3 7. THE SIX TRIGONOMETRIC FUNCTIONS OF THE REST OF THE SPECIAL ANGLES

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1 LESSON DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING THE UNIT CIRCLE Topics in this lesson:. DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING A CIRCLE OF RADIUS r. DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING THE UNIT CIRCLE. THE SPECIAL ANGLES IN TRIGONOMETRY. TEN THINGS EASILY OBTAINED FROM UNIT CIRCLE TRIGONOMETRY 5. THE SIX TRIGONOMETRIC FUNCTIONS OF THE THREE SPECIAL ANGLES IN THE FIRST QUADRANT BY ROTATING COUNTERCLOCKWISE. ONE METHOD TO REMEMBER THE TANGENT OF THE SPECIAL ANGLES OF 5 AND 7. THE SIX TRIGONOMETRIC FUNCTIONS OF THE REST OF THE SPECIAL ANGLES. DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING A CIRCLE OF RADIUS r P r r s - r r r - r r Coprighted b James D. Anderson The Universit of Toledo

2 Definition Let P r be the point of intersection of the terminal side of the angle with the circle whose equation is r. Then we define the following si trigonometric functions of the angle r r r sec provided that r csc provided that provided that cot provided that NOTE: B definition the secant function is the reciprocal of the ine function. The ecant function is the reciprocal of the e function. The cogent function is the reciprocal of the gent function. Back to Topics List. DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING THE UNIT CIRCLE Since ou can use an size circle to define the si trigonometric functions the best circle to use would be the Unit Circle whose radius r is. Ug the Unit Circle we get the following special definition. Definition Let P be the point of intersection of the terminal side of the angle with the Unit Circle. Since r for the Unit Circle then b the definition above we get the following definition for the si trigonometric functions of the angle ug the Unit Circle sec provided that Coprighted b James D. Anderson The Universit of Toledo

3 csc provided that provided that cot provided that P s - - The Unit Circle NOTE: The definition of the si trigonometric functions of the angle in terms of the Unit Circle sas that the ine of the angle is the -coordinate of the point of intersection of the terminal side of the angle with the Unit Circle. This definition also sas that the e of the angle is the -coordinate of the point of intersection of the terminal side of the angle with the Unit Circle. The gent of the angle is the -coordinate of the point of intersection of the terminal side of the angle with the Unit Circle divided b the -coordinate of the point of intersection. The secant function is still the reciprocal of the ine function the ecant function is still the reciprocal of the e function and the cogent function is still the reciprocal of the gent function. Eamples Find the eact value of the si trigonometric functions for the following angles.. Coprighted b James D. Anderson The Universit of Toledo

4 P sec csc = undefined cot = undefined. NOTE: This is the 9 angle in units of degrees. P Coprighted b James D. Anderson The Universit of Toledo

5 sec undefined csc = undefined cot. NOTE: This is the 8 angle in units of degrees. P sec csc = undefined Coprighted b James D. Anderson The Universit of Toledo

6 cot = undefined. 7 NOTE: This is the angle in units of radians. P 7 7 sec 7 = undefined 7 csc 7 7 = undefined cot 7 5. NOTE: This is the angle in units of degrees. Coprighted b James D. Anderson The Universit of Toledo

7 P sec csc = undefined cot = undefined. 9 NOTE: This is the angle in units of radians. P 9 Coprighted b James D. Anderson The Universit of Toledo

8 9 sec 9 = undefined 9 csc 9 9 = undefined cot NOTE: This is the angle in units of radians. P 8 8 sec 8 8 csc 8 = undefined 8 cot 8 = undefined Coprighted b James D. Anderson The Universit of Toledo

9 8. NOTE: This is the 7 angle in units of degrees. P sec = undefined csc = undefined cot 9. NOTE: This is the angle in units of radians. Coprighted b James D. Anderson The Universit of Toledo

10 P sec csc = undefined cot = undefined Back to Topics List. THE SPECIAL ANGLES IN TRIGONOMETRY The Special Angles in radians in trigonometr are Coprighted b James D. Anderson The Universit of Toledo

