Trigonometric functions are mathematical functions that relate angles to side lengths in right triangles. The three primary functions are sine, cosine, and tangent. Sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent. These ratios remain constant for any given angle, making them powerful tools for solving geometric problems.
The unit circle is a circle with radius one centered at the origin. It provides a powerful way to understand trigonometric functions. As we move around the circle, the x-coordinate gives us the cosine value, and the y-coordinate gives us the sine value. Watch how these values change as the angle increases from zero to two pi radians.
Now let's examine the graphs of trigonometric functions. The sine function starts at zero and oscillates between negative one and positive one with a period of two pi. The cosine function starts at one and has the same amplitude and period as sine, but is shifted by pi over two. Both functions are continuous and smooth, making them essential for modeling periodic phenomena in physics and engineering.
Trigonometric identities are fundamental relationships that hold true for all valid angle values. The most important is the Pythagorean identity: sine squared theta plus cosine squared theta equals one. We also have sum and difference formulas for sine and cosine, double angle formulas, and reciprocal identities. These identities are crucial for solving trigonometric equations and simplifying complex expressions in calculus and physics.
Let's solve a typical exam problem: find all solutions to sine of two x equals one half in the interval zero to two pi. First, we identify that sine equals one half when the angle is pi over six or five pi over six. Setting two x equal to these values and adding the general period, we get x equals pi over twelve plus pi k, or x equals five pi over twelve plus pi k. In our interval, this gives us four solutions: pi over twelve, five pi over twelve, thirteen pi over twelve, and seventeen pi over twelve. The graph shows these intersection points clearly.