Trigonometric Identities An identity is an equation that is satis ed by all the values of the variable(s) in the equation. We have already introduced the following: (a) tan x (b) sec x (c) csc x (d) cot x tan x The trigonometric identity sin x + cos x The trig identity sin x + cos x follows from the Pythagorean theorem. In the gure below, an angle x is drawn. The side oosite the angle has length a, the adjacent side has length b, and the hyotenuse has length h. h a x b Therefore, a h and b. It follows that h () + () a b + h h a h + b h a + b h The Pythagorean theorem asserts that a + b h. It follows that () + () a + b h h h For convenience, () and () are written more brie y as sin x and cos x resectively. Therefore we have veri ed that if x is a given angle then sin x + cos x You are exected to have this identity on your nger tis. Two useful identities are derived from sin x + cos x by dividing both sides of the identity by sin x or cos x. If we divide both sides of sin x + cos x by cos x, the result is sin x cos x + cos x cos x OR + Since tan x and sec x, it follows that (tan x) + (sec x). For convenience, we write (tan x) and (sec x) more brie y as tan x and sec x. Therefore we have veri ed that tan x + sec x
If we divide both sides of sin x + cos x by sin x, the result is sin x sin x + sin x sin x OR + Since cot x and csc x, it follows that + (cot x) (csc x). For convenience, we write (cot x) and (csc x) more brie y as cot x and csc x. Therefore we have veri ed that cot x + csc x You will be required to verify identities. One way to do so is to start with one side of the identity and through a series of algebraic oerations, derive the other side of the identity. Here is an examle: Examle To verify the identity + sec x. We start with the left hand side because there is more we can do with it than the right hand side. Denote the left hand side by LHS and the right hand side by RHS. Then LHS + + + cos x sec x RHS (since we need a common denominator in order to add the two fractions) (since + cos x ) We have veri ed that both sides of the identity are equal. This veri es the identity. Exercise. The trigonometric exression + + can be reduced to: (A) (B) (C) cos x (D) sec x. Verify each identity: a) sec x cot x tan x b) tan x c) tan x + cot x sec x csc x d) sec x sin x e) sec x tan x sec x csc x f) + tan x sec x g) + cot x csc x h) sec x tan x i) csc x cot x sec x csc x j) + cot x k) sin x tan x cos x l) tan x
m) sec x tan x n) tan x + + cot x o) tan x + cot x sec x csc x q) s) cot x + ) tan x + r) tan x + tan x sec x csc x cos x + t) sin x + u) tan x + + sec x v) sec x + csc x sec x csc x w) y) + sec x x) sin y cot y + + 0 cos y tan y sin y + tan y z) ( ) + ( + ). Write cos x sin x as a di erence of two squares, factor it and use the result to show that cos x sin x cos x sin x. Identities for di erences or sums of angles In this section we walk you through a derivation of the identity for cos(y x). This will be used to derive others involving sums or di erences of angles. Contrary to what one would exect, it is NOT TRUE that cos(y x) cos y for all angles x and y. Almost any air of angles x and y gives cos(y x) 6 cos y. For examle, the choice y 0 and x 0 gives cos (y x) cos (90 ) 0 which is not equal to cos 0 cos 0, (you can easily check that). To derive the identity, start with a circle of radius and center (0; 0), labelled P in Figure below. Figure
Consider a line P Q that makes an angle y, in the second quadrant, with the ositive horizontal axis. Figure Can you see why the coordinates of Q must be (cos y; sin y)? The gure below includes a line P R making an angle x in the rst quadrant, with the ositive horizontal axis. Figure The coordinates of R are (; ). A standard notation for the length of the line segment QR is jjqrjj. Use the distance formula to show that jjqrjj simli es to jjqrjj (cos y + sin y ) () If you rotate the circle in Figure counter-clockwise until the ray P R merges into the ositive horizontal
