Name: trigonometric wave graphs and asymptotes
Uploaded: 2026-02-12T10:41:06.000Z
Duration: 1 min 30 s
Channel: JoVE

Trigonometric graphs are related to the unit circle. As a point moves counterclockwise around the circle, its vertical projection traces the sine wave.Similarly, the horizontal projection of the point&#8217;s motion around the circle produces the cosine wave.Sine and cosine graphs have a period of 2&#960;, a domain of all real numbers, and a range from -1 to 1.On the unit circle, the intersection of a vertical tangent line and a secant on the circle that passes through the origin traces the tangent function graph. The tangent function is defined everywhere except where the cosine is zero. Its range is all real numbers.The cosecant graph repeats every 2&#960;. It is undefined wherever the sine is zero. Its values are always less than -1 or greater than 1.The secant graph also has a 2&#960; period. It is undefined where cosine is zero and has the same range as cosecant.The cotangent graph is undefined where the sine is zero and has a range of all real numbers.Trigonometric graphs model periodic motion&#8212;like the height of a passenger on a Ferris wheel rising and falling over time traces a sine wave as the wheel rotates around its center.

graphs of trigonometric functions

Trigonometric functions exhibit periodic and symmetrical behavior, deeply rooted in the unit circle. The sine and cosine functions correspond to the vertical and horizontal projections, respectively, of a point rotating counterclockwise around the circle. These functions trace smooth, repeating waveforms with identical periods and bounded ranges. The tangent function is defined as the ratio of sine to cosine and produces an unbounded curve that repeats every units, with vertical asymptotes where cosine is zero. Conversely, the cotangent function is the ratio of cosine to sine and behaves similarly but with asymptotes where sine is zero.The secant and cosecant functions, which are the reciprocals of cosine and sine, respectively, form curves that arch away from the x-axis, becoming undefined at the zeros of their base functions. These graphs display vertical asymptotes and do not intersect the x-axis. Despite their different appearances, all these functions exhibit regular periodicity and distinct symmetries&#8212;sine, tangent, and cosecant being odd functions symmetric about the origin, while cosine and secant are even functions symmetric about the y-axis. Their domain restrictions arise from division by zero in their definitions, and their ranges reflect whether they are bounded or unbounded.Graph Properties of Trigonometric Functions<table class='ck-table-resized'><tbody><tr><td>Function</td><td>Period</td><td>Domain</td><td>Range</td></tr><tr><td>Sine</td><td>2</td><td>All real numbers</td><td>[&#8722;1,1]</td></tr><tr><td>Cosine</td><td>2</td><td>All real numbers</td><td>[&#8722;1,1]</td></tr><tr><td>Tangent</td><td>&#160;</td><td>All real numbers x=&#960;2+n&#960;, where n is any integer</td><td>All real numbers</td></tr><tr><td>Cotangent</td><td>2</td><td>All real numbers x=n&#960;, where n is any integer</td><td>All real numbers</td></tr><tr><td>Secant</td><td>2</td><td>All real numbers x=&#960;2+n&#960;, where n is any integer</td><td>All real numbers &#8804; &#8211;1 or &#8804; &#8211;1</td></tr><tr><td>Cosecant</td><td>2</td><td>All real numbers x=n&#960;, where n is any integer</td><td>All real numbers &#8804; &#8211;1 or &#8804; &#8211;1</td></tr></tbody></table>

Graphs of Trigonometric Functions

Trigonometric functions exhibit periodic and symmetrical behavior, deeply rooted in the unit circle. The sine and cosine functions correspond to the ...

Education

JoVE Core

Math Fundamentals

Unit

Trigonometry

KEY TERMS AND CONCEPTS

trigonometric wave graphs and asymptotes

Trigonometric functions create repeating waves and curved graphs with clear symmetry. Their shapes come from the unit circle, where a point moves counterclockwise and its vertical and horizontal positions form the sine and cosine values.

Sine and cosine graphs are smooth waves with the same period, or repeat length. Their ranges stay bounded between -1 and 1. Sine, cosine, and the other trig functions all show regular patterns, but each one has its own graph shape.

Tangent is the ratio of sine to cosine, so its graph is unbounded. It repeats every period and has vertical asymptotes, which are lines the graph approaches but does not cross, where cosine equals zero. Cotangent is the ratio of cosine to sine and has similar behavior, with asymptotes where sine equals zero.

Secant and cosecant are reciprocal functions. Secant is the reciprocal of cosine, and cosecant is the reciprocal of sine. Their graphs arch away from the x-axis, become undefined at zeros of the base functions, and do not intersect the x-axis.

These trig graphs also have different symmetries. Sine, tangent, and cosecant are odd functions, so they are symmetric about the origin. Cosine and secant are even functions, so they are symmetric about the y-axis. Their domains and ranges depend on whether the function is bounded or unbounded, and the table of properties shows the period and domain restrictions for sine, cosine, tangent, cotangent, secant, and cosecant.

