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- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/11%3A_Analytic_TrigonometryThus, in the Mercator projection, when a globe is ”unwrapped” on to a rectangular map, the parallels need to be stretched to the length of the Equator. Because the radius of the circle of latitude \(\...Thus, in the Mercator projection, when a globe is ”unwrapped” on to a rectangular map, the parallels need to be stretched to the length of the Equator. Because the radius of the circle of latitude \(\theta\) is \(R \cos \theta\), the corresponding parallel on the map must be stretched by a factor of \(\frac{1}{\cos \theta}\). And because the secant is the reciprocal of the cosine, the scale factor is proportional to the secant of the latitude.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/13%3A_Non-Right_Triangle_Trigonometry/13.05%3A_Vectors_-_An_Algebraic_ApproachThis section covers vectors from an algebraic perspective, including expressing vectors in coordinate form, converting between geometric and coordinate forms, and performing vector operations such as ...This section covers vectors from an algebraic perspective, including expressing vectors in coordinate form, converting between geometric and coordinate forms, and performing vector operations such as scalar multiplication and addition. It introduces unit vectors, discusses their significance, and explores their use in practical applications like force and equilibrium problems. The section includes examples and exercises to reinforce understanding of algebraic vector manipulation.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/13%3A_Non-Right_Triangle_Trigonometry/13.03%3A_AreasThis section explains how to find the area of a triangle using trigonometric methods. It covers two such formulas - one for triangles with two known sides and the included angle, and the other (Heron'...This section explains how to find the area of a triangle using trigonometric methods. It covers two such formulas - one for triangles with two known sides and the included angle, and the other (Heron's Formula) for when all three sides are known. Practical examples and exercises illustrate the application of these formulas.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/13%3A_Non-Right_Triangle_Trigonometry/13.01%3A_The_Law_of_SinesThis section covers the Law of Sines, including its derivation, and how to use it to find missing sides and angles in oblique triangles. It includes examples, practical applications, and solving trian...This section covers the Law of Sines, including its derivation, and how to use it to find missing sides and angles in oblique triangles. It includes examples, practical applications, and solving triangles, addressing the ambiguous case and its implications. The section emphasizes step-by-step solutions and advises on efficient calculation techniques to minimize errors.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/12%3A_Trigonometric_Equations/12.02%3A_The_Remaining_Inverse_Trigonometric_FunctionsThis section introduces the inverse trigonometric functions for cotangent, secant, and cosecant. It covers their definitions, properties, and domains, along with examples of evaluating these functions...This section introduces the inverse trigonometric functions for cotangent, secant, and cosecant. It covers their definitions, properties, and domains, along with examples of evaluating these functions exactly and using technology. The section also explains how to simplify compositions and expressions involving these inverse functions, and includes practical exercises to reinforce learning.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/06%3A_Triangles_and_Circles/6.03%3A_Similar_TrianglesThis section focuses on similar triangles, highlighting their definition, congruence, and applications. It introduces the concept of similarity, demonstrates how to identify similar triangles through ...This section focuses on similar triangles, highlighting their definition, congruence, and applications. It introduces the concept of similarity, demonstrates how to identify similar triangles through examples, and explores their properties, including proportional sides and angles. Key topics include using proportions to solve problems involving similar triangles, understanding similar right triangles, and dealing with overlapping triangles.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/11%3A_Analytic_Trigonometry/11.01%3A_Basic_Trigonometric_Identities_and_Proof_TechniquesThis section reviews basic trigonometric identities and proof techniques. It covers Reciprocal, Ratio, Pythagorean, Symmetry, and Cofunction Identities, providing definitions and alternate forms. The ...This section reviews basic trigonometric identities and proof techniques. It covers Reciprocal, Ratio, Pythagorean, Symmetry, and Cofunction Identities, providing definitions and alternate forms. The section emphasizes understanding over memorization and offers a strategy for proving identities, including steps like simplifying complex expressions, converting to sines and cosines, and working with each side of an equation separately.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/12%3A_Trigonometric_Equations/12.01%3A_The_Fundamental_Inverse_Trigonometric_FunctionsThis section introduces inverse trigonometric functions, focusing on arcsine, arccosine, and arctangent. It covers their definitions, domains, and ranges, and how to evaluate these functions both exac...This section introduces inverse trigonometric functions, focusing on arcsine, arccosine, and arctangent. It covers their definitions, domains, and ranges, and how to evaluate these functions both exactly and approximately using technology. The section also explores compositions involving trigonometric and inverse trigonometric functions, simplifying expressions, and applying these concepts in modeling and problem-solving.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/11%3A_Analytic_Trigonometry/11.02%3A_Sum_and_Difference_IdentitiesThis section covers the Sum and Difference Identities for sine, cosine, and tangent. It explains how to use these identities to find the exact values of trigonometric functions at non-special angles, ...This section covers the Sum and Difference Identities for sine, cosine, and tangent. It explains how to use these identities to find the exact values of trigonometric functions at non-special angles, prove equations are identities, and simplify expressions. The section includes detailed examples and exercises for practical application, demonstrating the derivation and usage of these identities in various trigonometric problems.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/06%3A_Triangles_and_Circles/6.01%3A_Angles_and_Basic_GeometryBefore jumping into Trigonometry, we need to build a solid foundation. This section provides the fundamental building blocks for working with the most basic quantity in Trigonometry - the angle. We de...Before jumping into Trigonometry, we need to build a solid foundation. This section provides the fundamental building blocks for working with the most basic quantity in Trigonometry - the angle. We delve into as much detail about angles as we dare, without introducing unnecessary topics. We cover a little bit of required Geometry for success in Trigonometry, and wrap things up with a brief geometric review of circles (another foundational topic for Trigonometry).
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/10%3A_Graphs_of_the_Trigonometric_FunctionsIf two musical instruments play the same note, the notes have the same pitch, but they sound different because the amplitudes of each of the harmonics is different for the two instruments. Fourier ana...If two musical instruments play the same note, the notes have the same pitch, but they sound different because the amplitudes of each of the harmonics is different for the two instruments. Fourier analysis is also used in X-ray crystallography to reconstruct a crystal structure from its diffraction pattern, and in nuclear magnetic resonance spectroscopy to determine the mass of ions from the frequency of their motion in a magnetic field.
