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- https://math.libretexts.org/Bookshelves/Algebra/College_Algebra_and_Trigonometry_(Beveridge)/03%3A_Exponents_and_Logarithms/3.01%3A_Exponential_and_Logistic_ApplicationsAs mathematicians examined these graphs during the 17 th and 18 th centuries, they began to question what the value of the base " \(b^{\prime \prime}\) should be in the equation \(y=b^{x}\) so that th...As mathematicians examined these graphs during the 17 th and 18 th centuries, they began to question what the value of the base " \(b^{\prime \prime}\) should be in the equation \(y=b^{x}\) so that the slope of the tangent line at the point (0,1) would be equal to exactly \(1 .\) The answer was \(e \approx 2.71828\)
- https://math.libretexts.org/Bookshelves/Precalculus/Elementary_Trigonometry_(Beveridge)/02%3A_Graphing_the_Trigonometric_Functions/2.05%3A_Combining_the_Transformations\] Notice that the distance between the starting point \(-\frac{\pi}{4}\) and the ending point \(\frac{\pi}{4}\) is equal to the period we found at the beginning of the problem, which was \(\frac{\pi}...\] Notice that the distance between the starting point \(-\frac{\pi}{4}\) and the ending point \(\frac{\pi}{4}\) is equal to the period we found at the beginning of the problem, which was \(\frac{\pi}{2} .\) Now let's graph the function: Along the \(x\) -axis, the period for the graph will be \(\frac{2 \pi}{B}=\frac{2 \pi}{3},\) since the coefficient \(B\) in this problem is \(3 .\) To find the new starting point, we'll take the argument of the cosine function and set it equal to zero.
- https://math.libretexts.org/Bookshelves/Algebra/College_Algebra_and_Trigonometry_(Beveridge)/01%3A_Algebra_ReviewThis College Algebra text will cover a combination of classical algebra and analytic geometry, with an introduction to the transcendental exponential and \(\log\) arithmic functions. If mathematics is...This College Algebra text will cover a combination of classical algebra and analytic geometry, with an introduction to the transcendental exponential and \(\log\) arithmic functions. If mathematics is the language of science, then algebra is the grammar of that language. Like grammar, algebra provides a structure to mathematical notation, in addition to its uses in problem solving and its ability to change the appearance of an expression without changing the value.
- https://math.libretexts.org/Bookshelves/Algebra/College_Algebra_and_Trigonometry_(Beveridge)/04%3A_Functions/404%3A__TransformationsIf we take a given function, let's say \(f(x)=x,\) then this has the graph we see below - a straight line with a slope of 1 and a \(y\) -intercept of \(0 .\) If we add to the function \(f(x)+6=x+6,\) ...If we take a given function, let's say \(f(x)=x,\) then this has the graph we see below - a straight line with a slope of 1 and a \(y\) -intercept of \(0 .\) If we add to the function \(f(x)+6=x+6,\) then this will add 6 to all of the \(y\) -values which shifts the graph 6 places up. If we multiply the function by a constant outside of the parentheses: \(y=2 f(x)\) then this will have the effect of multiplying all of the \(y\) values by \(2 .\) In the table:
- https://math.libretexts.org/Bookshelves/Precalculus/Elementary_Trigonometry_(Beveridge)/04%3A_The_Law_of_Sines_and_The_Law_of_Cosines/4.03%3A_The_Law_of_CosinesBecause of the issue of the ambiguous case in using the Law of sines, it's often a good idea to find the angles that correspond to the two shortest sides in the triangle, because if there is an obtuse...Because of the issue of the ambiguous case in using the Law of sines, it's often a good idea to find the angles that correspond to the two shortest sides in the triangle, because if there is an obtuse angle in the triangle it will have to correspond to the longest side. If we had used the Law of sines to find \(\angle A\), the calculator would have returned the value of the reference angle for \(\angle A\), rather than the angle that is actually in the triangle described in the problem!
