Trigonometric Identities

Transcription

1 Trigonometric Identities Dr. Philippe B. Laval Kennesaw STate University April 0, 005 Abstract This handout dpresents some of the most useful trigonometric identities. It also explains how to derive new ones. Basic Trigonometric Identities. Quick Review You will recall that an identity is a statement which is always true. In contrast, an equation is a statement which is only true for certain values of the variable(s) involved. For example, 5x +=0, sinx + 3=0are equations; they are only true for certain values of x. (x + y) = x +xy + y is an identity; it is true no matter what x and y are. We already know some identities. Some are definitions. Others have been proven. We begin by listing all the identities we should know.. Known Identities. Pythagorean Identities sin α + cos α = +tan α = sec α + cot α = csc α

2 . Reciprocal Identities sin α = csc α cos α = sec α tan α = sin α cos α cot α = cos α sin α sec α = cos α csc α = sin α 3. Even-Odd Identities sin ( α) = sin α cos ( α) = cos α tan ( α) = tan α. Cofunction Identities ( π ) sin α ( π ) cos α = cosα = sinα We have already proven all these identities, except the cofunction identities. We have already mentioned them when we studied transformations of the graphs of sine and cosine. There is a nice way to prove them using a triangle. Consider the triangle below: In this triangle, we have: sin α = a c cos β = a c

3 Hence, But since It follows that sin α =cosβ α + β + π = π β = π α Therefore, we have ( π ) sin α =cos α The proof is similar for the other cofunction identity. Try it. These identities will be used as our starting point for proving more identities. Before we do this, you may have already asked yourself: what are identities used for? One answer is that learning how to prove identities is a good exercise for the brain. But identities are useful for other reasons. Very often, identities allow you to simplify expressions. The simpler an expression is, the easier it is to work with. Identities are also used in solving trigonometric equations..3 Guidelines for Proving Identities The primary strategy used is to transform one side of the equation into the other side. This transformation is made by using the rules of algebra as well as identities you already know. It may require several steps. During this transformation, keep the following in mind:. Memorize the basic identities. Known identities are often used to prove new ones.. It is usually easier to start with the more complicated side. 3. It is sometimes useful to rewrite everything in terms of sines and cosines.. Use algebra and the identities you know. In particular, factor, bring fractional expressions to a common fraction, rationalize the denominator,... We illustrate this with a few examples. Example Show that +tan x csc =tan x. x We start with the more complicated side, and transform it into the other side. +tan x csc x = sec x csc x = cos x sin x = sin x cos x = tan x 3

4 Example Show that cos x (sec x cos x) = sin x We start with the more complicated side, and transform it into the other side. ( ) cos x (sec x cos x) = cosx cos x cos x ( cos ) x = cosx cos x = cos x = sin x ( Example 3 Express ) + cos x in terms of sin x csc x ( ) +cos x csc x = ( sin x) + cos x = sinx +sin x +cos x = sinx Other Identities. Sum and Difference Identities.. The Identities Proposition Let α and β be two real numbers (or two angles). have: Then we. sin (α + β) = sin α cos β + cos α sin β. sin (α β) = sin α cos β cos α sin β 3. cos (α + β) =cosα cos β sin α sin β. cos (α β) =cosα cos β +sinα sin β 5. tan (α + β) = 6. tan (α β) = tan α +tanβ tan α tan β tan α tan β +tanα tan β.. Proof of cos (α β) =cosα cos β +sinα sin β We prove the fourth identity with the help of a graphical method. Given α and β, the angle α β can be represented as shown on the picture below.

5 We now concentrate on α β, and represent it for various values of α and β, in such a way that α β remains constant. Two possible such representations are shown in the picture below. Because a β remained constant, the distance between A and B, denoted d (A, B) is the same as the distance between A and B, denoted d (A,B ).The reader will recall that if the coordinates of A are (x, y) and those of B are (x,y ), then d (A, B) = (x x) +(y y). Therefore, we can write: d (A, B) = d (A B ) (d (A, B)) = (d (A B )) 5

6 Using the coordinates on the picture above, we can compute these distances. (d (A, B)) = (cosα cos β) + (sin α sin β) and = cos α cosαcos β +cos β +sin α sinαsin β +sin β = cos α +sin α + cos β +sin β (sin α sin β +cosαcos β) = + (sin α sin β +cosαcos β) = (sin α sin β +cosαcos β) (d (A B )) = (cos (α β) ) +(sin(α β) 0) = cos (α β) cos(α β) + + sin (α β) = cos (α β) + sin (α β) cos (α β)+ = cos(α β)+ = cos(α β) Since the two distances are equal, we have cos (α β) = (sin α sin β + cos α cos β) cos (α β) = sin α sin β + cos α cos β..3 Proof of cos (α + β) =cosα cos β sin α sin β We write α + β = α ( β) and use the identity for cos (α β). cos (α + β) = cos(α ( β)) = cosα cos ( β) + sin α sin ( β) = cosα cos β sin α sin β since cos ( α) = cos α and sin ( α) = sin α... Proof of sin (α + β) = sin α cos β + cos α sin β We use the cofunction identities. ( π ) sin (α + β) = cos (α + β) (( π ) ) = cos α β We now use the difference identity for cosine. sin (α + β) = ( π ) ( π ) cos α cos β +sin α sin β = sinα cos β +cosαsin β 6