11 The Special Angles in degrees in trigonometr are Here are the coordinates of the point of intersection of the terminal side of the positive Special Angles with the Unit Circle. These coordinates could be used to find the eact value of the si trigonometric functions of these Special Angles. Here are the coordinates of the point of intersection of the terminal side of the negative Special Angles with the Unit Circle. These coordinates could be used to find the eact value of the si trigonometric functions of these Special Angles. Back to Topics List. TEN THINGS EASILY OBTAINED FROM UNIT CIRCLE TRIGONOMETRY. The -coordinate of an point on the Unit Circle satisfies the condition. Since we get the ine of an angle from the -coordinate of the point of intersection of the terminal side of the angle with the Unit Circle then for all. Thus the range of the ine function is the closed interval [ ].. The -coordinate of an point on the Unit Circle satisfies the condition. Since we get the e of an angle from the -coordinate of the point of intersection of the terminal side of the angle with the Unit Circle then for all. Thus the range of the e function is also the closed interval [ ].. B definition where is the -coordinate and is the - coordinate of the point of intersection of the terminal side of the angle with Coprighted b James D. Anderson The Universit of Toledo

12 the Unit Circle. Since and b definition then for all.. Similarl ce 5. Similarl ce. Similarl ce sec b definition then csc b definition then cot b definition then sec for all. csc for all. cot for all. 7. The equation of the Unit Circle is. This equation sas that if ou take the -coordinate and the -coordinate of a point on the Unit Circle square them and then add ou will get the value of. Since we get the ine of an angle from the -coordinate and the e of the angle from the -coordinate of the point of intersection of the terminal side of the angle with the Unit Circle then we obtain the equation which we write as for all. This equation is the first of the three Pthagorean Identities. 8. Taking the equation equation b and dividing both sides of the we obtain the second Pthagorean Identit: Coprighted b James D. Anderson The Universit of Toledo

13 sec sec for all. 9. Similarl taking the equation and dividing both sides of the equation b ou will obtain the third Pthagorean Identit: csc cot for all.. The sign of the ine e and gent functions in the Four Quadrants: Back to Topics List 5. THE SIX TRIGONOMETRIC FUNCTIONS OF THE THREE SPECIAL ANGLES IN THE FIRST QUADRANT BY ROTATING COUNTERCLOCKWISE Coprighted b James D. Anderson The Universit of Toledo

14 THE POINT OF INTERSECTION OF THE ANGLE THE UNIT CIRCLE WITH P This picture shows us that the -coordinate is greater than the -coordinate of the point P which is the point of intersection of the angle Unit Circle. Thus with the sec csc Coprighted b James D. Anderson The Universit of Toledo

15 cot THE POINT OF INTERSECTION OF THE ANGLE 5 UNIT CIRCLE WITH THE P This picture shows us that the -coordinate is same as the -coordinate of the point P which is the point of intersection of the angle 5 Unit Circle. Thus with the Coprighted b James D. Anderson The Universit of Toledo

16 sec csc cot THE POINT OF INTERSECTION OF THE ANGLE UNIT CIRCLE WITH THE P Coprighted b James D. Anderson The Universit of Toledo

17 This picture shows us that the -coordinate is less than the -coordinate of the point P which is the point of intersection of the angle Unit Circle. Thus with the sec csc cot Back to Topics List. ONE METHOD TO REMEMBER THE TANGENT OF THE SPECIAL ANGLES OF 5 AND You will need to remember the three numbers and. Arrange the angles 5 and arrange the three numbers from smallest to largest: from smallest to largest. Then Coprighted b James D. Anderson The Universit of Toledo

18 5 Tangent In other words the gent of the smallest angle of number of the gent of the largest angle of is the smallest is the largest number of and the gent of the middle angle of 5 number of. is the middle Back to Topics List 7. THE SIX TRIGONOMETRIC FUNCTIONS OF THE REST OF THE SPECIAL ANGLES To find the trigonometric functions for the special angles in the II III and IV quadrants b rotating counterclockwise and for all the special angles in the IV III II and I quadrants b rotating clockwise we make use of the reference angle of these angles. Reference angles will be discussed in Lesson. Back to Topics List Coprighted b James D. Anderson The Universit of Toledo