Tangent in Geometry

In geometry, the tangent of an angle in a right-angled triangle is the ratio of the length of the side opposite the angle to the length of the adjacent side. It is one of the primary trigonometric functions and is denoted as tan(θ), where θ represents the angle.

Formula for Tangent

The formula for tangent is given by:

tan(θ) = Opposite / Adjacent

This formula can be used to calculate the tangent of an angle in a right-angled triangle when the lengths of the opposite and adjacent sides are known.

Properties of Tangent

Some important properties of the tangent function include:

The tangent function is periodic with a period of π (180 degrees).
It is an odd function, which means that tan(-θ) = -tan(θ).
The tangent function is undefined at odd multiples of π/2 (90 degrees) due to division by zero.

Applications of Tangent

The tangent function has various applications in real-world problems, particularly in fields such as engineering, physics, and architecture. It is used to solve for unknown angles and distances in right-angled triangles, as well as in the analysis of periodic phenomena such as sound and light waves.

Study Guide

To master the concept of tangent, it is important to practice using the formula and understanding its properties. Here are some key steps to include in your study guide:

Understand the definition of tangent and its relation to right-angled triangles.
Practice using the tangent formula to calculate the value of the tangent of different angles.
Explore the periodic nature of the tangent function and its behavior for different angle values.
Solve problems involving real-world applications of the tangent function, such as finding distances and angles in practical scenarios.
Review the properties of the tangent function, including its periodicity and behavior at specific angle values.

By following this study guide and practicing problems related to the tangent function, you can develop a strong understanding of this fundamental trigonometric concept.

◂Math Worksheets and Study Guides Seventh Grade. Exploring Area and Surface Area

Study Guide

Exploring Area and Surface Area

Worksheet/Answer key

Exploring Area and Surface Area

Worksheet/Answer key

Exploring Area and Surface Area

Worksheet/Answer key

Exploring Area and Surface Area

Worksheet/Answer key

Exploring Area and Surface Area

The resources above cover the following skills:

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.

Use visualization, spatial reasoning, and geometric modeling to solve problems.

Use geometric models to represent and explain numerical and algebraic relationships.

Measurement (NCTM)

Apply appropriate techniques, tools, and formulas to determine measurements.

Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.

Develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes.

Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders.

Grade 7 Curriculum Focal Points (NCTM)

Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areas and volumes of three-dimensional shapes

By decomposing two- and three-dimensional shapes into smaller, component shapes, students find surface areas and develop and justify formulas for the surface areas and volumes of prisms and cylinders. As students decompose prisms and cylinders by slicing them, they develop and understand formulas for their volumes (Volume = Area of base x Height). They apply these formulas in problem solving to determine volumes of prisms and cylinders. Students see that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram. They select appropriate two- and three-dimensional shapes to model real-world situations and solve a variety of problems (including multi-step problems) involving surface areas, areas and circumferences of circles, and volumes of prisms and cylinders.