7 ..5 Application: Finding the Exact Value of the Trigonometric Functions The sum and difference identities are often used to prove other identities, as we will see later. You will also use them in Calculus I, so you must know them. They can also be used to find the exact value of the trigonometric functions at certain angles. We know the exact value of the trigonometric functions at the following angles: π π π π t sin t 0 3 cos t 0 For the other angles, we rely on our calculator. The sum and difference formulas allow us to find the exact value of the trigonometric functions for additional angles. Example 5 Find the exact value of sin 75 sin 75 = sin (30 + 5) = sin 30 cos 5 + cos 30 sin 5 = = = Example 6 Find the exact value of cos π First, we express π intermsofanglesforwhichweknowthevalueofthe trigonometric functions. Since π = π 3 π,wehave cos π ( π = cos 3 π ) = cos π 3 cos π +sinπ 3 sin π = = 7

8 ..6 Application: Simplifying Expressions of the Form A sin α + B cos α If we could find an angle β such that cos β = A and sin β = B, the we would have A sin α + B cos α = cosβ sin α +sinβcos α = sin(α + β) If this is going to work, then we must have A +B =since cos β +sin β =. What about if A and B are such that A +B? Here is the trick to remember: A sin α + B cos α = ( ) A + B A A + B sin α + B A + B cos α ( ) ( ) A B First, we note that + =. Wecanfindanangle A + B A + B A β such that cos β = A + B and sin β = B. To see this, simply A + B draw a triangle in which one of the angles is β, the length of the side opposite β is A, the length of the side adjacent β is B. So, we have: Proposition 7 If A and B are real numbers, then where β satisfies A sin α + B cos α = A + B sin (α + β) cos β = A B and sin β = A + B A + B Example 8 Express 3sinα +cosα in the above form. From what we saw above, 3sinα + cos α = 3 + sin (α + β) = 5sin(α + β) where sin β = 5 and cos β = 3. Since both sin β and cos β are positive, β should 5 be in the first quadrant. Using a calculator, we find that β 53..Thus, 3sinα +cosα =5sin(α +53.). Double and Half-Angle Identities In this section, we derive identities for sin α, cos α, tan α, sin α, cos α,and tan α. The first three are known as the double-angle identities. The last three are the half-angle identities. 8

9 Proposition 9 (double-angle identities) Let α be any angle. Then. sin α = sin α cos α. cos α =cos α sin α = sin α =cos α 3. tan α = tanα tan α Proposition 0 (half-angle identities) Let α be any angle. Then. sin α cos α = ±. cos α + cos α = ± 3. tan α cos α = ± +cosα. tan α = sin α +cosα 5. tan α = cos α sin α We now show how these identities can be derived... Proof of sin α = sin α cos α The trick to remember here is to write α = α + α and use the sum identity for sine. sin α = sin(α + α) = sinα cos α +cosαsin α = sin α cos α The proof is similar for cosine and tangent, we leave it as an exercise... Note on the identity for cos α You will notice that this identity is in fact three identities. The first one is proven thesamewaytheidentityforsin α was proven. To go from cos α sin α to sin α or cos α, we simply use the Pythagorean identity. For example cos α sin α = ( sin α ) sin α = sin α 9

10 The identities are important. We can rewrite them as cos α = sin α cos α = cos α sin cos a α = cos α = +cosα These two new identities are often used in Calculus II. They allow us to decrease the power of either sine or cosine...3 Proof of sin α = ± cos α Since sin β = and cos α cos β, it follows that cos β sin β = ± This is true for every β, soitistrueforβ = α.so,weobtain sin α = ± cos α The identity for cos α is proven the same way. You will note that for these two identities, we will need to know the quadrant of the angle to determine the sign of the expression... Proof of the half-angle identities for tan α The first identity is proven by using the half-angle identities for sine and cosine, and the definition of tangent. The proof is left as an exercise. The proof of the remaining two identities can be found in your book...5 Examples Example Find the exact value of sin.5. Since.5 = 5,wehave sin.5 = sin 5 cos 5 = ± 0

11 Since.5 is in the first quadrant, sin.5 > 0. Therefore, cos 5 sin.5 = = = = Example Express sin x cos x in terms of the first power of cosine..3 Practice Problems sin x cos x = cos x + cos x = cos x = +cosx = cos x 8. Prove that sin (α β) = sin α cos β cos α sin β (hint: write α β = α +( β) and apply the identity for the sum of two angles and other identities) tan α +tanβ. Prove that tan (α + β) = (Hint: write tan (α + β) in terms tan α tan β of sine and cosine) 3. Prove that tan (α β) = tan α tan β +tanα tan β. Prove that cos α =cos α sin α = sin α =cos α 5. Prove that tan α = tanα tan α 6. Prove that cos α + cos α = ± 7. Prove that tan α cos α = ± +cosα

12 8. Prove that tan α = sin α +cosα 9. Prove that tan α = cos α sin α 0. Do #, 3, 5, 8,, 5, 7 on pages 58, 59.. Do # 9, 0, 7 on page 58.