axis the result is Figure below. Figure Angle QP R is y x, therefore Q has coordinates (cos (y c) ; sin (y x)). Clearly R has coordinates (; 0). It follows that the length of QR is also given by jjqrjj [cos(y x) ] + [sin (y )] Show that this simli es to Comaring () and () reveals that jjqrjj cos(y x) () What do you conclude about cos(y You should obtain cos(y x) (cos y + sin y ) x)? cos (y x) cos y + sin y This is our rst identity for the cosine of the di erence of two angles. Examle We know the exact values of cos 0, sin 0, cos and sin. We may use them to calculate cos. The result: cos cos ( 0 ) cos cos 0 + sin sin 0 + + 6 + Examle We are given that x is an angle in the rst quadrant with, y is an angle in the third quadrant with cos y and we have to determine cos (y x). Since cos (y x) cos y + sin y, we have to nd and sin y. Diagrams will hel. In Figure we have a right triangle with a hyotenuse of length and a horizontal side of length because horizontal coordinate length of hyotenuse
The vertical side of the triangle is labelled b because it is not known. Pythagorean theorem: + b We may determine b using the Therefore b, hence b. We take the ositive sign because the vertical coordinate is ositive in the rst quadrant. Therefore b and so. In Figure 6 we have drawn a rectangle with a hyotenuse with length and a horizontal coordinate because cos y horizontal coordinate length of hyotenuse The vertical coordinate is unknown, therefore we labelled it a. By the Pythagorean theorem ( ) + a + a Therefore a 8, hence a 8. In this case we must take the negative sign because the vertical coordinate is negative in the third quadrant. Therefore sin y 8 Figure Figure 6 It follows that cos (y x) cos y + sin y +!! 8 0 Other sum/di erence identities cos (y + x) cos y sin y. To derive this, we use the fact that cos( A) cos A and sin( B) sin B for all angles A and B. Therefore cos (y + x) cos (y ( x)) cos y cos( x) + sin y sin( x) cos y sin y Examle cos 7 cos (0 + ) cos 0 cos sin 0 sin 6 sin (y x) sin y + cos y. To derive it, we use the fact that cos(90 A) sin A and sin(90 B) cos B for any angles A and B. Therefore sin(y x) cos (90 (y x)) cos ((90 y) + x) cos(90 y) sin(90 y) sin y cos y Examle 6 sin sin ( 0 ) sin cos 0 cos sin 0 6 6
sin (y + x) sin y +cos y. We derive this using the fact that cos( A) cos A and sin( B) sin B for all angles A and B. Therefore sin (y + x) sin (y ( x)) sin y cos( x) cos y sin( x) sin y cos y( ) sin y + cos y Examle 7 sin 7 sin (0 + ) sin 0 cos + cos 0 sin + + 6 tan y + tan x tan (y + x) tan y tan x sin(x y) tan (y x) cos(x y). tan y tan x and tan (y x). These follows from the fact that + tan y tan x Examle 8 tan 7 tan (0 + ) tan 0 + tan tan 0 tan Exercise 9 + +. Given that x is an angle in the second quadrant with, and y is an angle in the fourth quadrant with cos y, the exact value of sin (x + y) is (A) (B) 8 (C) 7 (D). Given that x is an angle in the rst quadrant with, and y is an angle in the third quadrant with cos y, the exact value of cos (x y) is (A) 6 (B) 6 6. You are given that x and y are angles in the rst and fourth quadrants resectively, with and cos y. Draw the two angles then determine the exact values of the following exressions: (C) a) b) sin y c) tan x d) tan y e) sin (x + y) f) sin (x y) g) cos (x + y) h) cos (x y) i) tan (x y) j) tan (x y) k) csc (x y) l) cot (x + y). If x is an angle in the rst quadrant with, and y is an angle in the fourth quadrant with cos y, determine the exact value of each exression: a. sin (x y) b. cos (x + y) c. sin (x + y) d. tan (x y) e. sec (x y) f. cot (x + y) The Double Angle Formulas These are formulas that enable us to calculate,, tan x and their recirocals, once we know the values of, and tan x, hence the term "double angle formulas". They are derived from the identities for sums of angles. If we relace y by x in the identity sin (x + y) cos y + sin y, we get 6 6 (D) sin (x + x) + 6 6 Thus This is our rst "double angle formula" () 7
