Circular Trigonometry Notes April 24/25

Transcription

1 Circular Trigonometry Notes April 24/25 First, let s review a little right triangle trigonometry: Imagine a right triangle with one side on the x-axis and one vertex at (0,0). We can write the sin(θ) and the cos(θ): sin(θ) = cos(θ) = r y θ x If we place that triangle in a circle with its center at (0,0), we see that the hypotenuse of the triangle is the same length as the radius of the circle. Further, the terminal point tells us the length of the two legs of the right triangle.: (x, y) r y θ x Thus, in the diagram above, we can use the x and y value of the terminal point to write the trigonometric ratios of each angle: sin(θ) = cos(θ) =

2 Using the same ideas as in the example above, write the 6 trig ratios for each marked angle in each circle. Be sure to simplify as much as possible. OA = OB = OC = θ When you simplified the trig ratios, what did you notice? Why does this result make sense? What is special about the three triangles?

3 THE UNIT CIRCLE: You have just used right triangle trigonometry to find the trig functions of acute angles in any circle. You have also noticed that the trig ratios are the same no matter the radius if the circle! Thus, we can use any circle to find the trig ratios of any angles. Since we can use any circle, we will make our lives easy and use a circle whose radius is 1 unit. This circle is called the UNIT CIRCLE. When we use the unit circle, we can find the sine or cosine of any angle just from the x and y coordinates. HOW? You will need; the skills from the previous pages our knowledge of special triangles (from class notes 4-17/4-19) Find the missing side lengths of each right triangle below. 30 1unit 1unit 45 1unit 60

4 Superimposing those triangles on a unit circle: state the terminal points of each angle state the sine and cosine of each angle r = 1 (, ) 30 r = 1 r =

5 One of the great benefits of using circular trigonometry is that we are no longer limited to working with acute angles. We can calculate the sine and cosine of 0 degrees, 90 degrees and larger (or smaller) angles. When we want to find the sine or cosine (or other trig ratio) of an angle larger that 90degress, we use a reference triangle. See the examples below. Notice that some of the sines and cosines will be negative because some of the x and y values are negative! (-, ) (, ) So: Recaling that sin(θ) =, we have sin(θ) = = y We know that: sin(150 ) = Similarly, cos(θ) =, we have cos(θ) = = x We know that: cos(150 ) = -

6 Using what you ve discovered, fill in the terminal points of the 16-point Unit Circle below. 90 (, ) 120 (, ) 60 (, ) , (, ) 300 (, ) 270 (, ) Hint: Use the symmetry of a circle and your knowledge of where x and y-values are positive or negative to fill out the circle.

7 Using your unit circle, complete the tables below: y = sin(θ) θ (Angle) Sine (θ) y = cos(θ) θ (Angle) Cosine (θ) On graph paper, make a graph of y= sin(θ). On a second sheet (or on the back), graph y = cos(θ). When you are done graphing, answer the following questions: 1. What is the domain of each function? 2. What is the range of each function? 3. How are the graphs similar? 4. How are the graphs different?

8 Unit Circle Reprise with Radian Meaure. On Monday/Tuesday, we discussed Radian Measure. Radian Measure is the ratio of the arc length of the angle on a circle to the radius of the circle: Radian Measure = (Another way to consider this is: The number of radii that can fit along the arc length of an angle) We also found that we can convert to radian measure using the proportion: = For example, to convert 30 to radians: = Solving, we find: = = θ Convert the angles in the unit circle to Radians: Degrees Radians Now, go to your graphs and mark the angles in radians as well as degrees.