10. Trigonometric equations

Transcription

1 0 EP-Program - Strisuksa School - Roi-et Math : Trigonometry () Dr.Wattana Toutip - Department of Mathematics Khon Kaen University 00 :Wattana Toutip wattou@kku.ac.th 0. Trigonometric equations 0. The six trigonometric ratios The right angled triangle is labeled is shown. The six ratios between the sides are : sin OPP ;cos ADJ ; tan OPP HYP HYP ADJ cot ADJ ;sec HYP ;cos ec HYP OPP ADJ OPP Some relations between these functions are as follows: sin cos tan ;cot ;sec ;cos ec cos sin cos sin Pythagoras s Theorem gives rise to the three identities: ( Here cos means (cos ) etc cos sin sec tan cos ec cot ) Example Prove the identity: tan cot sincos Write the left hand side in terms of sin and cos, and add. sin cos sin cos tan cot cos sin sin cos Use the identity: cos sin tan cot sincos

2 EP.Program Strisuksa School Roi-et. Mathematics 0. Trigonometric Functions page 0... Exercise. Prove the following identities. (a) cos sin cos (b) 4 4 cos sin cos sin (c) 4 4 sec tan sec tan (d) sec cos ec sec cosec (e) tan sin tan (f) sec sin sin (g) cos tan sec sin. If sin x find, without using a calculator, the values of cos ecx,cos x, tan x.. If tan x find, in square root form, the values of sec x,cos x,sin x.

3 EP.Program Strisuksa School Roi-et. Mathematics 0. Trigonometric Functions page t 4. If sin x find cos x and tan x in terms of t. t. Simplify the following expressions: (a) (b) cos x tan x (c) sin x cot x cot x (d) cot x 6. If x cos and y sin show that x y Find an equation involving x and y, without, from the following. (a) xcos, y sin (b) x cos, y sin (c) xtan, y sec (d) x cot, y co sec (e) x cos, y sin (f) x cos, y sin (g) x cos sin, y cos sin (h) x sec tan, y sec tan. 0. Trigonometric functions for other angles The definition of the trigonometric functions can be extended to angles other than acute angles. For sin sin(80 ), cos cos 80, tan tan 80 For80 70 sin sin 80, cos cos 80, tan tan 80

4 EP.Program Strisuksa School Roi-et. Mathematics 0. Trigonometric Functions page 4 For sin sin 60, cos 60, tan tan 60. cos ec,sec and cot obey the same rule as sin,cos and tan respectively. For angles greater than 60 or less than 0, sin sin 60 hold for all the other functions..similar rules 0.. Example. Solve for 0 80, cos 0. Using the inv cos button, 76.y the formulae above, there is also a value in the range 70 to or or Exercises. Solve for Write 0 60, cos sin 0. cos as sin and rearrange. sin sin 0. This factorize, giving : sinsin The first bracket gives sin, so 90. The second bracket gives sin, which has solution in the ranges 80 to 70 and 70to or 0 or 0. Express the following in terms of rations of acute angles. (a) sin6 (b) cos 0 (c) tan (d) cot0 (e) sec70 (f) cos ec784. Solve the following equations, giving all the solutions in the range 0 to 60. (a) sin 0. x (b) tan x (c) sin x (d) cos x (e) sec x 4

5 EP.Program Strisuksa School Roi-et. Mathematics 0. Trigonometric Functions page (f) cos ec x 0. (g) (h) cot. x cos 0. x (i) sin x 0 0. (j) x tan 0.9. Find x in the range 0 x 60, if sin x and 4 cos x. 4. Find x in the range 0 x 60, if cos x and sin x.. If 6. If 8 sin x find cos x, given that x is between 90 and tan x find cosecx, given that x is between 90 and Compound angles 7. Solve the following equations, giving all the solutions in the solutions in the range 0 to 80 (a) sin x 0. (b) tan x (c) cos x sec x (d) tan x cot x (e) cos x sin x (f) sin x tan x (g) tan xsec x 7 0 (h) 4cos xsec x 4 (i) tan xcot x Trigonometric functions of sums and differences of angles can be found by the following formulae. sin A sin Acos cos Asin sin A sin Acos cos Asin

