 at 3pie/2, zero at 2pie
F.P is 2pie, D (-infinity, infinity), R [-1,1]
What does the graph of cos look like? (Unit circle, with fundamental period and domain and range)
Up at one at the origin, zero at pie/2, -1 at pie, zero at 3pie/2, one at 2pie
F.P is 2pie, D (-infinity, infinity), R [-1,1]
For Y=acosbx and Y=asinbx, how do you find the amplitude and fundamental period and what does the graph look like when a is negative?
Amplitude is |a|, fundamental period is 2pie/b, upside down
When the sin/cos graphs are Y=acosbx-c or Y=asinbx-c how do they change and how do you know how to change them?
They shift horizontally, new left and right endpoints are found with bx-c=0 and bx-c=2pie (solve for x)
When the sin/cos graphs are Y=acosbx-c + d or Y=asinbx-c + d how do they change and how do you know how to change them?
They move vertically d units up
What does the graph of tan look like? (Unit circle, with fundamental period and domain and range)
At the origin, at one at pie/4, approaching the asymptote of pie over two, negative one at -pie/4 approaching the asymptote of -pie/2
F.P is pie, D (theta not = pie/2 + pie*n, n e z), R (-infinity, infinity)
What does the graph of cot look like? (Unit circle, with fundamental period and domain and range)
Asymptote of 0 and pie, on the x-axis at pie/2, up at one at pie/4 and down to -1 at 3pie/4
F.P is pie, D (theta not = 0 + pie*n, n e z), R (-infinity, infinity)
For Y=atanbx and Y=atanbx, how do you find the height midway to the asymptote and fundamental period and what does the graph look like when a is negative?
A= + height midway to asymptote, F.P is pie/b, flipped (like negative slope)
When the tan/cot graphs are Y=atanbx-c or Y=acotbx-c how do they change and how do you know how to change them?
They shift horizontally, new left and right endpoints for tan are found with bx-c=-pie/2 and bx-c=pie/2 (solve for x); new left and right endpoints for cot are found with bx-c=0 and bx-c=pie (solve for x)
What is the period, domain, and range of Y=sec(x) and Y=csc(x), and what do they look like?
Sec- period is 2pie, domain is theta not equal to pie/2 plus pie*n (n E Z), range is (-infinity, -1] U [1, infinity)
Csc- period is 2pie, domain is theta not equal to pie*n (n E Z) range is (-infinity, -1] U [1, infinity)
They have asymptotes where cos (for sec) and sin (for csc) hit the x-axis and are u shaped going away from the slope of cos or sin
Where is the graph of arcsin restricted?
Quads 1 and 4
Where is the graph of arccos restricted?
Quads 1 and 2
Where is the graph of arctan restricted?
Quads 1 and 4
What is tan(30*)?
Square root of 3/3
What is tan(45*)?
1
What is tan(60*)?
square root of 3
