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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/00%3A_Front_Matter
- https://math.libretexts.org/Bookshelves/Precalculus/Trigonometry_(Yoshiwara)/00%3A_Front_Matter
- https://math.libretexts.org/Bookshelves/Precalculus/Trigonometry_(Yoshiwara)/03%3A_Laws_of_Sines_and_Cosines/3.02%3A_The_Law_of_CosinesIf we know two sides \(a\) and \(b\) of a triangle and the acute angle \(\alpha\) opposite one of them, there may be one solution, two solutions, or no solution, depending on the size of \(a\) in rela...If we know two sides \(a\) and \(b\) of a triangle and the acute angle \(\alpha\) opposite one of them, there may be one solution, two solutions, or no solution, depending on the size of \(a\) in relation to \(b\) and \(\alpha\), as shown below. Note 3.46 In the previous example, notice that we used the Law of Cosines instead of the Law of Sines to find a second angle in the triangle, because there is only one angle between \(0^{\circ}\) and \(180^{\circ}\) with a given cosine.
- https://math.libretexts.org/Bookshelves/Precalculus/Trigonometry_(Yoshiwara)/08%3A_More_Functions_and_Identities/8.03%3A_Chapter_Summary_and_Review2 it is not true in general that \(\cos (\alpha+\beta)\) is equal to \(\cos \alpha+\cos \beta\) for all angles \(\alpha\) and \(\beta\), or that \(\sin (\alpha+\beta)\) is equal to \(\sin \alpha+\sin ...2 it is not true in general that \(\cos (\alpha+\beta)\) is equal to \(\cos \alpha+\cos \beta\) for all angles \(\alpha\) and \(\beta\), or that \(\sin (\alpha+\beta)\) is equal to \(\sin \alpha+\sin \beta\). For Problems 85-86, use the fact that if \(\theta\) is one angle of a triangle and \(s\) is the length of the opposite side, then the diameter of the circumscribing circle is
- https://math.libretexts.org/Bookshelves/Precalculus/Trigonometry_(Yoshiwara)/03%3A_Laws_of_Sines_and_Cosines/3.03%3A_Chapter_3_Summary_and_Review5 If a triangle has sides of length \(a\) and \(b\), and the angle between those two sides is \(\theta\), then the area of the triangle is given by Use the result of Problem 57, the Law of Sines, and ...5 If a triangle has sides of length \(a\) and \(b\), and the angle between those two sides is \(\theta\), then the area of the triangle is given by Use the result of Problem 57, the Law of Sines, and the fact that every triangle has an acute angle to show that the quantities \(\frac{a}{\sin A}, \frac{b}{\sin B}\), and \(\frac{c}{\sin C}\) all represent the diameter of the circumscribing circle.
- https://math.libretexts.org/Bookshelves/Precalculus/Trigonometry_(Yoshiwara)/02%3A_Trigonometric_Ratios/2.00%3A_Side_and_Angle_RelationshipsIt is also true that the sum of the lengths of any two sides of a triangle must be greater than the third side, or else the two sides will not meet to form a triangle. In a right triangle, if \(c\) st...It is also true that the sum of the lengths of any two sides of a triangle must be greater than the third side, or else the two sides will not meet to form a triangle. In a right triangle, if \(c\) stands for the length of the hypotenuse, and the lengths of the two legs are denoted by \(a\) and \(b\), then The figure shows a triangle inscribed in a unit circle, one side lying on the diameter of the circle and the opposite vertex at point \((p, q)\) on the circle.
- https://math.libretexts.org/Bookshelves/Precalculus/Trigonometry_(Yoshiwara)/05%3A_Equations_and_Identities
- https://math.libretexts.org/Bookshelves/Precalculus/Trigonometry_(Yoshiwara)/06%3A_Radians/6.02%3A_Graphs_of_the_Circular_FunctionsNote 6.30 Because we know the basic shapes of the sine and cosine graphs, to make an adequate graph it is usually sufficient to plot the guide points at the quadrantal angles, and then draw a smooth c...Note 6.30 Because we know the basic shapes of the sine and cosine graphs, to make an adequate graph it is usually sufficient to plot the guide points at the quadrantal angles, and then draw a smooth curve through the points. Locate the four values \(t, 2\pi − t, \pi − t\), and \(\pi + t\) from Problem 63 on the graph of \(f(t) = \sin t\), on the graph of \(g(t) = \cos t\), and on the graph of \(h(t) = \tan t\) shown below.
