Search
- Filter Results
- Location
- Classification
- Include attachments
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/04%3A_Radian_Measure_and_the_Circular_FunctionsRadians connect the measure of an angle with the arc length it cuts out on a circle. The circumference of a circle is \(2 \pi\) times the length of its radius, so when the circle has turned through on...Radians connect the measure of an angle with the arc length it cuts out on a circle. The circumference of a circle is \(2 \pi\) times the length of its radius, so when the circle has turned through one complete revolution, it has traveled a distance of \(2 \pi\) units. In radian measure, the angle through which the circle has turned is equal to the distance the circle has traveled or, put another way, the length of the arc that has unrolled on the line.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/06%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/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/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.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/05%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.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/09%3A_Radian_Measure_and_the_Circular_FunctionsRadians connect the measure of an angle with the arc length it cuts out on a circle. The circumference of a circle is \(2 \pi\) times the length of its radius, so when the circle has turned through on...Radians connect the measure of an angle with the arc length it cuts out on a circle. The circumference of a circle is \(2 \pi\) times the length of its radius, so when the circle has turned through one complete revolution, it has traveled a distance of \(2 \pi\) units. In radian measure, the angle through which the circle has turned is equal to the distance the circle has traveled or, put another way, the length of the arc that has unrolled on the line.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/03%3A_Right_Triangle_TrigonometryUsing Trigonometry and the measured length of just one side, the lengths of the other sides can be calculated. Each calculated distance became the base side of another triangle used to calculate the d...Using Trigonometry and the measured length of just one side, the lengths of the other sides can be calculated. Each calculated distance became the base side of another triangle used to calculate the distance to another point, which in turn started another triangle. Because of the size of the area to be surveyed, the surveyors did not triangulate the whole of India.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/13%3A_Non-Right_Triangle_TrigonometryBy tracking the motions of the stars, early astronomers could identify the summer and winter solstices and the equinoxes. It gives the shortest distance between any two points on a sphere, and is the ...By tracking the motions of the stars, early astronomers could identify the summer and winter solstices and the equinoxes. It gives the shortest distance between any two points on a sphere, and is the analogue of a straight line on a plane. "You, who wish to study great and wondrous things, who wonder about the movement of the stars, must read these theorems about triangles. ... For no one can bypass the science of triangles and reach a satisfying knowledge of the stars.”
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/08%3A_Non-Right_Triangle_TrigonometryBy tracking the motions of the stars, early astronomers could identify the summer and winter solstices and the equinoxes. It gives the shortest distance between any two points on a sphere, and is the ...By tracking the motions of the stars, early astronomers could identify the summer and winter solstices and the equinoxes. It gives the shortest distance between any two points on a sphere, and is the analogue of a straight line on a plane. "You, who wish to study great and wondrous things, who wonder about the movement of the stars, must read these theorems about triangles. ... For no one can bypass the science of triangles and reach a satisfying knowledge of the stars.”
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/01%3A_Triangles_and_CirclesA circle is the simplest curve, and a triangle is the simplest polygon - the one with the fewest sides. The side opposite angle \(A\) is called \(a\), the side opposite angle \(B\) is called \(b\), an...A circle is the simplest curve, and a triangle is the simplest polygon - the one with the fewest sides. The side opposite angle \(A\) is called \(a\), the side opposite angle \(B\) is called \(b\), and the side opposite angle \(C\) is called \(c\). IF: \(\quad a, b\), and \(c\) are the sides of a right triangle, and \(c\) is the hypotenuse \(a^2+b^2=c^2\) The "if" part of the theorem is called the hypothesis, and the "then" part is called the conclusion.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/08%3A_Right_Triangle_TrigonometryUsing Trigonometry and the measured length of just one side, the lengths of the other sides can be calculated. Each calculated distance became the base side of another triangle used to calculate the d...Using Trigonometry and the measured length of just one side, the lengths of the other sides can be calculated. Each calculated distance became the base side of another triangle used to calculate the distance to another point, which in turn started another triangle. Because of the size of the area to be surveyed, the surveyors did not triangulate the whole of India.
