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FINDING EXACT TRIGONOMETRIC VALUES

FINDING EXACT TRIGONOMETRIC VALUES. Instructor Brian D. Ray. DRILL. DIRECTIONS : Solve each special right triangle shown below. 1). 2). y. 1. S. t. = 2. = . X. = 1. 1. =. What I NEED To Remember. In the 45 – 45 – 90 triangle, assume that a leg is 1.

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FINDING EXACT TRIGONOMETRIC VALUES

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  1. FINDING EXACT TRIGONOMETRIC VALUES Instructor Brian D. Ray

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

  4. Do you remember what kind of function we used to model each situation? OUR ULTIMATE GOAL • Why do we learn about functions?

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

  6. Do you remember what kind of function we used to model each situation? OUR ULTIMATE GOAL 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. Have you ever seen this before? HERE’S THE POINT Let’s look here: http://www.truveo.com/How-to-make-a-yoyo-sleep-Sleeper-yoyo-trick/id/2310084845 • What about these? • What function do we have to model this motion?

  8. 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. OBJECTIVE

  9. Find the six trigonometric values for . EXAMPLE 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. Find the six trigonometric values for . EXAMPLE 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. Find the six trigonometric values for . EXAMPLE 2 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? The Unit Circle!

  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. ( , ) ( , ) ( , ) ( , ) The Unit Circle!

  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. ( , ) The Unit Circle!

  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. ( , ) The Unit Circle!

  19. Quadrantal AnglesTry 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. ( , ) The Unit Circle!

  20. Quadrantal AnglesTry 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. ( , ) The Unit Circle!

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