Welcome to trigonometry! Trigonometry is the branch of mathematics that studies triangles and the relationships between angles and side lengths. In a right triangle, we have three sides: the hypotenuse, which is the longest side opposite the right angle; the adjacent side, which is next to our angle of interest; and the opposite side, which is across from our angle. These relationships form the foundation of trigonometric functions.
Now let's define the sine function. Sine of an angle theta is defined as the ratio of the opposite side to the hypotenuse. In mathematical notation, sine theta equals opposite divided by hypotenuse. For example, in our triangle with an opposite side of 3 and hypotenuse of 5, sine theta equals 3 divided by 5, which equals 0.6. An important property of sine is that this ratio remains constant for the same angle, regardless of the triangle's size. As the angle increases from 0 to 90 degrees, the sine value increases from 0 to 1.
Now let's explore the cosine function. Cosine of an angle theta is defined as the ratio of the adjacent side to the hypotenuse. In mathematical notation, cosine theta equals adjacent divided by hypotenuse. Using our same triangle example with an adjacent side of 4 and hypotenuse of 5, cosine theta equals 4 divided by 5, which equals 0.8. An important relationship to notice is that cosine and sine are complementary. As the angle increases from 0 to 90 degrees, cosine decreases from 1 to 0, which is the opposite behavior of sine. This complementary relationship is fundamental in trigonometry.
Now let's complete our trio of basic trigonometric functions with tangent. Tangent of an angle theta is defined as the ratio of the opposite side to the adjacent side. We can also express tangent as sine theta divided by cosine theta, which shows the relationship between all three functions. Using our triangle example with opposite side 3 and adjacent side 4, tangent theta equals 3 divided by 4, which equals 0.75. Tangent has unique behavior compared to sine and cosine. As the angle approaches 90 degrees, tangent approaches infinity because the adjacent side approaches zero while the opposite side remains finite. This makes tangent particularly useful for measuring steep angles and slopes.
Let's examine the exact values of trigonometric functions for commonly used angles. These special angles are 0, 30, 45, 60, and 90 degrees, or in radians: 0, pi over 6, pi over 4, pi over 3, and pi over 2. These values come from two special right triangles: the 30-60-90 triangle and the 45-45-90 triangle. In a 30-60-90 triangle, the sides are in the ratio 1 to square root of 3 to 2. In a 45-45-90 triangle, the sides are in the ratio 1 to 1 to square root of 2. From these triangles, we can derive all the exact trigonometric values. For example, sine of 30 degrees equals one half, cosine of 45 degrees equals square root of 2 over 2, and tangent of 60 degrees equals square root of 3. These values are fundamental in trigonometry and should be memorized.