TRIGONOMETRY INTRODUCTION s the title of the unit suggests, it deals with the calculation of angles or the length of their sides. In this unit, the trigonometric ratios of acute angles, general angles as well as special angles are explained with illustrations. The unit further gives explanation to other trigonometric identities like difference of two angles, tangents of compound angles and multiple angles. Objectives On completion of this unit, it is expected that you would be able to: differentiate angles based on the four quadrants explain the trigonometric ratios of acute and general angles apply trigonometric functions to solve problems involving all angles SESSION - NGLES positive angle measures a rotation in an anticlockwise direction. V O θ x 6 Fig.. One complete revolution of the arm OV (Fig.. starting from an initial positionox, is equivalent to 6. n angle can also include more than one complete revolution. For example, an angle of 9 o = Two complete revolutions (7 o + o. The o in this case will be called the basic angle. The basic angles will fall into one of four quadrants (Fig... 9 o
nd quadrant st quadrant 8 o o rd quadrant 4 th quadrant Fig.. 7 o -. Trigonometrical Ratios for cute ngles (First quadrant Fig.. shows the rotating arm OV (of unit length and the angle θ (measured in degrees is measured from the initialox. OP is called the x-component, PV the y-component, of OV. y V Y component θ O x component P x Fig.. We define the trigonometrical ratios as follows: sin y y, cos x x, tan y. x These definitions agree with those you have already used in dealing with right-angled triangles.
Fig..4 shows a right-angled triangle BC B c a θ C b Fig.. a x y Then sin, cos x, tan. (If c is of unit length, we have the above c x definitions a a sin lso, tan c c b cos c gain, by Pythagoras Theorem, a + b = c Then a c b c or sin cos Though we have derived these relationships using an acute angle, they are identities, i.e., true for any angle and they should be remembered. tan sin cos sin cos s you already know, the numerical values of the trigonometrical ratios for acute angles can be obtained from tables. -. Trigonometrical Ratios for a General ngle Since the tables only give numerical values of the ratios for an acute angle, we cannot directly find, say, tan9 o or cos7 o. We must first find a relationship between the ratios for such angles and a corresponding acute angle. s we shall see, this depends on which quadrant the
basic angle lies in. To do this, the definitions sin θ = y, cos θ = x and tan θ = x y, where x and y are the components of the unit rotating arm, are used throughout. Due regard is paid to the signs of x and y. -.. Second quadrant (Fig.. Here x is negative, y is positive. is obtuse and the corresponding angle (8 - θ is acute. V +y y P 8 - θ -x O θ x Fig. Then Sin y sin(8 Cos x Cos(8 y tan x y x tan(8. Hence, if θ lies in the second quadrant, then Sin Sin(8 Cos Cos(8 tan tan(8 s sin sin(8 cos68 tan98 sin4 cos( 8 68 cos tan(8 98 tan8.77.978 7.
-.. Third quadrant (Fig.6 When the rotating arm OV lies in the third quadrant, both components x and y are negative. The corresponding acute angle POV ˆ is 8. y P -x θ x O -y V Fig..6 Then sin y sin( 8 Cos x Cos( 8 tan y x y x tan( 8 So, if θ lies in the third quadrant, Sin Sin( 8 Cos Cos( 8 tan tan( 8 Note that the TN is s Sin6 Sin(6 Cos6 tan97 8 Sin6 cos(6 8 Sin6 tan(97 8 tan7.9.9.7
-.. Fourth quadrant (Fig..7 In this quadrant the x component is positive, the y component negative. The corresponding acute angle is P OV ˆ (6. y θ O +x 6 -θ P -y x V Fig.7 Then sin y sin(6 cos x cos(6 y y x x Note that the COS is + tan tan(6 s sin sin(6 sin7.68 cos cos(6 cos7.7986 tan tan(6 tan 7.76 -..4 Negative angles The rotating arm will describe a negative angle if it rotates in a clockwise direction. To convert a negative angle to a normal base angle, add 6 o or a multiple of 6 o. The value of a trigonometrical ratio of any negative angle can then be found. s n angle of 4 o is equivalent to a base angle of 4 o + 6 o = o. Then
sin( 4 sin( sin(6 sin4, cos( 4 cos( cos(6 cos4 tan( 4 tan( tan(6 tan 4 Hence in general: sin( sin cos( cos tan( tan nd these will be true for any negative angle. The relationships given above connecting any basic angle and a corresponding acute angle are summarized in Fig.8. These results are very important and should be remembered. Second Quadrant Sin sin positiive cos 8 Tan positive sin(8 cos(8 tan(8 Third Quadrant (8 (8 tan(8 Fig.8 9 First Quadrant sin ll Positive cos tan Cos positive sin(8 cos(8 tan(8 7 Fourth Quadrant, 6
Find sin494 494 6 4 sin494 sin4 sin(8 4 sin(46.79 If cosx = -.49, find the values of x between and 6. Solution Since the cosine is negative, x must lie in either the nd or rd quadrants. From tables, the corresponding acute angle is 6. 97. Hence, for the nd quadrant, 8 x 6.97 or x 9.77 For the rd quadrant, x 8 6.97 or x 4.97 Then the required values of x are 9.79 and 4.97 The sine of an angle is -.746 and its tangent is positive. Find the angle, if it lies between and 7. Solution The basic angle must lie in the third quadrant (negative sin and positive tan. The corresponding acute angle from tables is. Hence if x is the required basic angle, x 8 or x. The angle can therefore be or + 6 n (any multiple of 6 added, where n,,,... To obtain an angle between and 7 we choose n. Then the angle = + 6 = 6.