- https://math.libretexts.org/Bookshelves/Precalculus/Trigonometry_(Yoshiwara)/11%3A_Answers_to_Selected_Exercises/11.02%3A_Laws_of_Sines_and_Cosinesb \(C=58.99^{\circ}, B=81.01^{\circ}, b=9.22\) or \(C=121.01^{\circ}, B=18.99^{\circ}, b=3.04\) First the Law of Cosines: \(x^2=47^2+29^2-2 \cdot 47 \cdot 29 \cos 81^{\circ}\), then either the Law of ...b \(C=58.99^{\circ}, B=81.01^{\circ}, b=9.22\) or \(C=121.01^{\circ}, B=18.99^{\circ}, b=3.04\) First the Law of Cosines: \(x^2=47^2+29^2-2 \cdot 47 \cdot 29 \cos 81^{\circ}\), then either the Law of Sines: \(\dfrac{\sin \theta}{47}=\dfrac{\sin 81^{\circ}}{x}\) or the Law of Cosines: \(47^2 = x^2 + 29^2 - 2 \cdot x \cdot 29 \cos \theta\) so \(2a^2 = 2a(c \cos B + b \cos C\), and dividing both sides by \(2a\) yields \(a = (c \cos B + b \cos C)\).
- https://math.libretexts.org/Bookshelves/Precalculus/Trigonometry_(Yoshiwara)/11%3A_Answers_to_Selected_Exercises/11.07%3A_More_Functions_and_Identities25. \(\sin \theta=\dfrac{5}{\sqrt{74}}, \quad \cos \theta=\dfrac{-7}{\sqrt{74}}, \quad \tan \theta=\dfrac{-5}{7}, \quad \sec \theta=\dfrac{-\sqrt{74}}{7}, \quad \csc \theta=\dfrac{\sqrt{74}}{5}, \quad...25. \(\sin \theta=\dfrac{5}{\sqrt{74}}, \quad \cos \theta=\dfrac{-7}{\sqrt{74}}, \quad \tan \theta=\dfrac{-5}{7}, \quad \sec \theta=\dfrac{-\sqrt{74}}{7}, \quad \csc \theta=\dfrac{\sqrt{74}}{5}, \quad \cot \theta=\dfrac{-7}{5}\) 37. \(\sin \theta = \dfrac{-\sqrt{9 - a^2}}{3}, \cos \theta = \dfrac{a}{3}, \tan \theta = \dfrac{-\sqrt{9 - a^2}}{a}, \sec \theta = \dfrac{3}{a}, \csc \theta = \dfrac{-3}{\sqrt{9 - a^2}}, \cot \theta = \dfrac{-a}{\sqrt{9 - a^2}}\)
- https://math.libretexts.org/Bookshelves/Precalculus/Trigonometry_(Yoshiwara)/11%3A_Answers_to_Selected_Exercises/11.06%3A_Circular_Functions63. \(\dfrac{\pi}{12}, \dfrac{5 \pi}{12}, \dfrac{3 \pi}{4}, \dfrac{13 \pi}{12}, \dfrac{17 \pi}{12}, \dfrac{7 \pi}{4}\) 9. \(\dfrac{\pi}{18}, \dfrac{7 \pi}{18}, \dfrac{13 \pi}{18}, \dfrac{19 \pi}{18}, ...63. \(\dfrac{\pi}{12}, \dfrac{5 \pi}{12}, \dfrac{3 \pi}{4}, \dfrac{13 \pi}{12}, \dfrac{17 \pi}{12}, \dfrac{7 \pi}{4}\) 9. \(\dfrac{\pi}{18}, \dfrac{7 \pi}{18}, \dfrac{13 \pi}{18}, \dfrac{19 \pi}{18}, \dfrac{25 \pi}{18}, \dfrac{31 \pi}{18}\) 23. \(\dfrac{5 \pi}{12}, \dfrac{7 \pi}{12}, \dfrac{13 \pi}{12}, \dfrac{5 \pi}{4}, \dfrac{7 \pi}{4}, \dfrac{23 \pi}{12}\) 27. \(0, \dfrac{\pi}{4}, \dfrac{\pi}{2}, \dfrac{3 \pi}{4}, \pi, \dfrac{5 \pi}{4}, \dfrac{7 \pi}{4}, 2 \pi\)
