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Answers by Kevin C.

Hi, Leah.   Remember that 1-sin2x = cos2x   So if you multiply this fraction (cosx)/(1-sinx) by (1+sinx)/(1+sinx) you will get: (cosx)(1+sinx)/(1-sin2x) = (cosx)(1+sinx)/(cos2x) or (1+sinx)/(cosx)   or: 1/cosx + sinx/cosx =...

Find the perpendicular bisectors of segments AQ, QB, BT and TA.  If a point lies on the perpendicular bisector of a segment, all points on that line are equidistant from the endpoints.  Therefore, the point at which these lines meet is the center of a circle.   That is a nice,...

AB is a chord bisected by PO, the radius since OT is an angle bisector.  The arcs PA and PB are also congruent since the arc AB is bisected also.    Tangent TA and chord AB intercept arc PA and is half its measure.   Chord PA and chord AB intercept arc PB and...

Use the log and the equation becomes cos2x + tanx = 0   So, cos2x - sin2x + tanx = 0   Does this help??  Does using the identities to put this into sin and cos help?   Kevin

OK.   csc2x = cot2x + 1,  from the Pythagorean identities.   Therefore, csc2x - 1 = cot2x.   Substitute:  (csc2x - 1)/cot2x = cot2x/cot2x = 1   I hope this helps.

Find cos2x (answer)

Use the double angle formula:  cos2x = 1 - 2sin2x.   Therefore, if sinx = -3/5, and sin2x = 9/25,   cos2x = 1 - 2(9/25) = 1 - 18/25 = 25/25 - 18/25 = 7/25.   BUT: Since the angle lies in quadrant 3, (π < x < 3π/2), the double angle, 2x lies between...

... or using de Moivre's Theorem:  wk =  z1/n = (rcisθ)1/n = r1/n (cis(θ+360k)/n), where k = 0 and 1, and n = 2, z = 0 + 2i, so r = 2, and θ = 90°.   we get w0 = √2 (cis(90/2)), and w1 = √2 (cis(90 + 360)/2);   So, w0 = √2 cos 45 + √2 i sin...

Find cos theta/2 (answer)

Hi, David.   The half angle formula:  cos(θ/2) = ±√((1 + cosθ)/2).   Since sinθ = 3/5, cosθ = 4/5.   Therefore, (1+cosθ)/2  = 9/10, and cos(θ/2) = ±√.9.   Since 0 < θ < π,  we know that 0 < θ/2 < π/2, and cos(θ/2)...

Find sin theta/2 (answer)

Hi, David.   The half angle formula:  sin(θ/2) =± √((1-cosθ)/2).   Therefore, (1-cosθ)/2 = (1+.6)/2 = .8.   sin(θ/2) = ±√(.8)   and  since π/2 < θ < 3π/2, Π/4 < θ/2 < 3π/4,  sin(θ/2) is ±√(.8).   I hope...

The arithmetic mean is the average of 2 or more numbers.   The arithmetic mean of x and y is:  (x + y)/2.   The geometric mean is the square root of the product of two numbers.   For example:  the geometric mean of 4 and 9 is 6, the square root...

y = cos-1(1/2) = π/3 and, since the cos is also positive for (3π/2, 2π), the answer needs to include 5π/3.   The value of arccos0 has two values:  at π/2, and 3π/2.   Arccos(-15/17):  x = -15, r = 17, and y = 8.  Therefore, the cot must = x/y, or -15/8.   If...

x cos y = 2 (answer)

Hi, Ehsan.   Using the product formula, taking the derivative of: x cos y = 2,   We get:  1 cos y  + x (-sin y) dy/dx = 0   So -x sin y (dy/dx) = -cos y,   Divide by -xsiny, dy/dx = (cos y)/(x sin y) = (cot y)/x.   Since...

First:  multiply all equations by 10.     5a + 3b = 22 12c + 85b = -244 33c + 13a = 290   Eliminate b from the first 2 equations, getting an equation in a and c.   Use that and the 3rd equation to solve for a and c.    Finally,...

Plane figures (answer)

Q1.  Do you mean hextagon or hexagon?  The last three are parallelograms, and therefore, by definition, are polygons with parallel sides.   Also, all hexagons don't have any parallel sides unless they are regular or they may have only 2 parallel sides.