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Trig overview

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Trig overview

1. 1. The Unit Circle
2. 2. <ul><li>Sine t = y/r </li></ul><ul><li>Cosine t = x/r </li></ul><ul><li>Tangent t = y/x </li></ul><ul><li>Cotangent t = x/y </li></ul><ul><li>Secant t = r/x </li></ul><ul><li>Cosecant t = r/y </li></ul>
3. 3. <ul><li>The variable r is the Radius. </li></ul><ul><li>This will always equal 1 </li></ul><ul><li>on the unit circle. </li></ul>
4. 4. <ul><li>Here at 30° x = √3/2 </li></ul><ul><li>Also at 30° y = ½ </li></ul><ul><li>45° both x & y = √2/2 </li></ul><ul><li>At 60° x = ½ </li></ul><ul><li>Also at 60° y = √3/2 </li></ul>
5. 5. <ul><li>By using the Pythagorean Theorem the sides of a 45°, 45°, 90° can be found. </li></ul><ul><li>You will use these on to find all Six Trigonometric Functions. </li></ul><ul><li>The Hypotenuse must equal </li></ul><ul><li>1 therefore the sides opposite </li></ul><ul><li>45° is √2/2. </li></ul>
6. 6. <ul><li>Use the Pythagorean Theorem to find the sides of a 30°, 60°, 90° Triangle. </li></ul><ul><li>Remember the Hypotenuse must equal 1. </li></ul><ul><li>Therefore, the side opposite 30° = ½. </li></ul><ul><li>The side opposite 60° = √3/2. </li></ul>
7. 7. <ul><li>Here you can see Sine t = y/r. </li></ul><ul><li>Cosine t = x/r. </li></ul><ul><li>Notice that r = 1. </li></ul>
8. 8. <ul><li>Sin 30° = 1/2 </li></ul><ul><li>Cos 30° = √3/2 </li></ul><ul><li>Tan 30° = 3√3 </li></ul><ul><li>Cot 30° = √3 </li></ul><ul><li>Sec 30° = 2√3/3 </li></ul><ul><li>Csc 30° = 2 </li></ul>
9. 9. <ul><li>Sin 45° = √2/2 </li></ul><ul><li>Cos 45° = √2/2 </li></ul><ul><li>Tan 45° = 1 </li></ul><ul><li>Cot 45° = 1 </li></ul><ul><li>Sec 45° = √2 </li></ul><ul><li>Csc 45° = √2 </li></ul>
10. 10. <ul><li>Sin 60° = √3/2 </li></ul><ul><li>Cos 60° = 1/2 </li></ul><ul><li>Tan 60° = √3 </li></ul><ul><li>Cot 60° = 3√3 </li></ul><ul><li>Sec 60° = 2 </li></ul><ul><li>Csc 60° = 2√3/3 </li></ul>
11. 11. <ul><li>Notice that at 90° the (x,y) is (0,1). </li></ul><ul><li>Sin 90° = 1 </li></ul><ul><li>Cos 90° = 0 </li></ul><ul><li>Tan 90° = undefined </li></ul><ul><li>Cot 90° = 0 </li></ul><ul><li>Sec 90° = undefined </li></ul><ul><li>Csc 90° = 1 </li></ul>
12. 12. <ul><li>All y values stay the same as in Quadrant I. </li></ul><ul><li>All x values are negative here. </li></ul><ul><li>The Six Trig Functions are reflected across the y-axis here. </li></ul><ul><li>(-cos t, sin t) </li></ul>
13. 13. <ul><li>All x values are negative here. </li></ul><ul><li>All y values are negative here. </li></ul><ul><li>The Six Trig Functions are reflected across the origin from Quadrant I. </li></ul><ul><li>(-cos t, -sin t) </li></ul>
14. 14. <ul><li>All x values stay the same as in Quadrant I. </li></ul><ul><li>All y values are negative here. </li></ul><ul><li>The Six Trig Functions are reflected across the x-axis here. </li></ul><ul><li>(cos t, -sin t) </li></ul>
15. 15. <ul><li>Sin 135° = √2/2 </li></ul><ul><li>Cos 135° = - √2/2 </li></ul><ul><li>Tan 135° = -1 </li></ul><ul><li>Cot 135° = -1 </li></ul><ul><li>Sec 135° = -√2 </li></ul><ul><li>Csc 135° = √2 </li></ul>
16. 16. <ul><li>Sin 150° = -√3/2 </li></ul><ul><li>Cos 150° = 1/2 </li></ul><ul><li>Tan 150° = -√3 </li></ul><ul><li>Cot 150° = -3√3 </li></ul><ul><li>Sec 150° = 2 </li></ul><ul><li>Csc 150° = -2√3/3 </li></ul>
17. 17. <ul><li>Sin 225° = -√2/2 </li></ul><ul><li>Cos 225° = -√2/2 </li></ul><ul><li>Tan 225° = 1 </li></ul><ul><li>Cot 225° = 1 </li></ul><ul><li>Sec 225° = -√2 </li></ul><ul><li>Csc 225° = -√2 </li></ul>
18. 18. <ul><li>Sin 300° = -√3/2 </li></ul><ul><li>Cos 300° = 1/2 </li></ul><ul><li>Tan 300° = -√3 </li></ul><ul><li>Cot 300° = -3√3 </li></ul><ul><li>Sec 300° = 2 </li></ul><ul><li>Csc 300° = -2√3/3 </li></ul>
19. 19. <ul><li>Sin 315° = √2/2 </li></ul><ul><li>Cos 315° = - √2/2 </li></ul><ul><li>Tan 315° = -1 </li></ul><ul><li>Cot 315° = -1 </li></ul><ul><li>Sec 315° = -√2 </li></ul><ul><li>Csc 315° = √2 </li></ul>
20. 20. <ul><li>All the Trig Functions in the second Quadrant have the same values as Quad I except the x values are now negative. </li></ul><ul><li>The same thing happens in Quad III both the x and y values are negative here. </li></ul><ul><li>In Quad IV the x values are positive and the y values here are negative. </li></ul>
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