- https://math.libretexts.org/Bookshelves/Algebra/College_Algebra_and_Trigonometry_(Beveridge)/07%3A_Combinatorics
- https://math.libretexts.org/Bookshelves/Algebra/College_Algebra_and_Trigonometry_(Beveridge)/10%3A_Trigonometric_Identities_and_Equations/10.01%3A_Reciprocal_and_Pythagorean_Identities\[\frac{\tan \theta-\cot \theta}{\sin \theta \cos \theta}=\sec ^{2} \theta-\csc ^{2} \theta\] \[\frac{\tan \theta}{\sin \theta \cos \theta}-\frac{\cot \theta}{\sin \theta \cos \theta}=\sec ^{2} \theta...\[\frac{\tan \theta-\cot \theta}{\sin \theta \cos \theta}=\sec ^{2} \theta-\csc ^{2} \theta\] \[\frac{\tan \theta}{\sin \theta \cos \theta}-\frac{\cot \theta}{\sin \theta \cos \theta}=\sec ^{2} \theta-\csc ^{2} \theta\] \[\frac{\frac{\sin \theta}{\cos \theta}}{\sin \theta \cos \theta}-\frac{\frac{\cos \theta}{\sin \theta}}{\sin \theta \cos \theta}=\sec ^{2} \theta-\csc ^{2} \theta\] \[\frac{\sin \theta}{\cos \theta} \cdot \frac{1}{\sin \theta \cos \theta}-\frac{\cos \theta}{\sin \theta} \cdot \…
- https://math.libretexts.org/Bookshelves/Algebra/College_Algebra_and_Trigonometry_(Beveridge)This College Algebra text will cover a combination of classical algebra and analytic geometry, with an introduction to the transcendental exponential and logarithmic functions. If mathematics is the l...This College Algebra text will cover a combination of classical algebra and analytic geometry, with an introduction to the transcendental exponential and logarithmic functions. If mathematics is the language of science, then algebra is the grammar of that language. Like grammar, algebra provides a structure to mathematical notation, in addition to its uses in problem solving and its ability to change the appearance of an expression without changing the value.
- https://math.libretexts.org/Workbench/1250_Draft_3/04%3A_Polynomial_Functions/4.07%3A_Solution_of_Polynomial_Inequalities_by_GraphingIn Section \(2.2,\) we solved equations by graphing and finding the \(x\) -values which made \(y=0 .\) In solving an inequality, we will be concerned with finding the range of \(x\) values that make \...In Section \(2.2,\) we solved equations by graphing and finding the \(x\) -values which made \(y=0 .\) In solving an inequality, we will be concerned with finding the range of \(x\) values that make \(y\) either greater than or less than \(0,\) depending on the given problem. The solution to the given inequlaity \(2 x^{3}+8 x^{2}+5 x-3 \geq 0\) are \(\mathrm{A} \leq x \leq \mathrm{B}\) OR \(x \geq \mathrm{C}\) 5) Determine the interval(s) for which \(6 x^{4}-13 x^{3}+2 x^{2}-4 x+15 \geq 0\)
- https://math.libretexts.org/Bookshelves/Precalculus/Elementary_Trigonometry_(Beveridge)/01%3A_Right_Triangle_Trigonometry/1.02%3A_The_Trigonometric_RatiosGiven an angle situated in a right triangle, the sine function is defined as the ratio of the side opposite the angle to the hypotenuse, the cosine is defined as the ratio of the side adjacent to the ...Given an angle situated in a right triangle, the sine function is defined as the ratio of the side opposite the angle to the hypotenuse, the cosine is defined as the ratio of the side adjacent to the angle to the hypotenuse and the tangent is defined as the ratio of the side opposite the angle to the side adjacent to the angle.
- https://math.libretexts.org/Bookshelves/Algebra/College_Algebra_and_Trigonometry_(Beveridge)/02%3A_Polynomial_and_Rational_Functions/201%3A_Representing_IntervalsFor example, in the diagram below, we would represent the interval shown on the graph as \(x<-3\) Sometimes students try to represent the intervals above as \(6 \leq x<-4,\) however, this expression w...For example, in the diagram below, we would represent the interval shown on the graph as \(x<-3\) Sometimes students try to represent the intervals above as \(6 \leq x<-4,\) however, this expression would represent a single interval where \(x\) is both less than -4 and, at the same time, greater than or equal to \(+6 .\) This is simply not possible, and would result in the empty set, which is the reason that the OR portion is needed in the correct answer.
