"Sine, sine, tangent and cosine. Learning trigonometry's breaking my mind." "Trigonometry" comes from the Greek for "3-sided measure" and is the branch of mathematics dedicated to the relationships between sides and angles of a right triangle. Many people believe trig is difficult because of the math involved, but it needn't be. Here's how to do trigonometry.
Understanding the Basic Trig Functions
Label one of the angles in the triangle other than the right angle. Use a Greek letter if you're into Pythagoras or just a common English letter such as "A."
Measure the side opposite the angle and the length of the line segment (hypotenuse) connecting angle A to the other non-right angle. Divide the length of the opposite side by the hypotenuse. This trig function is called the "sine."
Get the length of the side connecting angle A to the right angle (the adjacent side) and divide this by the length of the hypotenuse. This trig function is called the "cosine."
Divide the length of the side opposite angle A to the side adjacent angle A. This trig function is called the "tangent." Divide the sine value by the cosine value. You should get the same result.
Draw triangles larger and smaller than the one you drew while keeping the angles the same. The sine, cosine and tangent values are the same; as long as the angle measures are the same, so is the math.
Graphing Trig Functions on the Unit Circle
Draw perpendicular axes, with the horizontal axis as "x" and the vertical axis as "y." Using the point where they intersect as the center, draw a circle with a radius of 1, then draw a line from the origin to the edge of the circle and another line from that point to the horizontal axis.
Define the horizontal distance to the right of where the axes cross as positive and to the left as negative. Define the vertical distance upward from the axes as positive and downward as negative.
Label the point where the radius intersects the circle as (x,y), with "x" representing the horizontal coordinate and "y" the vertical coordinate. Label the angle formed where the radius touches the axes as angle A.
Find the cosine of angle A by comparing the adjacent side to the hypotenuse. This will equal the value of x.
Compare the opposite side to the hypotenuse to find the sine of angle A. This will equal the value of y.
Imagine the radius as a line sweeping around the axes counterclockwise starting from the x axis. The value of the cosine (x) starts at 1 and declines to zero while the sine (y) starts at zero and climbs to 1 as the radius sweeps toward the y axis. On crossing the y axis, the cosine goes down to negative 1 as it reaches the x axis, while the sine goes down to zero. Passing the x axis, the cosine goes back toward zero as the sine goes down to negative 1 as the radius sweeps toward the y axis. Past the y axis, the cosine goes up toward 1 as the sine goes toward zero.
Divide the sine by the cosine to find the tangent. When the sine and cosine are both positive or both negative, the tangent will be positive; when only the sine or cosine is positive and the other is negative, the tangent will be negative.