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.
After completing this tutorial, you will be able to complete the following:
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.
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.
|Approximate Time||37 Minutes|
|Pre-requisite Concepts||adjacent, angle, hypotenuse, opposite, origin, ratio, right triangle, triangle|
|Type of Tutorial||Concept Development|
|Key Vocabulary||cosine, cotangent, right triangle|