Trigonometric functions

This course is a part of Algebra (all content), a 20-course Topic series from Khan Academy.

This topic covers: - Unit circle definition of trig functions - Trig identities - Graphs of sinusoidal & trigonometric functions - Inverse trig functions & solving trig equations - Modeling with trig functions - Parametric functions

This course contains 18 segments:

Introduction to radians

Learn about radians, which are the official unit of measurement for angles in algebra (in contrast to degrees, which are used in geometry).

The unit circle definition of sine, cosine, & tangent

Learn how the trigonometric ratios are extended to all real numbers using algebra. Start solving simple problems that involve this new definition of the trigonometric functions.

The graphs of sine, cosine, & tangent

Learn how the graphs of y=sin(θ), y=cos(θ), and y=tan(θ) look, using the unit circle definition of the functions.

Basic trigonometric identities

Learn about very useful trigonometric identities that arise by considering different properties of the unit circle definition.

Trigonometric values of special angles

Learn how to find the trigonometric values of some special angles without the use of a calculator.

Pythagorean identity

Prove the Pythagorean trigonometric identity for all real numbers and use it in order to solve problems.

Introduction to amplitude, midline, & extrema of sinusoidal functions

Learn about very important features of sinusoidal functions: the amplitude and the midline. Learn how they relate to the extremum points of the function.

Finding amplitude & midline of sinusoidal functions from their formulas

Learn how to find the amplitude and the midline of the graph of a sinusoidal function from its formula. For example, find the amplitude and the midline of f(x)=3*sin(2x-1)+5.

Period of sinusoidal functions

Learn about the period of sinusoidal functions: how it relates to extremum points and the midline, and how to find it from the formula of the function. For example, find the period of f(x)=3*sin(2x-1)+5.

Graphing sinusoidal functions

Learn how to draw the graph of sinusoidal functions. For example, draw the graph of f(x)=-2*cos(πx)-7.

Constructing sinusoidal functions

Learn how to find the formula of a sinusoidal function from its graph or a few selected features. Model real-world situations with sinusoidal functions.

The inverse trigonometric functions

Learn about the inverse functions of sine, cosine, and tangent, and how they are defined even though the functions are not really invertible. These functions will be very helpful when you will solve trigonometric equations.

Solving basic sinusoidal equations

Learn how to solve equations of the form sin(x)=d or cos(x)=d where d is any number. For example, solve sin(x)=0.6.

Solving advanced sinusoidal equations

Learn how to solve equations of the form a*sin(bx+c)=d or a*cos(bx+c)=d where a, b, c, and d are any number. For example, solve 5*sin(2x-1)=3.

Solving sinusoidal models

Solve word problems that involve sinusoidal modeling functions.

Introduction to the trigonometric angle addition identities

Learn about the trigonometric angle addition identities, which help us discuss the trig values of sums of angles in terms of the trig values of the individual angles. For example, we can express sin(x+y) in terms of sin(x), sin(y), cos(x), and cos(y).

Using trigonometric identities to solve problems

Learn about different ways in which trigonometric identities, such as the Pythagorean identity and the angle addition identities, can be used to solve problems.

Parametric equations

Here we will explore representing our x's and y's in terms of a third variable or parameter (often 't'). Not only can we describe new things, but it can be super useful for describing things like particle motion in physics.