ClipArt ETC
Angles used to illustrate the sum and difference of two angles and trig identities.

Angles Used to Illustrate Sum and Difference of Two Angles

Angles used to illustrate the sum and difference of two angles and trig identities.

Illustration of two triangles, showing the sine of the sum of two acute angles expressed in terms of the sines and cosines of the angles.

Sum of 2 Acute Angles

Illustration of two triangles, showing the sine of the sum of two acute angles expressed in terms of…

Lines drawn to horizontal to form triangle ratios (reference triangles, angles).

Reference Angles/Triangles Formed by Angles in Quadrants

Lines drawn to horizontal to form triangle ratios (reference triangles, angles).

Lines drawn to horizontal to form triangle ratios (reference triangles, angles). Axes, quadrants, abscissa, ordinate, and distance labeled.

Reference Angles/Triangles Formed by Angles in Quadrants With Labels

Lines drawn to horizontal to form triangle ratios (reference triangles, angles). Axes, quadrants, abscissa,…

Illustration of one possible outcome (no triangle occurs) when discussing the ambiguous case using the Law of Sines. In this case, side a is less than the height (bsinα).

Ambiguous Case

Illustration of one possible outcome (no triangle occurs) when discussing the ambiguous case using the…

Illustration of one possible outcome (1 triangle occurs) when discussing the ambiguous case using the Law of Sines. In this case, side a is equal to the height (bsinα).

Ambiguous Case

Illustration of one possible outcome (1 triangle occurs) when discussing the ambiguous case using the…

Illustration of one possible outcome (2 triangles occur) when discussing the ambiguous case using the Law of Sines. In this case, side a is greater than the height (bsinα).

Ambiguous Case

Illustration of one possible outcome (2 triangles occur) when discussing the ambiguous case using the…

Circle with 36 degree angles marked. This diagram can be used with the following trig problem: Locate the centers of the holes B and C by finding the distance each is to the right and above the center O. The radius of the circle is 1.5 inches. Compute correct to three decimal places.

Circle With 36 degree Angles and Radius 1.5 in.

Circle with 36 degree angles marked. This diagram can be used with the following trig problem: Locate…

Circle modeling the earth. O is the center of the earth, r the radius of the earth, and h the height of the point P above the surface; it is required to find the distance from the point P to the horizon at A.

Circle With Center o and Radius r with point P

Circle modeling the earth. O is the center of the earth, r the radius of the earth, and h the height…

Circle with chord AB=2 ft. and radius OA = 3 ft.. Triangle AOC is a right triangle. Angle AOC=half angle AOB, and the central angle AOB has the same measure as the arc AnB.

Circle With a Chord of 2 ft. and a Radius of 3 ft.

Circle with chord AB=2 ft. and radius OA = 3 ft.. Triangle AOC is a right triangle. Angle AOC=half angle…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. The circle is divided into four quadrants by the x- and y- axes. The circle can be labeled and used to find the six trigonometric values (sin, cos, tan, cot, sec, csc, cot) at each of the quadrantal angles.

Unit Circle

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. The…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. All quadrantal angles are given in radian measure in terms of pi.

Unit Circle Labeled At Quadrantal Angles

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. All quadrantal angles and angles that have reference angles of 30°, 45°, and 60° are given in radian measure in terms of pi. At each quadrantal angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled At Special Angles

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. All quadrantal angles and angles that have reference angles of 30°, 45°, and 60° are given in radian measure in terms of pi.

Unit Circle Labeled At Special Angles

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. At 30° increments, the angles are given in both radian and degree measure. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled In 30° Increments With Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. The circle is marked and labeled in both radians and degrees in 45° increments. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled In 45 ° Increments

