Unit 3, Lesson 1.3 Special Angles in the Unit Circle

Transcription

1 Unit, Lesson Special Angles in the Unit Circle Special angles exist within the unit circle For these special angles, it is possible to calculate the exact coordinates for the point where the terminal side intersects the unit circle The patterns of the triangle and the triangle can be used to find these points Recall the pattern for a triangle: Recall the pattern for a triangle: Notice the two legs are the same length and the Notice the otenuse is twice as long as the otenuse is equal to the length of a leg times short leg, and the longer leg is equal to the times length of the short leg times Each special angle can be viewed in radians as well as degrees: 60 radians, and 90 radians 0 radians, 5 6 radians, Thus, the patterns for the special triangles, noted in radians instead of degrees, are as follows: The special angles continue around the unit circle and can be identified as all angles whose reference angles are special angles MUnitLesson /6/08

2 Unit, Lesson Special Angles in the Unit Circle (continued) To find the coordinates of the point where the terminal side of a special angle intersects the unit circle, first identify the reference angle Then use the pattern to identify cos and sin Recall that the coordinates of the point where the terminal side intersects the unit circle are always ( cos, sin ) However, since the reference angle was used to find cos and sin, remember to account for negative coordinates based on which quadrant the point is located in The following illustration gives the coordinates of the points where the terminal side of each special angle intersects the unit circle While these coordinates can be memorized, it is helpful to understand how to derive them for a given problem Common Errors/Misconceptions forgetting to account for a negative cos or sin when using reference angles thinking that, for a triangle, the otenuse is and the longest leg is switching the coordinates of the point where the terminal side of the angle intersects the unit circle Example Find the coordinates of the point where the terminal side of a 0 angle intersects the unit circle Sketch the angle on a unit circle and identify the location of the terminal side A 0 angle is close to a full rotation (60 ) The terminal side falls in Quadrant IV MUnitLesson /6/08

3 Unit, Lesson Special Angles in the Unit Circle (continued) Example (continued) Identify the reference angle The reference angle is the angle that the terminal side makes with the x-axis Since the terminal side is located in Quadrant IV, subtract 0 from 60 to find the reference angle ref The reference angle for 0 is 0 Find the cosine and sine of the reference angle Remember the pattern for a triangle: Use the ratios for sine and cosine, substituting in the values from the triangle adjacent side cos Cosine ratio otenuse cos0 The cosine of the reference angle is Substitute opposite side sin Sine ratio otenuse sin 0 The sine of the reference angle is for the adjacent side, for the otenuse, and Substitute for the opposite side, for the otenuse, and 0 for 0 for Determine the coordinates of the point where the terminal side intersects the unit circle The coordinates of the point where the terminal side intersects the unit circle are ( cos, sin ) The sine and cosine of the reference angle are the same as the sine and cosine of the original angle except for the sign, which is based on the quadrant in which the terminal side is located Since the terminal side is in Quadrant IV, the x-coordinate ( cos ) must be positive and the y-coordinate ( sin ) must be negative Therefore, the coordinates of the point at which the terminal side intersects the unit circle are, MUnitLesson /6/08

4 Unit, Lesson Special Angles in the Unit Circle (continued) Example 5 Find the coordinates of the point where the terminal side of an angle with a measure of radians intersects the unit circle Sketch the angle on the unit circle and identify the location of the terminal side 5 is the same as and thus is of the way between radians and radians The terminal side falls in Quadrant III Identify the reference angle The reference angle is the angle that the terminal side makes with the x-axis Since it is 5 located in Quadrant III, subtract radians from radians to find the reference angle 5 ref Subtract from the original angle measure 5 Rewrite as a fraction over a common denominator 5 The reference angle for radians is radians Find the cosine and sine of the reference angle radian is the same as 5 Recall the pattern for a triangle measured in radians: MUnitLesson /6/08