-. Trigonometrical Ratios of Special ngles The trigonometrical ratios of the angles, 4, 6 are often used in mechanics, and other branches of mathematics, so it is useful to have their values in surd form. square, BCD, is drawn with sides of unit length (Fig.9. C, a diagonal, is drawn. 4 sin4 cos4 BC C B C BC B tan 4 D Fig..9 4 C B n equilateral triangle, BC, of side units in length, is drawn (Fig... D is drawn from perpendicular to BC. 6 ˆ CD 6 D sin6 C CD cos6 C tan 6 D CD 6 B D Fig.. C ˆ C D (Fig.. sin cos tan D C CD C D CD
-.4 The Remaining Trigonometrical Ratios The three remaining ratios are the reciprocal of the sine, cosine and tangent. They are: Secant ; Cosecant ; Co tan gent. cosine Sine tan gent The values of these ratios in the four quadrants are determined from the reciprocals of the principal ratios. These ratios are used in problems involving calculations where it is more convenient to use their values, read from tables, than it is to use the reciprocals of the three principal ratios. SESSION - FURTHER TRIGONOMETRICL IDENTITIES Sin Cos Sin Cos Sin Now divide both sides by Cos : ; tan Cos Cos Cos Cos but and sec. Cos Therefore tan Sec Similarly, by dividing the original identity by Sin, we obtain Cot Cosec -. Difference of Two ngles These results are very important and must be remembered: they are summarized below: sin( B sincosb cossinb sin( B sin cosb cos sinb cos( B coscosb sinsinb cos( B cos cosb sin sinb These formulae are identities, as they are true for any value of angles and B. The formulae can be use as alternative ways of determining the ratios of angles of any magnitude. For example:
sin sin(6 sin6 cos cos6 sin cos ( sin sin. cos 4 cos(8 6 cos8 cos6 sin8 sin6 ( cos6 sin6 -cos6. They can also be used to find the value of negative angles. For example: sin( sin( sin cos cos sin cos ( sin sin. cos cos ( ( cos cosb sin sin ( cos sin cos. The formulae can also be used to find the ratios of compound angles in a simple way. Care must be taken over the signs of the ratios involved. 4 If Sin and CosB, Find the value of Sin( B and Cos( B without using tables, (i If and B are both acute angles, (ii If is obtuse and B is acute.
Solution 4 then 6 if is cute. - if is Obtuse If Sin Cos Sin If Cos( B then SinB Cos B 44 - as B is Obtuse 69 ( i Sin( B SinCosB CosSinB 4 6 x x 6 Cos( B CosCosB SinSinB 4 6 6 4 ( ii Sin( B 6 4 6 Cos( B. 6 If special angles are used, the answer can usually be expressed in surd form. Find the value of Sin, leaving the answer in surd form. Sin Sin(4 Sin4 Cos 6 4 Cos4 Sin The formulae can also be used to reduce certain trigonometrical expressions to a single ratio.
Express as a single ratio: (a Cos Cos48 -Sin Sin48 ; (b. Sin46 Cos44 cos46 Sin44 Solution ( a Cos( B CosCosB SinSinB. and ( ; B 48 Cos Cos48 Sin Sin48 Cos( 48 Cos8 ( b Sin( B SinCosB CosSinB, and ( 46 ; B 44 Sin Cos Cos Sin Sin 46 44 46 44 (46 44 Sin9
-. Tangents of Compound ngles The tangents of compound angles can be deduced from the formulae for the sines and cosines of compound angles. Sin( B tan( B Cos( B Similarly tan( B SnCosB CosSinB CosCosB SinSinB SinCosB CosSinB CosCosB CosCosB CosCosB SinSinB CosCosB CosCosB tan tanb - tantanb Sin( B Cos( B Divide top and bottom by CosCosB SnCosB CosSinB CosCosB SinSinB SinCosB CosSinB Divide top and bottom CosCosB CosCosB CosCosB SinSinB by CosCosB CosCosB CosCosB tan tanb tantanb These formulae are important and must be remembered; they are summarized below tan tan B tan( B tan tan B tan tan B tan( - B tan tan B The tangent formulae are used in the same way as those for sine and cosine. If tan and tan B, both and B being acute angles, find the value of tan( B 4 without using tables.
Solution tan tan B tan( B tan tan B 4 9 4 4 4 4 9 4 If tan(-b and tan, find the value of tanb. Solution tan tan B tan( B tan tan B -tanb tanb i.e. tan B tan B 9 7 tan B 9. i.e. tan B 7 Express the following as a single trigonometrical ratio: tan. tan Solution tan tan B tan( B tan tan B hence, tan i.e. 6 (or 4 and B tan tan tan(6 or tan(4.
