11+Evaluating+trig+functions+(sin,+cos,+tan)

**Evaluating Trig Functions (sin, cos, tan)**    **1. Summary:** When you evaluate Trig Functions of sin, cos, and tan the first step is to draw the angle that you are given if you are given one. Once you draw the angle you always use the reference angle to find the answer. Most likely the reference angle with either be 30°,60° or 45°. You then use the rules of the special right triangles that are below. From here you should then be able to find the answer by using the sin, cos or tan. If you are not given an angle and you are given either the sin, cos or tan of an angle you have to graph the triangle depending on the numbers you are given (like example 2 below). Also in an example like this you have to use the "All Student's Take Calc" graph and you can see which trig functions are positive in which coordinate. If you need to find the third side of the triangle you can always use the pythagorean theorem. The unit circle can also help because the cosine is always the x coordinate and the sine is always the y coordinate. The goals of evaluating trig functions are being able to identify which special right triangle you have depending on the reference angle that you are given. Another goal is being able to use sin, cos and tan to figure out the answer to the solution depending on the given triangle. The last goal would be able to use the "All Students Take Calc" graph to realize which quadrant your triangle must be located in depending if you want it to be positive or negative.

**2. Rules and Properties**  **3. Sample Problems**

 1. **Find the exact value of** **cos120** ° The first step is to draw an 120 ° angle on the coordinate plane. Next you must find the reference angle which in this problem is 60°. Next you need to see which special right triangle it is. In this case it is a 30-60-90. In the last picture we drew in the side lengths which we got from our special right triangle chart above. Cosine is adjacent/ hypotenuse so the answer to the problem would be -1/2. (Also it is a **negative** one because its on the left of the coordinate plane)

  2. A**ssuming that sin****θ** **is with 90°**__**<**__**θ < 180****°, what is the value of cos****θ.**

The first step in this problem is to think about in which quadrants is sine positive. It is either the first or the second. Since theta is between 90 and 180 it must be in the second quadrant. Sine is opposite/ hypotenuse so below we were able to draw the picture of that. To figure out the third side you have to use the pythagorean theorem. The third side is equal to. (Its negative because you are on the left side of the coordinate plane). You then need to find the cosine of theta which would be adjacent/hypotenuse. The answer to the problem would be




 * 3. Find the exact value of cos **[[image:Pictudddsre_1.png]]**.**

<span style="font-family: 'Comic Sans MS',cursive;">The first step in this problem is to draw the angle. This angle is located in the 3rd quadrant. After you draw the angle you need to find the reference angle. We then realized that this is a special right triangle and it is a 30-60-90. We used the rules for this triangle to determine the side lengths. Cosine is adjacent/hypotenuse so the answer to this problem is.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">4. Extra Links http://www.themathpage.com/atrig/functions-angle.htm http://literacy.purduecal.edu/STUDENT/mrrieste/evaluating.html http://www.mathamazement.com/Lessons/Pre-Calculus/04_Trigonometric-Functions/trigonometric-functions-of-any-angle.html