3.8 Determining trigonometric functions given their graphs
In previous chapter, we learned the basic terminology and identities in Trigonometry. From there, we learned that all of the trigonometric identities are related to the right triangle formed in a unit circle. This defines the measurement of the radius to 1, sin $\theta$ to y and cos $\theta$ to x. For this chapter, we will use these basic concepts and learn how to graph the different Trigonometric functions.
For example, your given the trigonometric function f(t) = sin (t), the amplitude will have a measure of one, and the sine period is equal to 2$\pi$. This means that the sinusoidal curve will oscillate by an amplitude of one unit up, and one unit down in every 2$\pi$ unit interval. The sine wave passes through point zero. We will see more of this in session 1 of this chapter.
Cosine waves on one hand look very similar to sine waves except that they do not pass through zero. In this wave, y=1 and x=0. Examples of which would be given in the second part of the chapter. For the third part of this chapter, we will learn how to graph the tangent function, which looks very different from that of the sine and cosine graph. For the next three parts of this chapter, we will look at the graphs of other trigonometric function like the cotangent, secant, and cosecant which are the inverse of tangent, cosine and sine respectively. If you want to learn more on how to graph basic trig functions with Graphing calculators, click here.
In section 7 we will learn how to transform trigonometric functions through their graph. With our knowledge of the basic graphs of the six functions, we will easily follow through with the horizontal and vertical displacement/phase shifts for every transformation. For the last part of this chapter, we will apply all the things we learned about how to graph a trigonometric function so we could identify the trigonometric function reflected in a graph.
Determining trigonometric functions given their graphs
Lessons

1.
For the following graph, write an equation (with the smallest phase shift) in the form of:
(i) y = a sin b(x  c) + d
(ii) y =  a sin b(x  c) + d
(iii) y = a cos b(x  c) + d
(iv) y =  a cos b(x  c) + d