Trigonometric ratios are mathematical functions used to relate the angles and side lengths of a right-angled triangle. These ratios are essential in trigonometry and have various applications in fields such as engineering, physics, and architecture.

The Primary Trigonometric Ratios

There are three primary trigonometric ratios: sine, cosine, and tangent, which are abbreviated as sin, cos, and tan, respectively. These ratios are defined in relation to the acute angles of a right-angled triangle.

Sine (sin θ): The sine 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 hypotenuse.
Cosine (cos θ): The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
Tangent (tan θ): The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Study Guide

When studying trigonometric ratios, it's important to understand the following key concepts:

Understanding right-angled triangles: Familiarize yourself with the components of a right-angled triangle, including the hypotenuse and the two shorter sides (legs).
Defining the trigonometric ratios: Learn how to apply the sine, cosine, and tangent ratios to the angles of a right-angled triangle.
Calculating trigonometric ratios: Practice using the trigonometric ratios to solve for unknown side lengths or angles in right-angled triangles.
Applying trigonometric ratios: Explore real-world applications of trigonometric ratios, such as calculating heights, distances, or angles in various contexts.

Additionally, familiarize yourself with the unit circle and the reciprocal trigonometric ratios (cosecant, secant, and cotangent) to deepen your understanding of trigonometric functions.

Remember to practice solving a variety of problems involving trigonometric ratios to build your proficiency in this topic.

By mastering trigonometric ratios, you will develop a strong foundation in trigonometry and be better equipped to tackle more advanced concepts in mathematics and related fields.

◂Math Worksheets and Study Guides Eighth Grade. Functions

The resources above cover the following skills:

Grade 8 Curriculum Focal Points (NCTM)

Algebra: Analyzing and representing linear functions and solving linear equations and systems of linear equations

Students use linear functions, linear equations, and systems of linear equations to represent, analyze, and solve a variety of problems. They recognize a proportion (y/x = k, or y = kx) as a special case of a linear equation of the form y = mx + b, understanding that the constant of proportionality (k) is the slope and the resulting graph is a line through the origin. Students understand that the slope (m) of a line is a constant rate of change, so if the input, or x-coordinate, changes by a specific amount, a, the output, or y-coordinate, changes by the amount ma. Students translate among verbal, tabular, graphical, and algebraic representations of functions (recognizing that tabular and graphical representations are usually only partial representations), and they describe how such aspects of a function as slope and y-intercept appear in different representations. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines that intersect, are parallel, or are the same line, in the plane. Students use linear equations, systems of linear equations, linear functions, and their understanding of the slope of a line to analyze situations and solve problems.

Connections to the Grade 8 Focal Points (NCTM)

Algebra: Students encounter some nonlinear functions (such as the inverse proportions that they studied in grade 7 as well as basic quadratic and exponential functions) whose rates of change contrast with the constant rate of change of linear functions. They view arithmetic sequences, including those arising from patterns or problems, as linear functions whose inputs are counting numbers. They apply ideas about linear functions to solve problems involving rates such as motion at a constant speed.