5 Unit, Lesson Special Angles in the Unit Circle (continued) Example (continued) Substitute values from the triangle into the ratios for sine and cosine adjacent side cos ref Cosine ratio otenuse cos Substitute for the adjacent side, for the otenuse, and for ref cos The cosine of the reference angle is opposite side sin ref Sine ratio otenuse sin Substitute for the opposite side, for the otenuse, and for ref sin The sine of the reference angle is Determine the coordinates of the point where the terminal side intersects the unit circle The coordinates of the point where the terminal side intersects the unit circle are ( cos, sin ) The sine and cosine of the reference angle are the same as the sine and cosine of the original angle except for the sign, which is based on the quadrant in which the terminal side is located Since the terminal side is in Quadrant III, both the x-coordinate ( cos ) and the y-coordinate ( sin ) must be negative Therefore, the coordinates of the point at which the terminal side intersects the unit circle are, Example Find the coordinates of the point where the terminal side of an angle with a measure of radians intersects the unit circle Sketch the angle on the unit circle and identify the location of the terminal side is the same as and thus is of the way between radians and radians The terminal side is located along the y-axis MUnitLesson 5 /6/08

6 Unit, Lesson Special Angles in the Unit Circle (continued) Example (continued) Determine the coordinates of the point where the terminal side intersects the unit circle Example Since the point is located on the y-axis, the x-coordinate must be 0 Since the radius of the unit circle is, the y-coordinate must be The coordinates of the point where the terminal side intersects the unit circle are ( 0, ) Sketch the three special angles that are located in Quadrant II Label the coordinates of the points where their terminal sides intersect the unit circle Use degrees Identify the special angles that are located in Quadrant II The special angles of a unit circle are 0, 5, 60, 90, and their multiples For the angle to fall in Quadrant II, its measure must be larger than 90 and smaller than 80 The multiples of of 0 (up to 80 ) are 60, 90, 0 that fall in Quadrant II are 0 and 50 The multiples of 5 (up to falls in Quadrant II is 5 The multiples of 60 and 80 ) are 90, 0, 50, and 80 The only multiples 5, and 80 The only one of these that 90 are included in the multiples of 0 Therefore, the special angles that are located in Quadrant II are Sketch 0, 5, and 50 angles on the unit circle 0, 5, and 50 Identify the reference angles for the 0, 5, and 50 angles The reference angle is the angle that the terminal side makes with the x-axis Since these angles are located in Quadrant II, subtract each original angle measure from 80 to find its reference angle 80 0 The reference angle for 0 is 60 ref The reference angle for 5 is 5 ref The reference angle for 50 is 0 ref 0 MUnitLesson 6 /6/08

7 Unit, Lesson Special Angles in the Unit Circle (continued) Example (continued) Find the sine and cosine of each reference angle Remember the patterns for a triangle and a triangle: Use the ratios for cosine and sine, substituting in the values from the special right triangles for each angle measure Recall that the cosine ratio is opposite side sin otenuse cos adjacent side otenuse, and the sine ratio is For a 60 reference angle: For a 5 reference angle: adj opp adj cos60 sin 60 cos 5 sin 5 opp For a 0 reference angle: adj cos0 sin 0 opp 5 Determine the coordinates of the point where each terminal side intersects the unit circle and label the coordinates on the sketch The coordinates of the point where the terminal side intersects the unit circle are ( cos, sin ) The sine and cosine of the reference angle are the same as the since and cosine of the original angle except for the sign, which is based on the quadrant in which the terminal side is located Since the terminal sides are in Quadrant II, the x-coordinate ( cos ) must be negative and the y-coordinate ( sin ) must be positive MUnitLesson 7 /6/08

8 Unit, Lesson Special Angles in the Unit Circle (continued) Example (continued) The terminal side of the at, The terminal side of the at, The terminal side of the at, Label these coordinates on the sketch 0 angle (whose reference angle is 5 angle (whose reference angle is 50 angle (whose reference angle is 60 ) intersects the unit circle 5 ) intersects the unit circle 0 ) intersects the unit circle MUnitLesson 8 /6/08