-. Multiple ngles The formulae for compound angles can be used to find values of multiple angles. In the first instance, angle B is made equal to angle. Hence sin sin( sin cos cos sin sincos cos cos( cos cos sin sin cos sin But cos sin sin cos cos cos a ( cos cos lso cos sin sin cos (-sin -sin tan tan tan tan( tan tan tan -tan The formulae for multiple angles are important, and should be memorised; they are summarised as sin sin cos cos cos sin tan tan tan cos sin Higher multiple angles can be found by building up on the results already found, e.g. sin = sin (+; the right hand expression is expanded, using the compound angle formula, then sin is substituted from the results above. Similarly, cos4 = cos( + in this way multiples of can be found in terms of ratios of single angles.
-.4 Half ngles Half angles can be substituted in the compound angle formulae: sin sin sin cos similarly,cos cos sin cos - sin tan and tan tan useful substitution in later trigonometrical work can be derived by expressing the trigonometrical ratios in terms of the tangent of the half angle. sin sin cos sin cos sin cos sin cos original expressionis unaltered sin cos sin cos cos cos cos dividing top and bottom by cos tan Now put tan t tan t we have sin t cos cos sin cos sin cos sin - t after dividing top and bottom by cos t
tan tan tan t, t To summarize the results: t sin t t cos t t tan t These are called the t formulae, where t tan. Express sin in termsof sin. sin sin( sin cos cossin (sin cos cos ( sin sin cos sin ( sin sin sin sin 4sin ( sin ( sin sin sin. sin sin sin Express simply sin 4. Now cos sin ; put 4. cos84 sin 4.
tan Evaluate, without using tables,. tan tan a Now tan ; tan put tan tan 6 tan. If tan, find ( i sin ; ( ii tan without using tables. 4 Now sin cos ie.. tan sec (dividing throughout by cos tan cos tan cos 9 6 6 4 If θ is in the st quadrant, cos 4 ; if θ is in the rd quadrant, cos 4. sin cos 6 9 (sinθ, similarly, can be in the st quadrant, and in the rd quadrant. 4 sin sin cos (Only a positive result as, if θ is in the st quadrant, both sin θ and cos θ are positive; if θ is in the rd quadrant, both ratios are negative. 6 tan 4 4 4 tan as tan is givenas 9 7 4. tan 7 6 6
θ is an obtuse angle and tan. Without using tables, find the value of (a tan θ; (b cos θ; (c cos4 θ. tan tan tan Let tan t then t tan t i. e. 4t t which is t 4t i. e. (t ( t whence t Since θ is obtuse, tan θ = -. or -. If θ is obtuse, and tanθ is positive, θ must lie in the rd quadrant. cos cos4 cos tan 44 (Hence 4θ lies in the st or 4 th quadrant. 44 69 44 69 9. 69 -. The Factor Formulae Using the compound angle formulae, the sum of two sines can be written as; sin( B sin( B sin cos B cos sin Bcos B cos sin B sin cos B Now let B X and B Y Then X Y X Y and X-Y B X-Y B Substitution the values of X and Y for and B, we have: sin X siny X Y X Y sin cos
The formula for the sum of two cosines can be written as: cos( B cos( B cos cosb sin sin B cos cosb sin sin B cos cosb Substituting the values of X and Y for and B, we have: cos X cosy X Y X Y cos cos Express in factors; sin sin. The factors are obtained from the formula for the sum of two sines. X Y X Y sin X siny sin cos (here X and Y sin sin sin cos sin cos The difference of two sines can be written as: sin( B sin( B sin cosb cos sin B (sin cosb cos sin B cos sin B Substituting the values of X and Y for and B, we have: sin X siny X Y X Y cos sin The difference of two cosines can be written as: cos( B cos( B sin cosb cos sin B (sin cosb cos sin B cos sin B Substituting X and Y for and B, we have: cos X X Y X Y cosy sin cos. Remember the negative sign in this formula.
Express in factors: cos cos. X Y X Y cos X cosy sin sin cos cos sin sin sin 4 sin The factor formulae are important and should be memorised; they are summarised below: X Y X y sin X siny sin cos X Y X Y sin X siny cos sin X Y X Y cos X cosy cos cos X Y X Y cos X cosy sin sin These formulae can also be remembered verbally, as under: sin sin B sin sin sin B cos cos cosb cos cos cosb sin ( Bcos ( Bsin ( Bcos ( Bsin ( B ( B ( B ( B Express sin cos as a differencebetween X Y X Y sin X siny cos sin i. e X Y X Y.........( and X - Y X - Y 4.........( dding ( and (, X 4 X 7 Subtracting ( from(, Y 6 Y sin cos cos sin sin7 - sin. two trigonometrical ratios.
-.6 Simple Identities trigonometrical identity is an expression that is valid for all values of the angles contained in the expression, e.g. sin cos is true for values of θ. The three basic identities, already found, are: sin cos tan sec cot cosec These three basic identities, together with the identities for compound and multiple angles and the identities of the factor formulae, can be used to manipulate trigonometric expressions into different forms.