If we relace y by x in the identity cos (x + y) cos y sin y, we get Thus cos (x + x) cos x sin x cos x sin x () There are two other versions of this formula obtained by using the identity sin x + cos x. If we solve for sin x to get sin x cos x then substitute into () we get I.e. cos x sin x cos x ( cos x) cos x cos x If, on the other hand, we solve for cos x to get cos x I.e. sin x then substitute into () we get cos x sin x sin x sin x sin x sin x Which one of these three formulas for to use deends on the roblem at hand. If we relace y by x In the identity tan(x + y) Thus tan(x + x) tan x tan x tan x + tan y tan x tan y we get tan x + tan x tan x tan x tan x tan x tan x tan x Examle 0 To determine, and tan x given that x is an angle in the second quadrant with. The angle is shown below. The horizontal coordinate a is unknown but we easily calculate it using the Pythagorean theorem. Thus + a which translates into a. This imlies that a. Since the horizontal coordinate must be negative, we must choose a. From the gure, a and tan x a. Therefore 9 : cos x 0 9 9. tan x tan x tan x tan x 0 8
Rules for Changing Product to Sum/Di erences In some roblems, it is easier to deal with sums of two trigonometric functions than their roduct. In such cases, we use the "Product to Sum" rules to convert roducts like cos y, sin y and cos y into sums or di erences involving sine or cosine terms. We use the sum/di erence rules we have already derived which are: sin A cos B + cos A sin B sin (A + B) () sin A cos B cos A sin B sin (A B) (6) cos A cos B + sin A sin B cos (A B) (7) Adding () to (6) gives cos A cos B sin A sin B cos (A + B) (8) sin A cos B [sin(a + B) + sin(a B)] Adding (7) to (8) gives cos A cos B [cos(a B) + cos(a + B)] Subtracting (8) from (7) gives sin A sin B [cos(a B) cos(a + B)] Examle To write, cos x cos x and as sums/di erences of trig functions. We use sin A sin B [cos(a B) cos(a + B)] to write as a sum of trig functions. Take A x and B x. Then [cos( x) cos 8x] [cos x cos 8x] For cos x cos x, we use cos A cos B [cos(a B) + cos(a + B)] with A x and B x. The result is cos x cos x [ + cos 9x] For, we rst write it as and use sin A cos B [sin(a + B) + sin(a A x and B x. The result is B)] with Sum/Di erence to Products [ + ] [ + ] There are roblems that require us to change a sum or di erence of sine or cosine terms into a roduct of sine and/or cosine terms. In other words, we may want to do the oosite of what we did in the above section. We use the same identities as above but now we write them as: sin (A + B) sin A cos B + cos A sin B (9) sin (A B) sin A cos B cos A sin B (0) cos (A + B) cos A cos B sin A sin B () cos(a B) cos A cos B + sin A sin B () We then introduce variables P A + B and Q A B. We have to write A and B in terms of these new variables. It turns out that A (P + Q) and B (P Q) 9
Adding (9) to 0 gives a sum of sines which is sin (A + B) + sin(a B) sin A cos B Using the new variables P and Q this result may be written as sin P + sin Q sin (P + Q) cos (P Q) Subtracting 0 from (9) to gives a di erence of sines which is Adding () to gives a sum of cosines which is sin P sin Q cos (P + Q) sin (P Q) cos P + cos Q cos (P + Q) cos (P Q) Subtracting from () to gives a di erence of cosines which is cos P cos Q sin (P + Q) sin (P Q) Examle To write cos x + cos x, +, and cos x as roducts of trig functions: Using the above identities: x + cos x cos (x + x) cos (x x) cos x + sin (x + x) cos (x x) cos (x + x) sin (x x) cos 7 x sin x The Half Angle Formulas cos x sin (x + x) sin (x x) sin x sin x These are formulas that enable us to calculate sin x, cos x, tan x and their recirocals, once we know the values of, and tan x, hence the term "half angle formulas". They are derived from the two identities. cos y sin y and cos y cos y Take cos y sin y. Rearrange it as sin y cos y. Now solve for sin y. The result is r sin y cos y Finally, relace y with x to get sin x r The sign is dictated by the osition of the angle x. If it is in the rst or second quadrant, (where the sine function is ositive), take the ositive sign. If it is in the third or fourth quadrant, take the negative sign. To derive the half angle formula for the cosine function, take the identity cos y cos y and solve for cos y. The result is r + cos y cos y 0
