Overview: Introduction to Trigonometric Functions
An angle is formed when a ray is rotated in a coordinate plane around the x axis (the horizontal axis). The measurements of the length of the sides of a triangle have a special relationship to one another, called the sine, the cosine, and the tangent. They are ratios, similar to the slope and y intercept.
Review of Right Triangles
When a ray is rotated in the coordinate plane around the x axis, one can always make the angle into a triangle by dropping a perpendicular line from a point on the ray to the x axis. The perpendicular line will form a 900 angle for one of the angles of the triangle. By the Euclidean definition, the sum of all three angles will equal 180o. Therefore, the other two angles will equal 90o. In addition, the sides of the right triangle are in a special relationship, the Pythagorean theorem, a2 +b2 = c2. By definition, a2 is the opposite side to the right angle, b2 is the adjacent side, and c2 is the hypotenuse.
Quadrants and Coordinates
The triangle will be in different quadrants in the coordinate plane depending on where the original ray is located. Since the triangle starts at the origin of the x axis and the y axis, hypothetically speaking, if all points along the ray are positive, the triangle is in Quadrant 1. The angle formed by the opposite side to the right angle will be from 0o to 90o. If the x coordinate is negative but the y coordinate is positive, the triangle will be in Quadrant 2, and the angle formed by the opposite side to the right angle will be from 90o to 180o. If both the x and y coordinates are negative, the triangle formed will be in Quadrant 3, and the angle formed by the opposite side to the right angle will be from 180o to nearly 270o. When the x coordinate is positive but the y coordinate is negative, the triangle will be in Quadrant 4, and the angle formed by the opposite side to the right angle will be from 270o to nearly 360o.
Relationship Between Sines, Cosines, and Tangents
The sine is the ratio of the opposite side to the hypotenuse. If the angle measures from 0o to 90o, that ratio will increase from 0 to nearly 1. At 90o, the sine equals 1. From just over 90o to 180o, it decreases from 1 to nearly 0. From 180o to nearly 270o, the sine decreases further from 0 to -1. At 270o, the sine measures -1, and from 270o to 360o, the sine increases from -1 to 0.
The cosine is the ratio of the adjacent side (the b side) to the hypotenuse. If that angle measures from 0o – 90o, the cosine decreases from 1 to nearly 0. At 90o, the cosine will equal 0. If the adjacent angle measures between 90o and 180o, the cosine will decrease from 0 to -1. From 180o to 270o, the cosine increases from -1 to 0, and from 270o to 360o, the cosine increases from 0 to 1.
The tangent is the ratio of the opposite side of the triangle to the adjacent side of the triangle. By definition, sines and cosines are real numbers, between -1 and 1. Tangents are undefined for degrees 90o and 2700, but they increase from 0 to very large numbers in all other quadrants.
Applications of Sines, Cosines, and Tangents
Trigonometric functions have many applications in the real world, because they are the result of mathematical laws. Harmonics such as sound waves or light waves follow sines and cosines. Triangulation and navigation apply trigonometry to determine unknown distances. Fields as diverse as building bridges and astronomy depend upon trigonometry.
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