c arc length radius a r radians degrees The proportion can be used to

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1 Advanced Functions Page of Radian Measures Angles can be measured using degrees or radians. Radian is the measure of an angle. It is defined as the angle subtended at the centre of the circle in the ratio of the radius (r) to the arc (a). c arc length radius a r c O To prove 80 Proof: When a C r c r, since angle at centre 0, r rad 0 rad 80 Special Angles in Radians r a r radians degrees The proportion can be used to 0 80 convert between radian and degree measures. Degrees Radians Eample Change from degrees to radians a) 0 o b) 00 o Eample Changing from radians to degrees 5 a) rad b) rad

2 Advanced Functions Page of Radian Measures Eample Change from degrees to radians a) 5.5 o b) 77.5 o Eample Change from radians to degrees ( decimal places) a).57 radians b) 5. radians Coterminal Angles measured in radians 0 () revolution= 0. () ( ) the angle is drawn with the -ais as the initial arm and rotated in an anti-clockwise manner. () ( ) the angle is drawn with the -ais as the initial arm and rotated in a clockwise manner. () Coterminal angles= n, n I o o Eample 5 Sketch and name coterminal angles: a) rad b) rad 5

3 Advanced Functions Page of Radian Measures Application: the arc length of a circle: We have alread defined c arc c a radius r Same ratios a r r c r An two of the three variables are given, the third unknown Can be found. a Eample Find the unknown value in each figure. a) b) c) 8 m cm rad 5 rad r 5.rad cm a 8.7 cm Eample 7 The area of a circle is cm. Find the length of one-third of the circumference. Homework P. 08 # 8 Worksheet: Fractions of a circle

4 Advanced Functions Page of Fractions of a Circle r r a a = arc length C = circumference r = radius # a C r C = r a r Fraction of a Circle Radian Measure: Arc length epressed as number of radii. (in deg) (in rad)

5 Advanced Functions Page of Angular Velocit An important application is angular velocit. The rate at which a central angle changes. The common unit for angular velocit is radians per second. The calculation for angular velocit is the same as for an velocit. - How far did ou go? and How long did it take ou? Recall that the difference between speed and velocit is direction. Speed cannot be negative, BUT velocit can be. Eample : Angular Velocit A small electrical motor turns at 00 rpm (revolutions per minute). a) Epress this angular velocit in radians per second. b) Find the distance a point. cm from the centre of rotation travels in s. Eample : Rotational Frequenc A wheel turns with an angular velocit of 0 rad/s. a) What is the rotational frequenc in revolutions per minute of this wheel? b) How far will the wheel roll in 7 seconds if the radius of the wheel is 5 cm? Eample : Angular velocit with velocit given A truck travels at 80 km/h. a) Find the angular velocit of a tire with radius 50 cm in radians per second. b) How far the truck travelled in 0 sec? Homework: P ,-9,0- Optional: Angular velocit

6 Advanced Functions Page of Angular Velocit. A wheel turns at 50 rev/min. a) Find the angular velocit in radians per seconds. b) How far does a point 5 cm from the point of rotation travel in 5 s?. A ferris wheel with radius m makes rotations in one minutes. a) Find the angular velocit in radians per second. b) How far has a rider travelled if the ride is 0 min long?. An automobile travels at 00 km/h. a) Find the angular velocit of a tire with radius cm in radians per second. b) Through what angle will the tire turn in 0 s at this speed?. a) A biccle wheel has a diameter of 0cm. Determine its angular velocit, in radians per second, when the biccle travels at 0km/h. b) Determine a general epression for the angular velocit, in radians per second, for a biccle wheel with diameter d centimetres when the biccle travels at km/h. 5. The hpotenuse of an isosceles right triangle is the chord of a circle. The legs of the triangle are the radii of the circle. The length of the chord is 8cm. Determine the length of the arc subtended b the chord.. The restaurant at the top of a well-known tower rotates in a counter-clockwise direction to allow its patrons an opportunit to see a view of the entire cit. It takes 8 minutes for one complete revolution. The diameter of the restaurant is 0 metres. A woman places her purse on the window ledge b her table when she sits down. Eighteen minutes later, she realizes that her table has moved but her purse has not. How far will her husband have to walk to retrieve her purse, assuming he can t walk over tables and chairs? 7. An electric motor turns at 000 r/min. Determine the angular velocit in radians per second in eact form and in approimate form, to the nearest hundredth. 8. The measures of the acute angles in a right triangle are in the ratio :. Write the radian measures of the acute angles in eact form. 9. The revolving restaurant in the CN Tower completes 5 of a revolution ever hour. If Shani and Laszlo ate dinner at the restaurant from 9:5 to :5, through what angle did their table rotate during the meal? Epress our answer in radian measure in eact form and in approimate form, to the nearest tenth. 0. A music CD rotates inside an audio plaer at different rates. The angular velocit is 500 r/min when music is read near the centre, decreasing to 00 r/min when music is read near the outer edge. a) When music is read near the centre, through how man radian does the CD turn each second? Epress our answer in eact form and in approimate form, to the nearest hundredth. b) When data are read near the outer edge, through how man radian does a CD turn each second in a X drive? Epress our answer in eact form and in approimate form, to the nearest radian.. A circle with centre at (0,), radius, rolls along the positive -ais. The centre moves to the point (5,). Point P, initiall at the origin will roll along the circumference, as the circle turns, to the point Q(,). What are the co-ordinates of Q, accurate to two decimal places? Q(,) Answers a) 5 rad/s b) 5.cm a) P(0,0) rad/s b) 895. m a) 77. rad/s ).8 rad a) 8.5 rad/s b) rad/s 9d 5) cm ) or 5. rad b) 5 m 7) 00 rad/s or. rad/sec 8) 0 or 50 rad ) Q(.8,.) rad & rad 9) rad or. rad 0a) 9 50 rad

