Module 2 builds on students' previous work with units and with functions from Algebra I, and with trigonometric ratios and circles from high school Geometry. The heart of the module is the study of precise definitions of sine and cosine (as well as tangent and the co-functions) using transformational geometry from high school Geometry. This precision leads to a discussion of a mathematically natural unit of rotational measure, a radian, and students begin to build fluency with the values of the trigonometric functions in terms of radians. Students graph sinusoidal and other trigonometric functions, and use the graphs to help in modeling and discovering properties of trigonometric functions. The study of the properties culminates in the proof of the Pythagorean identity and other trigonometric identities.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

This 8-minute video lesson looks at finding the equation of the line tangent to f(x)=xe^x when x=1.

An interactive applet and associated web page that demonstrate how to construct a tangent to a circle at a given point on the circle, using compass and straightedge. It starts with the given circle and point, and in an animation that can run without stopping or single stepped, shows how to construct the tangent line through that point. The text on the page contains written step-by-step instructions that can be printed. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that demonstrate how to construct the two possible tangents to a circle from an external given point, using compass and straightedge. It starts with the given circle and external point, and in an animation that can run without stopping or single stepped, shows how to construct the two tangent lines. The text on the page contains written step-by-step instructions that can be printed. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

The purpose of this lesson is to discover the relationship between a radius and a tangent, then solve related problems.

Included is a YouTube video to support Grade 9 Blended Learning Math - Unit 8.1: Circle Geometry - Properties of Tangents to a Circle.

Students explore the concept of similar right triangles and how they apply to trigonometric ratios. Use this lesson as a refresher of what trig ratios are and how they work. In addition to trigonometry, students explore a clinometer app on an Android® or iOS® device and how it can be used to test the mathematics underpinning trigonometry. This prepares student for the associated activity, during which groups each put a clinometer through its paces to better understand trigonometry.

An interactive applet and associated web page that demonstrate a tangent to a circle. (not trig). The applet shows a circle and a tangent line. The center point and the tangent contact point are both draggable. As you drag each, the figure changes to ensure that the line is always tangential to the circle. The line from the center to the tangent point is shown and the angle is shown to be always 90 degrees no matter what you do. The perpendicular and its angle can be turned off for class discussion. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that demonstrate the tangent to an ellipse. An ellipse is shown and a tangent line that touches it at the one point. The user can drag the tangent point around the ellipse and the tangent follows. The ellipse can be reshaped by dragging the foci. The applet shows the fact that the two lines from the foci to the tangent meet it at equal angles. The web page has the written properties of the tangent and links to other pages defining the various properties of an ellipse and to some ellipse constructions. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

The Tippy Tap hand-washing station is an inexpensive and effective device used extensively in the developing world. One shortcoming of the homemade device is that it must be manually refilled with water and therefore is of limited use in high-traffic areas. In this activity, student teams design, prototype and test piping systems to transport water from a storage tank to an existing Tippy Tap hand-washing station, thereby creating a more efficient hand-washing station. Through this example service-learning engineering project, students learn basic fluid dynamic principles that are needed for creating efficient piping systems.

Students investigate the relationships between angles and side lengths in right triangles with the help of materials found in the classroom and a mobile device. Using all or part of a meter stick or dowel and text books or other supplies, students build right triangles and measure the angles using a clinometer application on an Android® (phone or tablet) or iOS® device (iPhone® or iPad®). Then they are challenged to create a triangle with a given side length and one angle. The electronic device is used to measure the accuracy of their constructions.

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

El módulo 2 se basa en el trabajo previo de los estudiantes con unidades y con funciones del álgebra I, y con relaciones y círculos trigonométricos de la geometría de la escuela secundaria. El corazón del módulo es el estudio de definiciones precisas de seno y coseno (así como tangente y las cofunciones) utilizando geometría transformacional de la geometría de la escuela secundaria. Esta precisión lleva a una discusión de una unidad matemáticamente natural de medida rotacional, un radian, y los estudiantes comienzan a desarrollar fluidez con los valores de las funciones trigonométricas en términos de radianes. Los estudiantes grafican funciones trigonométricas sinusoidales y otras, y usan los gráficos para ayudar a modelar y descubrir propiedades de las funciones trigonométricas. El estudio de las propiedades culmina en la prueba de la identidad pitagórica y otras identidades trigonométricas.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.

English Description:
Module 2 builds on students' previous work with units and with functions from Algebra I, and with trigonometric ratios and circles from high school Geometry. The heart of the module is the study of precise definitions of sine and cosine (as well as tangent and the co-functions) using transformational geometry from high school Geometry. This precision leads to a discussion of a mathematically natural unit of rotational measure, a radian, and students begin to build fluency with the values of the trigonometric functions in terms of radians. Students graph sinusoidal and other trigonometric functions, and use the graphs to help in modeling and discovering properties of trigonometric functions. The study of the properties culminates in the proof of the Pythagorean identity and other trigonometric identities.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.