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The Unit Circle

The Unit Circle. Unit Circles : A unit circle typically has three features:. A coordinate plane. A circle with its center at the origin and a radius of 1 unit. An angle in standard position that intersects the circle. Standard Position. Of an angle when the vertex is at the origin.

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The Unit Circle

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  1. The Unit Circle

  2. Unit Circles: A unit circle typically has three features: • A coordinate plane. • A circle with its center at the origin and a radius of 1 unit. • An angle in standard position that intersects the circle.

  3. Standard Position • Of an angle when the vertex is at the origin.

  4. Initial Side • Is when the ray is on the x – axis

  5. Terminal Side • The other ray of the angle • Now for your example

  6. Example: • What is that symbol ? First of all it is called theta This symbol is used for the measure of an angle in standard position

  7. Measuring angles in standard position • The measure is positive when the rotation from the initial side to the terminal side is in the counterclockwise direction.

  8. Measuring angles in standard position • The measure is negative when the rotation from the initial side to the terminal side is in the clockwise direction.

  9. Co-terminal Angles • Two angles in standard position are co-terminal if they have the same terminal side.

  10. Things to know about Unit Circles • The absolute values of the (x,y) coordinates of the point where the inscribed angle intersects the circle are the lengths of the sides of the inscribed right triangle. • The y coordinate is equal to the sine function of the inscribed angle. • The x coordinate is equal to the cosine function of the inscribed angle. • The slope of the hypotenuse is equal to the tangent function of the inscribed angle.

  11. The six trigonometry functions can be expressed as:

  12. The Pythagorean Theorem • says that the sum of the squares of the lengths of the sides of any right triangle inscribed in such a unit circle will equal 1.

  13. Examples - Find the exact values of cos(-120) • Step 1 – Sketch an angle of –120 in standard position. • Sketch a right triangle • Find the length of each side

  14. Expanding the Circle • When a unit circle is dilated (expanded or shrunk) so that its radius is not equal to 1, it is no longer called a unit circle, and the 1 unit in the six trigonometry functions must be expressed in terms of the radius of the circle, as follows

  15. NOTE: • If you know any point through which the angle passes, you can find the radius of a circle with center at the origin and passing through the given point by using the equation of a circle (a derivation of the Pythagorean Theorem). Once you know the value of r (the distance of the point from the origin), you can use the trigonometric identities.

  16. Relating Signs (pos. or neg.) of Trigonometric Functions to Quadrants • 1. Build a grid using knowledge of the quadrants and SOH-CAH-TOA.

  17. Relating Signs (pos. or neg.) of Trigonometric Functions to Quadrants • 2. Add to the grid the signs of the x and y values for each quadrant.

  18. Relating Signs (pos. or neg.) of Trigonometric Functions to Quadrants 3. Use SOH-CAH-TOA to recall the basic trigonometric functions. NOTE. The reciprocal functions use the same inputs, so the positive vs negative outcomes will be the same.

  19. Relating Signs (pos. or neg.) of Trigonometric Functions to Quadrants 4. Determine which trig functions will have positive values in each quadrant. NOTE: The reciprocal functions are the same signs.

  20. Relating Signs (pos. or neg.) of Trigonometric Functions to Quadrants • Once derived, the mnemonic All Students Take Calculus can be used to remember the signs of the various trigonometric functions in each quadrant. • So then what happens to the sign for 0,90, 180, 270, 360? The easy solution is to use a graphing calculator and input the given function (or its co-function) and solve for 0, 90, 180, 270, or 360 degrees.

  21. More Examples – Find the measure of an angle between 0 and 360 that is coterminal with the given angle • 600 degrees • 180 degrees

  22. Homework P412-4132-18evens, 21-25

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