7 Advanced functions Page of Coterminal and Related Angles in Radians C.A.S.T. Rule : - : + : - : - Sin Tan All Cos : + : + : + : - 0 / sin cos tan Related Acute Angle Rule General rule: The given angle must be within revolution (), add or subtract multiples of to modified the given angle. (Fi the quadrant) sin sin cos cos tan tan use C.A.S.T. rule to determine the sign Eample : Epress the following into related acute angle, then evaluate. 80 rad rad 80 Do not convert to Degree first!!!!! Use to represent Radian Angles!! a) 5 sin b) tan c) cos

8 Advanced functions Page of Coterminal and Related Angles in Radians Eample : Epress the following into related acute angle, then evaluate. 9 5 a) sin b) cos c) 7 tan 5 Eample : Epress the following into related acute angle, then evaluate. 5 a) csc 5 b) sec c) cot 7 d) sec 7 Eample Find without using calculators, the value of 5 tan tan cot sec csc sin sin 9 9 Eample 5 Given A, B, C, find the value of sin A cos C tanb sec B. Homework WS: Coterminal (related & Corelated Angles (Part i onl)

9 Advanced functions Page of Coterminal and Related Angles in Radians. Evaluate each of the following b epressing each of the following in terms of i) a related acute angle. (Da ) ii) a correlated acute angle. (Da ) a) 9 tan b) cos c) 8 sin 7 d) cos 95 8 e) cos f) sin g) 8 cos 7 h) sin 7 i) sin j) 5 cos 8 k) tan l) sin m) sin 7 n) cos o) 9 tan 9 p) 7 cos q) 9 sin 7 r) cos s) tan t) 7 cos 7 u) sec v) 7 cot w) 7 csc 5 ) cot ) 5 sec z) 7 cot Answers: a) b) c) d) e) f) 0 g) h) i) j) k) - l) m) n) o) p) q) r) s) t) u) v) - w) - ) ) z)

10 Advanced functions Page of Coterminal and Corelated Angles in Radians C.A.S.T. Rule Sin Tan All Cos 0 / sin cos tan Corelated Acute Angle Rule General rule: The given angle must be within revolution (), add or subtract multiples of to modified the given angle. (Fi the quadrant) Eample : Epress the following into corelated acute angle, then evaluate sin cos cos sin tan cot 80 rad rad 80 use C.A.S.T. rule to determine the sign a) 5 cos b) 7 tan c) sin

11 Advanced functions Page of Coterminal and Corelated Angles in Radians Eample : Epress the following into corelated acute angle, then evaluate 9 5 a) cos b) csc c) 8 0 cot Eample Simplif tan sin cos tan Homework: WS: Coterminal (Related & Corelated Angles) ii

12 Advanced functions Page of Coterminal and Corelated Angles in Radians. Evaluate each of the following b epressing each of the following in terms of i) a related acute angle. (Da ) ii) a correlated acute angle. (Da ) a) 9 tan b) cos c) 8 sin 7 d) cos 95 8 e) cos f) sin g) 8 cos 7 h) sin 7 i) sin j) 5 cos 8 k) tan l) sin m) sin 7 n) cos o) 9 tan 9 p) 7 cos q) 9 sin 7 r) cos s) tan t) 7 cos 7 u) sec v) 7 cot w) 7 csc 5 ) cot ) 5 sec z) 7 cot Answers: a) b) c) d) e) f) 0 g) h) i) j) k) - l) m) n) o) p) q) r) s) t) u) v) - w) - ) ) z)

13 Advanced functions Page of Addition and Subtraction Formulas C.A.S.T. Rule Sin Tan All Cos 0 / sin cos tan sin( ) sin cos cos sin sin( ) sin cos cos sin cos( ) cos cos sin sin cos( ) cos cos sin sin tan tan tan( ) tan tan tan tan tan( ) tan tan Eample Find the eact value of the followings 5 a) sin b) tan

