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7.4 Evaluating and Graphing Sine and Cosine Using Reference Angles to Evaluate Trig Functions at Special Angles Special right triangles While working in the unit circle we come across angles that are multiples of 30, 60, and 45. If you remember from geometry there was a need to work with 30-60-90 and 45-45-90 right triangles, that need resurfaces within the unit circle. 1st quadrant special angles 1 2n 1 2 n 3 2 n 𝟑 What does this picture look like if our radius is 1? What are the coordinates of this point? Remember that this is the unit circle, and in the unit circle for any P(x,y), the x value is cosine and the y value is sine. 1 𝟏 So what is the sin(30)? cos(30)? 𝟐 𝟑 𝟐 Lets try the same thing with a 60 degree angle for ϴ Can it be done with a 45 degree • If we look at the unit circle note that it is symmetric about both the x and y axis. Because of this symmetry we can learn a lot about the first quadrant and then use that information as a reference (or guide) for the remaining three quadrants. Take an angle that has a terminal ray in the 1st quadrant, what will the sign of cos(ϴ) be and what will the sin(ϴ) be? Will those signs be true for any angle that has a terminal ray in the 1st quadrant? Now do the same procedure for the 2nd, 3rd, and 4th quadrants. S Sine is positive T Tangent is positive A All are positive C Cosine is positive To create a reference angle. • Draw out the angle ϴ. • Take a perpendicular line to the x axis from the terminal ray • Note what quadrant the terminal ray is in. • Determine the angle of the new triangle. • Reproduce the same triangle in the first quadrant, label that angle α (alpha) 0 sin(120 ) ϴ α Now if we can figure out what sin(60) is we can determine what sin(120) is. Making sure that our sign matches up to that of a terminal angle in the 2nd quadrant. Practice Creating Reference angles sin(135o ) cos(330o ) tan(210o ) 5 sin 4 5 cos 6 2 tan 3 Use reference angles to find the following values 2 cos 3 Use reference angles to find the following values 19 sin 4 Use reference angles to find the following values tan 3 P.280 11-18