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. At 45° increments, the angles are given in both radian and degree measure. At each quadrantal angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled In 45° Increments With Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. At 45° increments, the angles are given in both radian and degree measure. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled In 45° Increments With Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. All quadrantal angles are given in radian measure in terms of pi. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Quadrantal Angles And Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. All…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. All quadrantal angles are given in degree measure. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Quadrantal Angles And Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. All quadrantal angles are given in both radian and degree measure. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Quadrantal Angles And Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. At each quadrantal angle, the coordinates are given, but not the angle measure. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Quadrantal Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. At each…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. At each quadrantal angle, the coordinates are given, but not the angle measure. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Quadrantal Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. The circle is marked and labeled in radians. All quadrantal angles and angles that have reference angles of 30°, 45°, and 60° are given in radian measure in terms of pi. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Special Angles And Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. The circle is marked and labeled in both radians and degrees at all quadrantal angles and angles that have reference angles of 30°, 45°, and 60°. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Special Angles And Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with the x- and y-axes indicated. The circle is marked and labeled in both radians and degrees at all quadrantal angles and angles that have reference angles of 30°, 45°, and 60°. At each angle, the coordinates are given. These coordinates can be used to find the six trigonometric values/ratios. The x-coordinate is the value of cosine at the given angle and the y-coordinate is the value of sine. From those ratios, the other 4 trigonometric values can be calculated.

Unit Circle Labeled With Special Angles And Values

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane with…

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. All quadrantal angles and angles that have reference angles of 30°, 45°, and 60° are marked from the origin, but no values are given.

Unit Circle Marked At Special Angles

Illustration of a unit circle (circle with a radius of 1) superimposed on the coordinate plane. All…

The coversedsine is equal to 1 minus the sine. In this figure, AE is parallel to OB; hence, EO = AC = sine of angle AOC, when radius OA - 1. Therefore, Coversedsine = 1 - sine = 1 - AC/OA = 1 - AC = 1 - EO = ER, when OA = 1.

Versedsine and Coversedsine in Circle

The coversedsine is equal to 1 minus the sine. In this figure, AE is parallel to OB; hence, EO = AC…

Illustration of the projection of point P as it moves around a vertical circle of radius 3 in. in a counterclockwise direction. It start with the radius in a horizontal position and moves with an angular velocity of one revolution in 10 seconds.

Projection of Points in Circular Motion

Illustration of the projection of point P as it moves around a vertical circle of radius 3 in. in a…

Illustration of the projection of point P as it moves around a vertical circle of radius 2 ft. in a counterclockwise direction. It start with the radius in a horizontal position and moves with an angular velocity of one revolution in .5 seconds.

Projection of Points in Circular Motion

Illustration of the projection of point P as it moves around a vertical circle of radius 2 ft. in a…

Coordinate axis with angle XOP equal to theta, Θ, and angle XOQ=180 - Θ. From any point in the terminal side of XOP, as B, a perpendicular can be drawn, AB, to the x-axis; and from D, any point in the terminal side o f XOQ, perpendicular CD can be drawn to the x-axis. The right triangles OAB and OCD are similar. Also, OA, AB, OB, CD, and OD are positive, while OC is negative.

Coordinate Axis With Angles, Lines, and Perpendiculars Drawn

Coordinate axis with angle XOP equal to theta, Θ, and angle XOQ=180 - Θ. From any point in the terminal…

Angle XOP=Θ and angle XOQ=- Θ. From a point in the terminal side of each a perpendicular line is drawn to the x-axis. The right triangles OAB and OAC thus formed are similar, and have all their sides positive except AC, which is negative.

Coordinate Axis With Perpendiculars Drawn To Form Similar Right Triangles From Positive and Negative Theta, Θ

Angle XOP=Θ and angle XOQ=- Θ. From a point in the terminal side of each a perpendicular line is drawn…

Angle XOP=Θ and angle XOQ=90+Θ. From a point in the terminal side of each a perpendicular line is drawn to the x-axis. The right triangles AOB and OCD thus formed are similar, and have all their sides positive except OC

Coordinate Axis With Perpendiculars Drawn To Form Similar Right Triangles

Angle XOP=Θ and angle XOQ=90+Θ. From a point in the terminal side of each a perpendicular line is…

Cosine curve plotted from negative pi to 2 pi. Graph of y=cos x.

Cosine Curve y=cos x

Cosine curve plotted from negative pi to 2 pi. Graph of y=cos x.

Illustration that can be used to show that if the cotangent of an angle is negative the angle must terminate in either the second or fourth quadrant.

Negative Cotangent Angles

Illustration that can be used to show that if the cotangent of an angle is negative the angle must terminate…

Illustration showing coterminal angles of 0° and -360°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, both angles are labeled with the proper degree measure.