14 Advanced functions Page of Addition and Subtraction Formulas c) 7 cos d) sec Eample 5 If sin a, a and cos b, b, evaluate tan( a b). 5 Eample Find the eact value of 8 cos cos sin 7 sin 8 7 Eample Prove that sin cos Homework: P. #-8,0

15 Advanced Functions Page of Double Angle Formulas Double Angle Formulas ) sin sin cos ) cos cos ) tan cos sin sin tan tan Half Angle Formulas (Optional) sin sin cos Proof sin sin( ) sin cos cos sin sin cos tan tan tan Proof tan tan( ) tan tan tan tan tan tan cos cos cos cos cos sin Proof cos cos( ) cos cos sin sin cos or cos cos sin sin ( cos or ( sin sin ) sin ) Eample Epress as a single trigonometric function. a) sin 7cos 7 b) cos c) sin Eample If cos,, find 5 a) sin b) cos

16 Advanced Functions Page of Double Angle Formulas Eample If cos, epress a) cos, b) cos and c) cos in terms of cos then find the eact values. Eample Find the eact value of sin. 8 Eample 5 5 If cos sin, find the value of sin. Homework: P. #0,-5,0,,,5

17 Advanced Functions Page of 5 Trigonometric Identities Steps in Proving Identities () Start from the more complicated side. () Epress all functions in terms of sine and cosine. () Epand the compound angles. () Look for squares and the use of Pthagorean Identities. sin cos tan Related Acute Identities sin sin cos cos tan tan Corelated Acute Identities sin cos cos sin tan cot Half Angle Formulas (Optional) cos sin cos tan cos cos cos Angles sta the same Reciprocal Identities csc sin sec cos cot tan Addition & Subtraction Formulas sin( ) sin cos cos sin sin( ) sin cos cos sin cos( ) cos cos sin sin cos( ) cos cos sin sin tan tan tan( ) tan tan tan tan tan( ) tan tan Quotient Identities sin tan cos cos cot sin Pthagorean Identities sin cos sin cos cos sin tan sec cot csc Double Angle Formulas sin sin cos cos cos sin tan Product Formulas (Optional) cos( a b) cos( a b) sin asin b cos( a b) cos( a b) cos acos b cos( a b) cos( a b) tan a tan b cos( a b) cos( a b) S T A C cos sin tan tan 0 /

18 Advanced Functions Page of 5 Trigonometric Identities Eample sin Prove that cos. cos Eample Prove that cos( )cos( ) cos cos. Eample sin Prove that csc tan. cos Eample cos Prove that tan( ). sin Question of the Da!! Prove: sin sin sin cos Package. Da : #0 - (Even onl) Da : #- 5 (Odd Onl) Da : #7-5

19 Advanced Functions Page of 5 Trigonometric Identities Prove the identities. sin tan sec cos. cos sin sin. csc sec csc sec. cos cos sin sin sin cos sin cos 5. sec sec tan tan tan tan cot cot. tan tan 7. sec cos csc sin tan tan 8. cos sin sin sin 9. sec tan tan sec sin 0. cot tan sin cos. cos cos sin sin cos sin. sin tan cos cos. cos sin 0 tan tan. sin cos tan tan 5. sin sin cos cos. tan tan 7. sin cos sin sin tan tan tan tan tan 8. sin5 sin cos sin cos cos sin 9. sin cot sin cos cos cos 0. cos

20 Advanced Functions Page of 5 Trigonometric Identities. sin tan cot cos sin tan sin tan sin cos. sin cot sin csc cos. cot sec cos cos sec tan. sec sin cot sin cos tan 5. sin sec csc cot sin. tan cos cos 7. cot sin 8. csc sec csc 9. cot cot tan cos 0. tan sin cos sin. sec tan cos sin cos sin. tan cos sin. sin cos sin cos. cos sin cos 5. sec tan tan sin cos. tan cos cos 7. sin cos cos sin tan cot tan cot cos sin tan cot sin

21 Advanced Functions Page 5 of 5 Trigonometric Identities 0. sin cos tan cot sec csc. sin cos sin csc cos. sin cos sin cos sin cos. cos sec sin csc sin. tan tan z 5. sin8 8sin cos cos cos. sin sin sin sin sin cos sec tan tan z tan tan tan tan z 7. sin sin z sinz 8. 0 sin sin sin sin z sin zsin 9. tan tan cot tan sin cos cot sin sin cot 50. sec tan tan tan tan z tan tan tan z tan z tan tan tan tan z tan tan z 5. tan cot tan cot csc sin csc 5. tan tan sec sin cos 55. cos sin sin cos 5. tan cos cos

22 Advanced Functions Page of Trigonometric Identities Eample Trigonometric Identities (Using Product Formula) Eample Trigonometric Identities Eample (Eg of +/- Formulas) Prove that sin sin sin cos Eample Develop a formula for cos( a b c).

23 Advanced Functions Page of Trigonometric Identities Eample sin If sin cos and cos prove: sin( ) 7 sin cos Eample If tan,, find the value of tan.