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# Class Note for MATH 1330 at UH

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This 11 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 18 views.

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Date Created: 02/06/15

Math 1330 Section 43 Unit Circle Trigonometry An angle is in standard position if its Vertex is at the origin and its initial side is along the positive axis Positive angles are measured counterclockwise from the initial side Negative angles are measured clockwise We Will typically use the Greek letter 9 to denote an angle Angles that have the same terminal side are called coterminal angles Measures of coterminal angles differ by a multiple of 360 if measured in degrees or by a multiple of 27 ifmeasured in radians If an angle is in standard position and its terminal side lies along the x or y axis then we call the angle a quadrantal angle You will need to be able to work with reference angles Suppose 9 is an angle in standard position and 9 is not a quadrantal angle The reference angle for 9 is the acute angle of positive measure that is formed by the terminal side of the angle and the x aXis We previously de ned the siX trigonometric functions of an angle as ratios of the lengths of the sides of a right triangle Now we will look at them using a circle centered at the origin in the coordinate plane This circle will have the equation x2 y2 r2 Ifwe select a point Px y on the circle and draw a ray from the origin through the point we have created an angle in standard position The length of the radius will be r The siX trig functions of 9 are de ned as follows using the circle above sint9Z csct9 y 0 V y cos19i sect9x 0 r x tanHXx 0 cott9 y 0 x 3 If 9 is a first quadrant angle these de nitions are consistent with the definitions given in Section 41 An identity is a statement that is true for all values of the variable Here are some identities that follow from the definitions above tan 9 Sin 9 cost9 cott9 cfm g s1nt9 csct9 1 s1nt9 1 sect9 cost9 We will work most often with a unit circle that is a circle with radius 1 In this case each value of r is 1 This adjusts the definitions of the trig functions as follows 1 sin19y cscHy 0 y cost9x secH x 0 l x tanHZx 0 cott9 y 0 x y Trig Functions of Quadrantal Angles and Special Angles You will need to be able to nd the trig functions of quadrantal angles and of angles measuring 30 45 or 60 without using a calculator Since sin 19 y and cos 9 x each ordered pair on the unit circle corresponds to cos 9 sin 19 of some angle 9 We ll show the values for sine and cosine of the quadrantal angles on this graph i my Using the identities given above you can nd the other four trig functions of an angle given just sine and cosine Note that some values are not defined You ll also need to be able to find the six trig functions of the special angles For a 300 angle For a 600 angle csc30 2 5 O 2 7 sec30 E 2 cot30 sin60 csc60 iamp 2 JE 3 cos60 sec60 2 tan60 B cot60 i 73 JE 3 For a 450 angle sin45 Q csc45 Q 2 2 cos45 g sec45 g tan45 1 cot45 1 Here s a method for nding the sine and cosine of the special angles in the first quadrant and the quadrantal angles on the positive x and y axes You can use the location of any angle and its reference angle to nd an exact value of multiples of the special angles and the quadrantal angles Write down the angle measures starting with 00 and continue until you reach 900 Under these write down the equivalent radian measures Under these write down the numbers from 0 to 4 Next take the square root of the values and simplify if possible Divide each value by 2 This gives you the sine value of each of the angles you need To nd the cosine values write the previous line in the reverse order Now you have the sine and cosine values for the quadrantal angles and the special angles From these you can find the rest of the trig values for these angles Write the problem in terms of the reference angle Then use the chart you created to nd the appropriate value Here s how you can use a reference angle to nd the exact value of a trig function for a multiple of one of the special angles First locate the angle in the coordinate plane Find the reference angle Determine the signs of the coordinates of the point where the terminal ray of the angle intersects the unit k Next rewrite the problem in terms of the reference angle Finally use the exact values for the trig functions in the rst quadrant or on the positive x and y axes to finish the problem Example 1 Sketch each angle in standard position a 2100 b 1350 2 07 Example 2 Find three angles two positive and one negative that are coterminal with each angle a 5120 157r Example 3 Name the quadrant in which both conditions are true cos 6 gt 0 and cot 6 lt 0 Example 4 Let Px y denote the point where the terminal side of an angle 9 intersects the unit circle If P is in quadrant I and y nd the six trig functions of angle 9 Example 5 Sketch an angle measuring 900 in the coordinate plane Then give the siX trigonometric functions of the angle Note that some of the functions may be unde ned Example 6 Rewrite each of the following in terms of its reference angle including the appropriate sign a cos l35 b tan250 127 C sm 7 7 Example 7 Sketch an angle measuring 1500 in the coordinate plane Give the coordinates of the point where the terminal side of the angle intersects the unit circle Then state the siX trigonometric functions of the angle Example 8 Evaluate each E sin240 sec2100 0 Example 9 Use a calculator to evaluate each of the following to the nearest ten thousandth a cosl48 d sec 217 e sinl75quot f cot 52

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