As you would exect, relace y by x to get cos x r + Again the sign is dictated by the osition of the angle x. If it is in the rst or fourth quadrant, (where the cosine function is ositive), take the ositive sign. If it is in the second or third quadrant, take the negative sign. We do not need to do any heavy lifting to determine the half angle formula for the tangent function. We simly use the identity tan x sin x cos x q q + r + : Likewise, you should exect the sign to be dictated by the osition of x. If it is in the rst or third quadrant, take the ositive sign. If it is in the second or fourth quadrant, take the negative sign. Examle To determine,, tan x, sin x, cos x and tan x given that x is an angle in the fourth quadrant with. The angle is shown below. The vertical coordinate a may be obtained from + a which translates into a. This imlies that a. Since the coordinate must be negative, we must take a. From the gure, a and tan x. Therefore 0 69 : cos x sin x 69 69 9 69. tan x tan x tan x tan x 0 8 0 Before calculating sin x, cos x and tan x, we note that x is an angle between and 80, (because x is between 70 and 60 ), therefore it is in the second quadrant. This imlies that sin x is ositive but cos x and tan x are both negative. Therefore r r q sin x 6
cos x r + tan x r + Exercise r + s + q 6 q. In the gure below, x and y are angles in the rst and fourth quadrants resectively, with 7 and cos y. Determine, without using a calculator, the exact value of: (a) (b) sin y (c) cos(y x) (d) sin (x + y) (e) (f) cos y. You should know the exact value of cos. Use it and a half angle formula to determine the exact value of cos :.. x is an angle in the third quadrant with. Draw the angle then use a half angle formula to determine the exact values of sin x and cos x.. If x is an angle in the rst quadrant with, and y is an angle in the fourth quadrant with cos y, determine the exact value of each exression: (a) sin(x y) (b) (c) tan(x + y) (d) tan y. If x is an angle in the rst quadrant with, and y is an angle in the fourth quadrant with cos y, determine the exact value of each exression: (a) cos(x y) (b) (c) tan y 6. Verify the identity ( + ) + 7. You are given that x is an angle in the second quadrant and
(a) Draw the angle and calculate the exact values of and tan x: (b) Now calculate the exact values of the following: (a) (b) (c) tan x (d) tan x (c) You have enough information to calculate sec x. Calculate it. 8. You are given that y is an angle in the third quadrant and tan y. (a) Draw the angle and calculate the exact value of cos y and sin y. (b) Calculate the exact value, (no calculator), of the following: (a) sin y (b) cos y (c) cos y (c) You have enough information to calculate the exact value of sec y. Calculate it. 9. You are given that y is an angle in the third quadrant and tan y. (a) Draw the angle and calculate the exact value of cos y and sin y. (b) Now calculate the exact value, (no calculator), of the following: (i) (ii) (iii) sin y cos y tan y You have enough information to calculate the exact value of csc y. Calculate it. 0. You are given that y is an angle in the third quadrant and tan y. (a) Draw the angle and calculate the exact value of cos y and sin y. (b) Calculate the exact value, (no calculator), of the following: (i) (ii) (iii) sin y cos y tan y You have enough information to calculate the exact value of cot y. Calculate it.. If x is an angle in the rst quadrant with, and y is an angle in the fourth quadrant with cos y, determine the exact value of each exression: (a) sin (x y) (b) cos y (c) (d) tan (x + y)