Mathematics, 23.07.2019 14:30, Lauraemia43

Rewrite the expression in terms of the first power of the cosine: tan^4x

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answered: cdradlet2001

Your answer is tan⁴x = (3 - 4 cos 2x + cos 4x) / (3 + 4 cos 2x + cos 4x) $\tan^4 x = \dfrac{3 - 4 \cos 2x + \cos 4x}{3 + 4 \cos 2x + \cos 4x}$ ======note that tan⁴ x = (tan x)⁴; all exponent properties apply normally.so we can break down tan⁴ x into tan² x · tan² x since we have the exponent property bˣ · bʸ = bˣ ⁺ ʸ. then we can use an identity that involves the square of tangent: $\tan^4 x = \tan^2 x \cdot \tan^2 x$ next, we use a tangent power-reducing identity. we do this because we want to, quite literally, reduce the power until it gets to the first power of cosine. so we have $\tan^2 \theta = \dfrac{1 - \cos 2\theta}{1 + \cos 2\theta} \quad \text{\scriptsize tangent power-reducing identity}$ to substitute into our original expression. $\begin{aligned} \tan^4 x & = \tan^2 x \cdot \tan^2 x \\ & = \left(\dfrac{1 - \cos 2x}{1 + \cos 2x}\right)\left(\dfrac{1 - \cos 2x}{1 + \cos 2x}\right) \end{aligned}$ expand this fraction using methods for expanding a product of two binomials $\begin{aligned} \tan^4 x & = \left(\dfrac{1 - \cos 2x}{1 + \cos 2x}\right)\left(\dfrac{1 - \cos 2x}{1 + \cos 2x}\right) \\ & = \frac{(1-\cos 2x)(1-\cos 2x)}{(1 + \cos 2x)(1 + \cos 2x)} & & \text{\scriptsize fraction }\\ & = \frac{1 - 2\cos 2x + \cos^2 2x}{1 + 2\cos 2x + \cos^2 2x } & & \text{\scriptsize use foil or other methods} \\ & = \frac{1 - 2\cos 2x + \boxed{\cos^2 2x}}{1 + 2\cos 2x + \boxed{\cos^2 2x} } \end{aligned}$ in order to lower the power of those second powers of cosine, we use the cosine power-reducing identity for the cos²2x terms, except with $\theta = 2x$ . the original identity: $\begin{aligned} \cos^2 \theta & = \dfrac{1 + \cos 2\theta}{2} & & \text{\footnotesize original identity} \\ \cos^2 2x & = \dfrac{1 + \cos 2(2x)}{2} & & \text{\footnotesize now with $\theta = 2x$} \\ & = \frac{1 + \cos 4x}{2} \end{aligned}$ so back into our original work: $\begin{aligned} \tan^4 x & = \frac{1 - 2\cos 2x + \boxed{\cos^2 2x}}{1 + 2\cos 2x + \boxed{\cos^2 2x} } \\ & = \frac{1 - 2\cos 2x + \frac{1 + \cos 4x }{2}}{1 + 2\cos 2x + \frac{1 + \cos 4x}{2} } \end{aligned}$ multiply numerator and denominator of the entire fraction by 2. the 2s will distribute so that we get rid of those little two denominators: $\begin{aligned} \tan^4 x & = \frac{2\left(1 + 2\cos 2x + \frac{1 + \cos 4x }{2} \right)}{2\left(1 + 2\cos 2x + \frac{1 + \cos 4x}{2} \right)} \\ & = \frac{2 - 4\cos 2x + 2\left(\frac{1 + \cos 4x }{2} \right)}{2 + 4\cos 2x +2\left( \frac{1 + \cos 4x}{2}\right) } \\ & = \frac{2 - 4\cos 2x + (1+ \cos 4x) }{2 + 4\cos 2x + (1 + \cos 4x) } \\ & = \frac{3 - 4 \cos 2x + \cos 4x}{3 + 4 \cos 2x + \cos 4x} & & \text{\footnotesize combine like terms} \end{aligned}$ we've now rewritten in terms of first power of cosine.

answered: Guest

answer: learning objective(s)

· describe the creation and use of systems of equations.

· graph a system of linear equations on the coordinate plane and identify its solution.

introduction

sometimes graphing a single linear equation is all it takes to solve a mathematical problem. other times, one line just doesn’t do it, and a second equation is needed to model the situation. this is often the case when a problem involves two variables. solving these kinds of problems requires working with a system of equations, which is a set of two or more equations containing the same unknowns.

let’s take a look at systems of equations, and see what the graphs of individual equations within a system reveal about the mathematical relationship of the variables.

systems of equations

a system of equations contains two or more linear equations that share two or more unknowns. to find a solution for a system of equations, we must find a value (or range of values) that is true for all equations in the system.

the graphs of equations within a system can tell us how many solutions exist for that system. look at the images below. each show two lines that make up a system of equations (in the graph on the right the two lines are superimposed and look like a single line). how many points in common does each system of lines reveal?

one solution

no solutions

infinite solutions

if the graphs of the equations intersect, then there is one solution that is true for both equations.

if the graphs of the equations do not intersect (for example, if they are parallel), then there are no solutions that are true for both equations.

if the graphs of the equations are the same, then there are an infinite number of solutions that are true for both equations.

remember, the graph of a line represents every point that is a possible solution for the equation of that line. so when the graphs of two equations cross, the point of intersection lies on both lines, meaning that it is a possible solution for both equations. when the graphs of two equations never touch, there are no shared points and there are no possible solutions for the system. when the graphs of two equations lie on top of one another, they share all their points and every one is a possible solution.

graphing as a solution method

graphing equations in order to identify a specific point of intersection is usually not a precise way to solve systems because it is often difficult to see exactly where two lines intersect (unless you are using a computer-based graphing program that allows you to zoom in on a point). however, the graph of a system of equations can still give a good idea of what type of solution, if any, exists. let’s graph a system, and see how it works.

graph the system y = 3x and x + 2y = 4. how many solutions does this system have?

a graph of the two lines y = 3x and x + 2y = 4 shows us that the lines intersect, meaning that there is a single (x, y) value that satisfies both equations. looking at the graph does not tell us exactly where that point is, but we don’t need to know that information here, because the question only asks for the number of solutions.

so a system made of two intersecting lines has one solution. now let’s look at a different situation:

how many solutions exist for the system y -0.5x = 7 and 2y = x − 3?

plotting both equations, it looks like there is no solution—the lines are parallel. to check this finding, we can compare the slopes of the equations. to make that easy, we’ll rearrange the equations into slope-intercept form, or y = mx + b. that gives us the equations y = 0.5x + 7 and y = 0.5x − 1.5. yes, the slope of both lines is 0.5, which means the lines really are parallel. they never intersect, so there is no point that lies on both lines, and no solution to the system.

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answered: Guest

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hope this !