0° and -360° Coterminal Angles

Illustration showing coterminal angles of 0° and -360°. Coterminal angles are angles drawn in…

Illustration showing coterminal angles of 0° and -360°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, neither angle is labeled.

0° and -360° Coterminal Angles

Illustration showing coterminal angles of 0° and -360°. Coterminal angles are angles drawn in…

Illustration showing coterminal angles of 0° and -360°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, only the negative angle is labeled with the proper degree measure.

0° and -360° Coterminal Angles

Illustration showing coterminal angles of 0° and -360°. Coterminal angles are angles drawn in…

Illustration showing coterminal angles of 0° and -360°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, only the positive angle is labeled with the proper degree measure.

0° and -360° Coterminal Angles

Illustration showing coterminal angles of 0° and -360°. Coterminal angles are angles drawn in…

Illustration showing coterminal angles of 10° and -350°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, both angles are labeled with the proper degree measure.

10° and -350° Coterminal Angles

Illustration showing coterminal angles of 10° and -350°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 10° and -350°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, neither angle is labeled.

10° and -350° Coterminal Angles

Illustration showing coterminal angles of 10° and -350°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 10° and -350°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, only the negative angle is labeled with the proper degree measure.

10° and -350° Coterminal Angles

Illustration showing coterminal angles of 10° and -350°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 10° and -350°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, only the positive angle is labeled with the proper degree measure.

10° and -350° Coterminal Angles

Illustration showing coterminal angles of 10° and -350°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 100° and -260°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, both angles are labeled with the proper degree measure.

100° and -260° Coterminal Angles

Illustration showing coterminal angles of 100° and -260°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 100° and -260°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, neither angle is labeled.

100° and -260° Coterminal Angles

Illustration showing coterminal angles of 100° and -260°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 100° and -260°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, only the negative angle is labeled with the proper degree measure.

100° and -260° Coterminal Angles

Illustration showing coterminal angles of 100° and -260°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 100° and -260°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, only the positive angle is labeled with the proper degree measure.

100° and -260° Coterminal Angles

Illustration showing coterminal angles of 100° and -260°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 105° and -255°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, both angles are labeled with the proper degree measure.

105° and -255° Coterminal Angles

Illustration showing coterminal angles of 105° and -255°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 105° and -255°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, neither angle is labeled.

105° and -255° Coterminal Angles

Illustration showing coterminal angles of 105° and -255°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 105° and -255°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, only the negative angle is labeled with the proper degree measure.

105° and -255° Coterminal Angles

Illustration showing coterminal angles of 105° and -255°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 105° and -255°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, only the positive angle is labeled with the proper degree measure.

105° and -255° Coterminal Angles

Illustration showing coterminal angles of 105° and -255°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 110° and -250°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, both angles are labeled with the proper degree measure.

110° and -250° Coterminal Angles

Illustration showing coterminal angles of 110° and -250°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 110° and -250°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, neither angle is labeled.

110° and -250° Coterminal Angles

Illustration showing coterminal angles of 110° and -250°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 110° and -250°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, only the negative angle is labeled with the proper degree measure.

110° and -250° Coterminal Angles

Illustration showing coterminal angles of 110° and -250°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 110° and -250°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, only the positive angle is labeled with the proper degree measure.

110° and -250° Coterminal Angles

Illustration showing coterminal angles of 110° and -250°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 115° and -245°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, both angles are labeled with the proper degree measure.

115° and -245° Coterminal Angles

Illustration showing coterminal angles of 115° and -245°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 115° and -245°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, neither angle is labeled.

115° and -245° Coterminal Angles

Illustration showing coterminal angles of 115° and -245°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 115° and -245°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, only the negative angle is labeled with the proper degree measure.

115° and -245° Coterminal Angles

Illustration showing coterminal angles of 115° and -245°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 115° and -245°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, only the positive angle is labeled with the proper degree measure.

115° and -245° Coterminal Angles

Illustration showing coterminal angles of 115° and -245°. Coterminal angles are angles drawn…

Illustration showing coterminal angles of 120° and -240°. Coterminal angles are angles drawn in standard position that have a common terminal side. In this illustration, both angles are labeled with the proper degree measure.

120° and -240° Coterminal Angles

Illustration showing coterminal angles of 120° and -240°. Coterminal angles are angles drawn…