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- https://k12.libretexts.org/Bookshelves/Mathematics/Trigonometry/05%3A_Polar_System_and_Complex_Numbers/5.02%3A_Polar_Curves_and_Rectangular_Conversions/5.2.03%3A_Rectangular_to_Polar_Form_for_Equations\(\begin{aligned} (x+4)^{2}+(y-1)^{2}&=17 \\ (r \cos \theta+4)^{2}+(r \sin \theta-1)^{2}&=17 && x=r \cos \theta \text{ and } y=r \sin \theta \\ r^{2} \cos ^{2} \theta+8 r \cos \theta+16+r^{2} \sin ^{2...\(\begin{aligned} (x+4)^{2}+(y-1)^{2}&=17 \\ (r \cos \theta+4)^{2}+(r \sin \theta-1)^{2}&=17 && x=r \cos \theta \text{ and } y=r \sin \theta \\ r^{2} \cos ^{2} \theta+8 r \cos \theta+16+r^{2} \sin ^{2} \theta-2 r \sin \theta+1&=17 && \text{ expand the terms} \\ r^{2} \cos ^{2} \theta+8 r \cos \theta-2 r \sin \theta+r^{2} \sin ^{2} \theta&=0 && \text{ subtract 17 from each side} \\ r^{2} \cos ^{2} \theta+r^{2} \sin ^{2} \theta&=-8 r \cos \theta+2 r \sin \theta && \text{isolate the squared terms}…
- https://k12.libretexts.org/Bookshelves/Mathematics/Trigonometry/03%3A_Trigonometric_Identities/3.02%3A_Basic_Trig_Identity_Applications/3.2.07%3A_Trigonometric_Equations_Using_the_Quadratic_FormulaThe quadratic formula with a trigonometric function in place of the variable.
- https://k12.libretexts.org/Bookshelves/Mathematics/Trigonometry/05%3A_Polar_System_and_Complex_Numbers/5.03%3A_Complex_Numbers/5.3.03%3A_Quadratic_Formula_and_Complex_SumsWhen solving for the roots of a function algebraically using the quadratic formula, you will end up with a negative under the square root symbol. With your knowledge of complex numbers, you can still ...When solving for the roots of a function algebraically using the quadratic formula, you will end up with a negative under the square root symbol. With your knowledge of complex numbers, you can still state the complex roots of a function just like you would state the real roots of a function. Consider when you use the quadratic formula-- if you have a negative under the square root symbol, both the + version and the - version of the two answers will end up being complex.
- https://k12.libretexts.org/Bookshelves/Mathematics/Trigonometry/02%3A_Trigonometric_Ratios/2.05%3A_Radians/2.5.01%3A_Radian_MeasureRecall from geometry that the arc length of a complete rotation is the circumference, where the formula is equal to \(2\pi\) times the length of the radius. \(2\pi\) is approximately 6.28, so the circ...Recall from geometry that the arc length of a complete rotation is the circumference, where the formula is equal to \(2\pi\) times the length of the radius. \(2\pi\) is approximately 6.28, so the circumference is a little more than 6 radius lengths. A radian is a unit of angle that is equal to the angle created at the center of a circle whose arc is equal in length to the radius.
- https://k12.libretexts.org/Bookshelves/Mathematics/Trigonometry/05%3A_Polar_System_and_Complex_Numbers/5.01%3A_The_Polar_Coordinate_System/5.1.03%3A_Graph_Polar_EquationsPolar coordinates represent the same point, but describe the point by its distance from the origin (\(r\)) and its angle on the unit circle (\(\theta \)). The polar coordinate system is a special coor...Polar coordinates represent the same point, but describe the point by its distance from the origin (\(r\)) and its angle on the unit circle (\(\theta \)). The polar coordinate system is a special coordinate system in which the location of each point is determined by its distance from the pole and its angle with respect to the polar axis. The location of each point is determined by its distance from the pole and its angle with respect to the polar axis.
- https://k12.libretexts.org/Bookshelves/Mathematics/Trigonometry/05%3A_Polar_System_and_Complex_Numbers/5.01%3A_The_Polar_Coordinate_System/5.1.04%3A_Transformations_of_Polar_GraphsFor this problem, the \(r\) value, or radius, is arbitrary. \(\theta \) must equal \(30^{\circ} \), so the result is a straight line, with an angle of elevation of \(30^{\circ} \). The shape of the li...For this problem, the \(r\) value, or radius, is arbitrary. \(\theta \) must equal \(30^{\circ} \), so the result is a straight line, with an angle of elevation of \(30^{\circ} \). The shape of the limaçon depends upon the ratio of ab where a is a constant and b is the coefficient of the trigonometric function. As we've seen with cardioids, it is possible to create transformations of graphs of limaçons by changing values of constants in the equation of the shape.
- https://k12.libretexts.org/Bookshelves/Mathematics/Trigonometry/02%3A_Trigonometric_Ratios/2.04%3A_Inverse_Trigonometric_Functions/2.4.01%3A_Inverse_Trig_FunctionsIn this concept we will use the inverses of these functions, \(\sin^{-1}\), \(\cos^{-1}\) and \(\tan^{-1}\), to find the angle measure when the ratio of the side lengths is known. Now, let's find the ...In this concept we will use the inverses of these functions, \(\sin^{-1}\), \(\cos^{-1}\) and \(\tan^{-1}\), to find the angle measure when the ratio of the side lengths is known. Now, let's find the measures of the unknown angles in the triangle shown and round answers to the nearest degree. Now we can find B by subtracting \(m\angle A\) from \(90^{\circ}\): \(90^{\circ} −30^{\circ} =60^{\circ}\) since the acute angles in a right triangle are always complimentary.
- https://k12.libretexts.org/Bookshelves/Mathematics/Trigonometry/02%3A_Trigonometric_Ratios/2.02%3A_Solving_Triangles/2.2.04%3A_Solve_Right_TrianglesUsing the inverse trigonometric functions to solve for missing information about right triangles.
- https://k12.libretexts.org/Bookshelves/Mathematics/Trigonometry/03%3A_Trigonometric_Identities/3.01%3A_Trig_Identities
- https://k12.libretexts.org/Bookshelves/Mathematics/Trigonometry/02%3A_Trigonometric_Ratios/2.03%3A_Trig_in_the_Unit_Circle/2.3.10%3A_Exact_Values_for_Inverse_Sine_Cosine_and_TangentThe brace is a right triangle, and the length of one side of the bracket is \(\sqrt{3}\approx 1.732\) and it is connected to the other side at a right angle. Using your knowledge of the values of trig...The brace is a right triangle, and the length of one side of the bracket is \(\sqrt{3}\approx 1.732\) and it is connected to the other side at a right angle. Using your knowledge of the values of trig functions for angles, you can work backward to find the angle that the brace makes: It can also be used to find a missing angle of a triangle from the ratio of two sides of the triangle.
- https://k12.libretexts.org/Bookshelves/Mathematics/Trigonometry/zz%3A_Back_Matter
