1. FINDING EXACT TRIGONOMETRIC VALUES Instructor Brian D. Ray

2. DRILL DIRECTIONS: Solve each special right triangle shown below. 1) 1 X y 1 St = 1 = = 2= 2) In the 45 – 45 – 90 triangle, assume that a leg is 1. The other leg is 1 since the 45 – 45 – 90 is isosceles! The hypotenuse, by the Pythagorean Theorem is units long.

3. DRILL DIRECTIONS: Solve each special right triangle shown below. 1) 1 x y 1 St = 1 = = 2= 2) In the 30 – 60 – 90 triangle, assume that the short leg is 1. How do we know which leg is the short leg? The short leg is opposite the angle. The hypotenuse is 2 units according to the derivation we did in our previous unit. The hypotenuse is units long by the Pythagorean Theorem.

4. OUR ULTIMATE GOAL Do you remember what kind of function we used to model each situation? Time (in hrs) 0.511.52 Distance (miles) 306090120 Why do we learn about functions?

5. OUR ULTIMATE GOAL Do you remember what kind of function we used to model each situation? Ground zero Path of baseball

6. OUR ULTIMATE GOAL Do you remember what kind of function we used to model each situation? Verizon charges me $0.45 for each additional minute that I use beyond my plan. I used 7:28 additional minutes, but of course, Verizon will round up, rather than round down. What function can I use to model this the additional cost I would pay?

7. HERE’S THE POINT Have you ever seen this before? What about these? Let’s look here: http://www.truveo.com/H ow-to-make-a-yoyo-sleep- Sleeper-yoyo- trick/id/2310084845 http://www.truveo.com/H ow-to-make-a-yoyo-sleep- Sleeper-yoyo- trick/id/2310084845 What function do we have to model this motion?

8. OBJECTIVE To model the situations given in the last slides, we need to learn more trigonometry! Our objective is to calculate the trigonometric value of any angle, particularly those having special reference angles.

9. EXAMPLE Find the six trigonometric values for. Step 1. Draw the angle. Step 2. Find the reference angle. Step 3. Set up the special right triangle. Be careful to use the correct signs. Step 4. Apply the definitions we learned from the reference angle to find the trigonometric values. opposite adjacent

10. EXAMPLE Find the six trigonometric values for. Step 1. Draw the angle. Step 2. Find the reference angle. Step 3. Set up the special right triangle. Be careful to use the correct signs. Step 4. Apply the definitions we learned from the reference angle to find the trigonometric values. opposite adjacent

11. EXAMPLE 2 Find the six trigonometric values for. Step 1. Draw the angle. Step 2. Find the reference angle. Step 3. Set up the special right triangle. Be careful to use the correct signs. Step 4. Apply the definitions we learned from the reference angle to find the trigonometric values. opposite adjacent

12. EXAMPLE 2 Find the six trigonometric values for. Step 1. Draw the angle. Step 2. Find the reference angle. Step 3. Set up the special right triangle. Be careful to use the correct signs. Step 4. Apply the definitions we learned from the reference angle to find the trigonometric values. opposite adjacent

13. EXAMPLE Find the six trigonometric values for. Step 1. Draw the angle. Step 2. Find the reference angle. Step 3. Set up the special right triangle. Be careful to use the correct signs. Step 4. Apply the definitions we learned from the reference angle to find the trigonometric values. opposite adjacent

14. EXAMPLE Find the six trigonometric values for. Step 1. Draw the angle. Step 2. Find the reference angle. Step 3. Set up the special right triangle. Be careful to use the correct signs. Step 4. Apply the definitions we learned from the reference angle to find the trigonometric values. opposite adjacent

15. Quadrantal Angles Definition. A quadrantile angle is an angle whose initial side lies on one of the coordinates axes. Examples. How do we find trig values in this case?

16. Trigonometric Values of Quadrantal Angles Definition. The unit circle is a circle whose radius is 1 unit long. (, ) Identify the ordered pair for each quadrantal angle. We will now find out how to find calculate the trigonometric values of these angles.

17. EXAMPLE: Quadrantal Angles (, ) Find the six trigonometric values for. Step 1. Draw the angle. Step 2. Find the ordered pair from the unit circle.. Step 3. Apply the definitions we learned from the reference angle to find the trigonometric values.

18. EXAMPLE: Quadrantal Angles (, ) Find the six trigonometric values for. Step 1. Draw the angle. Step 2. Find the ordered pair from the unit circle.. Step 3. Apply the definitions we learned from the reference angle to find the trigonometric values.

19. Quadrantal Angles Try This (, ) Find the six trigonometric values for. Step 1. Draw the angle. Step 2. Find the ordered pair from the unit circle.. Step 3. Apply the definitions we learned from the reference angle to find the trigonometric values.

20. Quadrantal Angles Try This (, ) Find the six trigonometric values for. Step 1. Draw the angle. Step 2. Find the ordered pair from the unit circle.. Step 3. Apply the definitions we learned from the reference angle to find the trigonometric values.