Trigonometric Wave Graphs and Asymptotes

radian and degree measure

Degrees to Radians in Circle Measure

Degrees and radians are the two main units used to measure angles. They both describe rotation around a fixed point. A full turn equals 360 degrees or 2π radians, depending on the unit used. Degrees divide a circle into 360 equal parts. This makes them easy to use in everyday math. They are common in navigation, design, and basic geometry. Radians come from the geometry of a circle. One radian is the angle made when the arc length matches the radius of the circle. Since a circle’s...

trigonometric functions of real numbers

Unit Circle Trig: Signs and Symmetry

The unit circle is the basis for trigonometric functions on real numbers. It is a circle with radius one centered at the origin of the coordinate plane. On this circle, arc length means the distance measured along the circumference between two points. Each real number t is shown as an arc length measured counterclockwise from the positive x-axis. A point on the unit circle has coordinates (x, y). These coordinates define cosine and sine. The x-coordinate equals cos t, and the y-coordinate...

Trigonometric functions create repeating waves and curved graphs with clear symmetry. Their shapes come from the unit circle, where a point moves counterclockwise and its vertical and horizontal positions form the sine and cosine values. Sine and cosine graphs are smooth waves with the same period, or repeat length. Their ranges stay bounded between -1 and 1. Sine, cosine, and the other trig functions all show regular patterns, but each one has its own graph shape. Tangent is the ratio of...

trigonometry of right triangles

Trig Ratios for Right Triangle Sides

Right triangle trigonometry uses sine, cosine, and tangent to connect an acute angle with the sides of the triangle. These trig ratios describe how side lengths relate to one another in right-angled geometry. They give a clear way to measure the links between angles and sides. Sine compares the side opposite an angle to the hypotenuse. Cosine compares the side adjacent to the angle to the hypotenuse. Tangent compares the opposite side to the adjacent side. Each ratio depends on the angle, not...

trigonometric functions: problem solving

Measuring Rocket Height with Trigonometry

Measuring rocket height with trigonometry uses a fixed ground point and a changing angle of elevation. An observer stands a known horizontal distance from the launch site and watches the object rise. The angle between the ground and the line of sight gives the key information for finding height. The tangent function is the main tool for this calculation. Tangent is the ratio of the vertical side to the adjacent side in a right triangle. When the ground distance is fixed, the object’s height...

the law of sines

Solving Oblique Triangles with Sines

Solving oblique triangles uses the Law of Sines. An oblique triangle is a triangle with no right angle. These triangles need a different trigonometric method because the Pythagorean theorem works only for right triangles. The Law of Sines links each side of a triangle to the sine of its opposite angle. In triangle ABC, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. The ratios a/sin A, b/sin B, and c/sin C are constant. This relationship helps find...

the law of cosines

Using Cosines to Find Triangle Sides

The Law of Cosines helps find unknown sides and angles in any triangle. It is a key trig formula for non-right triangles, where the Pythagorean Theorem does not give a direct answer. The law links side lengths to the cosine of an angle, so students can solve triangles with missing measurements. For a triangle with sides a, b, and c and opposite angles A, B, and C, the formula is written as a squared equals b squared plus c squared minus 2bc cosine A. Similar equations can be used for the other...

trigonometric identities i

Unit Circle Trig Identities and Sum Formulas

Trigonometric identities link trigonometric functions and stay true for every angle in their domains. A key example is the Pythagorean identity, which comes directly from the geometry of the unit circle. For an angle θ, a point on the unit circle has coordinates (cos θ, sin θ). Because the radius is 1, the Pythagorean Theorem gives the basic identity. That same identity can be rearranged to create another useful form. If it is divided by cos²θ, assuming cos θ is not 0, it leads to a second...

trigonometric identities ii

Double-Angle and Half-Angle Formulas

Double-angle and half-angle trigonometric formulas connect angles that are multiplied or divided by two. They come from the sum and difference formulas, so they build on earlier trigonometry skills. These identities help simplify expressions, solve equations, and evaluate integrals. The double-angle formulas come from substituting equal angles into the sum formulas. For sine and cosine, they link the value at 2θ to the value at θ. The tangent double-angle identity follows the same idea. These...

trigonometric identities iii

Cofunction Identities and Complementary Angles

Cofunction identities show how trigonometric functions change for complementary angles. Complementary angles add up to 90°. These identities are a useful part of trigonometry for grades 9–12. The idea becomes clear on the unit circle. Every angle θ, measured counterclockwise from the positive x-axis, matches a point with coordinates (cos θ, sin θ). The cosine gives the horizontal component, and the sine gives the vertical component of the terminal side. The same point can also be described...

inverse trigonometric functions

Inverse Trig Functions and Angle Finding

Inverse trigonometric functions are used to find angles from trig ratios. Standard trigonometric functions turn angles into ratios, while inverse trigonometric functions reverse that process and map a ratio back to its angle. That makes them useful whenever a problem gives distances or side lengths instead of an angle. A function must be one-to-one to have a valid inverse. This means each input must match only one output. Trigonometric functions are not naturally one-to-one over their full...

trigonometric equations

Solving Trigonometric Equations

Trigonometric equations use one or more trigonometric functions, and they often appear in mathematical modeling. Some are identities, which are true for every value of the variable. Others are conditional equations, which are true only for specific values. Solving these equations usually combines algebra with the basic properties of trigonometric functions. Some equations look like standard algebra problems, so factoring can work well. For example, a quadratic form in sin x can be factored...

key terms and concepts

Explore sine, cosine, tangent, cotangent, secant, and cosecant graphs, including periods, asymptotes, symmetry, domains, and ranges for high school math.

Trigonometric Wave Graphs and Asymptotes | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Trigonometric Wave Graphs and Asymptotes

Description

Transcript

Tags