Overview: What Are Right Triangles?
Right triangles have one angle with a measure of 90o. As can be shown from the axiom in geometry that the sum of all three angles equals 180o, the sum of the other angles have to equal 90o. In special right triangles, such as 45o-45o-90o, or 30o-60o-90o, the lengths of sides are in a special relationship. Therefore, the sines, cosines, and tangents are in a close relationship and are easily found.
The Pythagorean Theorem and Right Triangles
The Pythagorean Theorem, a2 +b2 = c2, holds true for right triangles. This means that the lengths of the sides opposite and adjacent to the right angle, as well as the hypotenuse, can be solved using the relationship between measurements. Suppose in a right triangle one side is equal to 5 cm and the hypotenuse is equal to 13. What is the measurement of the other side? Using the Pythagorean theorem , a2 = 132 -52 , a2=169-25, or a2 =144. Therefore, a = 12, the square root of 144.
What Are the Sines, Cosines, and Tangents of that Triangle?
In that example triangle, the sine is the ratio of the opposite side to the hypotenuse, or 5/13. The cosine is the ratio of the adjacent side to the hypotenuse of 12/13, and the tangent is the ratio of the opposite side to the adjacent side or 5/12. These relationships follow the rules for the values of sine, cosine, and tangent, no matter in which quadrant the triangle can be graphed.
What Is the Relationship of Sines, Cosines, and Tangents?
When the triangle is a 45o-45o-90o triangle, the sine and cosine will be equal, because the measurements of the opposite side and the adjacent side of the right angle are the same length. Similarly, the tangent will equal 1, because the sine and cosine are equal. In a 30o-60o-90o triangle, the hypotenuse is twice the length of the opposite side of the right angle. Therefore, the sine — the ratio of the opposite side to the hypotenuse — will equal 1/2. The cosine — the ratio of the adjacent side to the hypotenuse, will equal √3/2, and the tangent will equal √3/3.
Using a Reference Triangle
Every angle in the coordinate plane can be referenced to a similar right triangle that contains it. What that means for the purposes of using trigonometric functions is that the sign, cosine, and tangent can be measured with an example. Those measurements are contained in many different reference tables.
Interested in math tutoring services? Learn more about how we are assisting thousands of students each academic year.
SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Amherst, NH visit: Tutoring in Amherst, NH
