Math quiz second semester #2
Simplify the expression: 4-tan^2x
(2 + tanx) (2-tanx) 4-2tanx+2tanx-tan^2x = 4-tan^2x
factor: cos^2+2cosx+1
(cosx + 1)(cosx +1) = (cosx + 1)^2
Simplify the expression: (sin(x) + cos(x))^2
(sinx+cosx)(sinx+cosx) sin^2x+sinxcosx+cosxsinx+cos^2x sin^2+2sinxcosx+cos^2x sin^2+cos^2= 1 = 1+2sinxcosx
Verifying trigonometric identities rules:
1. choose 1 side to work with and DO NOT Touch the other side 2. strategies: -substitute identities - factor - distribute (multiply) - combine fractions (add/subtract) -split fractions
Simplify the expression: cot(x ) + cot(x) →
1cot(x) + 1cot(x) = 2cot(x)
Solve: [0, 2pi) 2sin(x) + 1 = 0
2sin(x) + 1 = 0 2sinx = -1 sinx = -1/2 y = -1/2 x= 7pi/6, 11pi/6
What's the exrema/turns of: f(x) = 3x^4 -6x^2:
3
solve: 3cot^2x-1=0
3cot^2x-1=0 3cot^2x = 1 cot^2x = 1/3 tan^2x = 3/1 √tan^2x = √3 tanx = +-√3 x= pi/3, 2pi/3, 4pi/3, 5pi/3
Multiplicity
Even - graph bounces at x-intercept Odd - graph crosses at x-intercept
Real Zero's of a Polynomial
Real Zero's of a Polynomial = x-intercept. x2 = 2 real zeros
domain of a polynomial
all real numbers
Simplify the expression: cos^2 (x)/cos(x)
cosxcosx/ cosx = cosx
Use trig identities to evaluate the remaining trig functions given: tan x = √3/3 cos x = -√3/2
cotx = 3/√3 (√3/√3) cotx = 3√3/3 = √3 secx = -2/√3 (√3/√3) secx = -2√3/3 tanx = sinx/cosx cosx tanx = sinx/cosx cosx cosx * tanx = sinx sinx = -√3/2 * √3/3 sinx = -3/6 = -1/2 cscx = -2/1 cscx = -2
Division Algorithm
f(x) = d(x)q(x) + r(x) or f(x)/d(x) = g(x) + r(x)/d(x) Where f(x) is the dividend, d(x) is the divisor, q(x) is the quotient, and r(x) is the remainder.
y-intercept
location where the graph crosses the y-axis, the y-value when x=0
Turns
relative extrema, relative maximum/minimum, point where graph ranges from increasing to decreasing to increasing EX: x^2 -1 = possible turns
Simplify the expression: sec(w)cos(w)/sec(w)
sec(w)cos(w)/ sec(w) = cos(w)
solve: sec^2x-1 = 0
sec^2x-1 = 0 sec^2x = 1 cos^2x = 1 √cos^2x = √1 cosx = +-1 x = 0 and pi
Reciprocal Identities
sin0= 1/csc0 csc0= 1/sin0 cos0= 1/sec0 sec0= 1/cos0 tan0= 1/cot0 cot0= 1/tan0
verify: sinB/tanB=cosB
sinB/sinB/cosB = cosB sinB / sin/cosB = cosB sinB * cosB/sinB cosB=cosB
Pythagorean Identities
sin^2 + cos^2 = 1 tan^2 + 1 = sec^2 1 + cot^2 = csc^2
Simplify the expression: FOIL (sin(y)-2)(sin(y)+3)
sin^2y+3siny-2siny-6 = sin^2y + 1siny -6
Simplify the expression: sinx + 1/sinx
sinx/sinx + 1/sinx 1 + 1/sinx = 1 + cscx
Simplify the expression: sinxtanx-1/tanx
sinxtanx/tax - 1/tanx sinx - 1/tanx = sinx - cotx
Quotient Identities
tan0 = sin0/cos0 cot0 = cos0/sin0
leading coefficient
the coefficient of the first term of a polynomial in standard form
degree of a polynomial
the greatest degree of any term in the polynomial
terms of a polynomial
the parts that are separated by an addition or subtraction sign.
What's the End Behavior of: f(x) = 3x^4 -6x^2:
x → -∞ y → ∞ x → ∞ y → ∞