6 EP.Program Strisuksa School Roi-et. Mathematics 0. Trigonometric Functions page 6 cos A cos Acos sin Asin cos A cos Acos sin Asin tan tan A A tan A tan tan Atan tan A tan tan Atan Important case occur when A. sin A sin Acos A cos A cos A sin A cos A sin A tan A tan A tan A The following are known as factor formulae: sin C sin D sin C D cos C D sin C sin D sin C D cos C D cos C cos D cos C D cos C D cos C cos D sin C D sin D C 0.. Examples. If and are acute angles, for which without using a calculator.] sin and sin, find sin

7 EP.Program Strisuksa School Roi-et. Mathematics 0. Trigonometric Functions page 7 Using the identity above: cos A sin A, cos and 4 cos.apply the formula sin sin cos cos sin 6 sin 6 4. Use the fact that cos 0 to find an expression for cos, without using a calculator. Let A in the formula for cos A above. cos 0 cos. Make cos the subject of this formula.. Prove the identity : cos 4 sin x sin y tan x y sin x sin y tan x y Factorize both top and bottom of the left hand side : sin x y cos x y sin x sin y sin x sin y cos x y sin x y sin x ycos x y cos( x y)sin( x y) Use the fact that tan sin / cos. sin x sin y tan x y sin x sin y tan x y 0.. Exercises. Simplify the following expression, without using a calculator. (a) sin0 cos0 cos0 sin 0 (b) cos0 cos0 sin0sin0 (c) sin A cos A cos Asin A (d) cos A sin A

8 EP.Program Strisuksa School Roi-et. Mathematics 0. Trigonometric Functions page 8. Simplify (a) sin 40cos40 tan 0 (b) tan 0. Letting be the angle such that 4.. (a) 4 sin cos (b) 4 cos sin 4 sin,cos, express in terms of and cos 4 sin 4,sin 0 cos 60,sin 60 cos 0. Use these values to obtain square root expressions for the following.do nor use a calculator. (a) sin 7 (b) sin (c) cos7 (d) tan (e) cos. (f) tan.. Without using a calculator find the value of tan x, given that x 6. If and are acute angles so that the following. (a) sin (b) sin (c) cos (d) tan tan 4 sin and sin, find the values of 7. Solve the following equations, for values of between 0 and 80. (a) sin.7sin (b) cos cos (c) sin 0 cos 60 (d) 4sin 4 cos 4 (e) tan tan (f) tan 4 tan 8. Prove the following identities. sin( A ) sin A sin Acos (a)

9 EP.Program Strisuksa School Roi-et. Mathematics 0. Trigonometric Functions page 9 cos A cos A cos Acos (b) 9. Use the factor formulae to prove the following identities: cos A cos (a) tan A sin A sin sin X siny (b) tan X Y cos X cosy (c) x x x sin sin 0 sin Use the factor formulae to solve the following equations, for 0 x 80: (a) x x sin sin 60 (b) cos xcosx 0 (c) sin x sin cos x (d) cos x0 cos x0 (e) sin x 0sin x0 (f) cos x cosx sin x sin x. Writing A a A, find an expression for cosa in terms of cos A only.. Find an expression for sin A in terms of sin A only. 0.4 The form asin bcos asin bcos b a b sin, where tan. a asin bcos a a b sin, where tan. b 0.4. Examples. Find the range of the function y 8sin x cos x. Use the formula above. 8 sin 7 sin y x x Whatever the value of, sin x 7 y 7 lies between and.. Solve the equation sin x4cos x,for 0 x 60. Use the formula above to rewrite the equation. sin x, where tan.. x sin. or 68. or 7.. When is subtracted the first value becomes out of range. x or 8 4

10 EP.Program Strisuksa School Roi-et. Mathematics 0. Trigonometric Functions page Exercises. Write the following in the form rsin x, stating the values of r and. (a) sin x 4cos x (b) 4sin x cos x (c) 6cos x sin x (d) sin x cos x (e) sin x cos x 4 (f) sin x cos x 4 8. Find the greatest value of the functions in Question, and the value of x which gives the maximum.. Write down the range of values of the function in Question. 4. Solve the following equations, foe values of x between 80 and 80. (a) sin xcos x (b) 4sin xcos x (c) 7sin x8cos x 9 (d) sin xcos x. Write y sin x 4cos x in the form rsin( x ).Show that y can also be written as scos( x ),giving the values of s and. Solve the equation y. in the range 80 x 80, using both forms of y. Check that your answer are the same. 0. Examination questions. Prove the identity: sin x cos x sec x cos x sin x. Solve the equations below, giving all the answer in the range 0 x 60 : (i) cos x 0. (ii) tan x4cot x (iii) 8sin x 8sin x cos x 0. (i) On the same diagram, sketch the graphs of y cos xand y sin x in the interval 0 x 60. (ii) y using the formula for sin x cos x in the same interval.

