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Trigonometric Ratios on the Unit Circle

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Trigonometric Ratios and Circles

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Lesson Focus

Trigonometric Ratios on the Unit Circle


Learning Made Easy

Calculate and represent ratios and values of sine, cosine, tangent, and cotangent based on principles of the unit circle. Learn the Pythagorean Theorem and sum of squares.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Interpret the trigonometric functions sine, cosine, tangent, and cotangent on the unit circle.
  • Interpret the Pythagorean Identity on the unit circle.
  • Calculate the trigonometric ratios of 0?, 90?, 180?, 270?, and 360.
  • Interpret the ranges of sine and cosine on the unit circle.
  • Specify the signs of the trigonometric ratios in the four quadrants of a unit circle.
  • Calculate trigonometric ratios of various angles.

Everything You'll Have Covered

This Activity Object introduces learners to the history of trigonometry. Trigonometry is a branch of mathematics that deals with triangles, especially plane 90 degree (right) triangles. The trigonometric functions if sin, cos, tan, and cot are addressed.

Unit circle

The unit circle's radius measures 1 unit. The unit circle is used to calculate the ratios for tangent, sine, and cosine. Each point on the unit circle can be used to express an angle. In this Activity Object, learners move the hypotenuse to see all the angles in the unit circle and their trigonometric functions. The unit circle is shown in the following illustration. Notice the ratios of tangent, sine, and cosine in the lower right corner.

X, Y values on the unit circle

With x, y values assigned to the right triangle in the unit circle, the sin , cos , tan , and cot x, y values are found. According to the Pythagorean Identity, the sum of the square of x and the square of y equals the square of the hypotenuse, which is 1. This is expressed in the following equation:


For an angle, the sum of the square of the cosine and the sum of the square of the sine is equal to one. This is expressed as:

The following graphic provides visual detail.

Signs on the unit circle

The signs of trigonometric ratios differ depending on which quadrant they are in. See the graphic which follows for details.

There is a mnemonic trick that can help your learners remember the signs in the quadrants. It is "All Students Take Calculus". In the first quadrant, A is for All indicating that, in the quadrant, all trigonometric functions are positive. In the second quadrant, S for Sine, sine functions are positive. In the third quadrant, T for Tangent, tangent functions are positive. And in the fourth quadrant, C for Cosine, cosine functions are positive. The cotangent is always the same sign as the tangent.

The following key vocabulary terms will be used throughout this Activity Object:

These words could be used in a vocabulary matching, word bank, or fill in quiz for the learners after completing this Activity Object. Selected vocabulary words can be chosen.

  • abscissa -the horizontal coordinate of a point in a Cartesian coordinate system measured parallel to the x-axis; the abscissa is known as the x-coordinate of a point
  • cosine - the ratio of the length of the adjacent side to the length of the hypotenuse of a right triangle.
  • cotangent - the reciprocal of the tangent of an angle in a right triangle; the ratio of the cosine to the sine
  • Hipparchus - recorded stars around 134 B.C.; he calculated the rate of movement, classified stars by magnitude, and began the discipline of trigonometry
  • ordinate - the vertical y value in a pair of coordinates; the ordinate and abscissa taken together are the coordinates and define the position of a point on two axes
  • origin - the intersection of two axes; the values of the coordinates are zero
  • Pythagorean Theorem - the sum of the squares of the lengths of the two legs of a right triangle is equal to the square of the length of the hypotenuse; this can be expressed by
  • quadrant - any of four equal areas made by dividing a plane by an x and y axis; also a quarter of the circumference of a circle making a 90 arc, which is one-fourth of the circle
  • sine - the ratio of the length of the opposite side to the length of the hypotenuse of a right triangle
  • tangent - the ratio of the opposite to the adjacent side of a right triangle; also the ratio of the sine to the cosine of a right triangle
  • trigonometric ratio - a ratio that describes the relationship between the sides and angles of triangles
  • trigonometry - a branch of mathematics dealing with the relations between the sides and angles of plane or spherical triangles and the related calculations
  • unit circle - used to define the trigonometric ratios of angles 0 to 360 degrees; the unit circle has a radius of 1 unit; sine is the y-coordinate of the intersection point of the intersection point of the terminal arm of the angle; cosine is the x-coordinate of the same point

Tutorial Details

Approximate Time 37 Minutes
Pre-requisite Concepts adjacent, angle, hypotenuse, opposite, origin, ratio, right triangle, triangle
Course Geometry
Type of Tutorial Concept Development
Key Vocabulary cosine, cotangent, right triangle