11 EP.Program Strisuksa School Roi-et. Mathematics 0. Trigonometric Functions page Common errors (iii)solve the inequality sin x cos x in this interval. 4. (i) Without the use tables or a calculator, find the values of (a) sin 40 cos0 cos 40 sin0 (b) cos sin (c) tan tan (ii)find, to one decimal place, the elements of the set x: 0 x 60 which satisfy the equation sin x cos x 60 sin A. Show that tan Atan and use it to find all the angles in the cos Acos range 0 x 80 which satisfy the equation tan x tanx tan 4x. 6. Determine all of the angles between 0 and 80 which satisfy the equation cos cos sin sin 7. Express cos x 4sin x in the form Rcos x, where R is positive and lies between 0 and 90. What are the maximum and minimum values of cos x 4sin x, and for what values of x in the range 0 to 60 do they occur? Draw the graph of y cos x 4sin x for values of x, at intervals of 0, in the range 0 to 60, taking cm to represent 0 on the x axis and cm to represent unit on the y axis. Using your graph, and giving solutions in range 0 to 60 : (i) Solve the equation 6cos x8sin x 7 (ii) Solve the inequality 6cos x8sin x 7.. Identities Do not confuse identities and equations. An identity is true for all values of x. An equation is only true for some values, and we find these values when we solve the equation. When proving an identity, do not assume the answer, i.e. do not start with the expression you are asked to show true.start with one side, and show that it can be reduced to the other side. Alternatively, show that both sides van be reduced to the same expression.. Solving equations When you are solving an equation in the range 0 to60, you must give all the solutions. The trig functions take each value twice in that range.

12 EP.Program Strisuksa School Roi-et. Mathematics 0. Trigonometric Functions page (to exercise) When solving say sin x 0. in the range 0 to 60, you must consider the values of sin 0. up to 70.When you divide x by the value will be less than 60. When solving an equation of the form x sin 0. in the range 0 to 60, you may have to consider values of sin 0. greater than 60, for when is subtracted, the result may be within the correct range. When you solve a quadratic in cos x, you might get the answer cos x or.only the value is relevant here-do not worry about the impossible value of.. Compound angles These formulas must be used when finding sin or cos of compound angles. e aware of the following : ,, 4,, t t, t t. (a) sin x 7. (a) x y 4 sin A sin A sin (b) sec x x y y 9 (b) (c) y 4 9 4x (e) x y (g) x (c) cos xsin x (d)sin xcos x x (d) x y (f) y (h) xy. (a) sin 4 (b) cos 40 (c) tan (d) cot0 (e) sec0 (f) cos ec64. (a), (b)6,6,6,06 (c)0,7 (d) 86,4 (e) 76,84 (f) 9,88 (g) 44 (d)

13 EP.Program Strisuksa School Roi-et. Mathematics 0. Trigonometric Functions page (i) 4,6 (j)68, (a) 4, (b) 60,0 (c), (d), (e) 79 (f) 0,80 (g) 60,7 (h) 60 (i) 4. (a) sin 40 (b) cos 0 (c) sin A (d) cos 4A. (a) sin80 (b) tan 40]. (a) sin x (b) cos 4. (a). (e) (b) (f) (c) 6. (a) 6 (b) 6 (c) 6 (d) (a) 0,80, (b)9 (c) all angles (d) 4 (e) 0,80,0,0 (f) no solutions 0. (a) 6,7 (b)90,4, (c) 4,,0,070. (d) 7 (e) 60,0 (f) 4cos A cos A. sin A 4sin A (d) 90,,. (a),. (b) 4, (c) 4,7 (d), 4 (e),7 6 (f) 4, 6. (a),7 (b) 4,4 (c) 4, 7 (d), (e),6 6 (f) 4,9 6

14 EP.Program Strisuksa School Roi-et. Mathematics 0. Trigonometric Functions page 4. (a) f (b) 4 f 4 (c) 4 f 4 (d) f (e).9 f 0.9 (f).0 f.0 4. (a) 40,08 (b) 79, 7 (c) 9,7 (d) 96, 70. r,.. s, 6.9. x,97. =========================================================== References: Solomon, R.C. (997), A Level: Mathematics (4 th Edition), Great ritain, Hillman Printers(Frome) Ltd. More: (in Thai)