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Module 6 Lesson 4 Using Trigonometric Identities – Part 2 There are more identities for Trigonometry. We won’t be getting into specifics on how they originated. They are divided into two groups for you. Identities you need to know how to use and be familiar with and identities that you just need to be aware of their existence. Identities you’ll need to know of and how to use: Sum and Difference IdentitiesSum and difference identities are used to find trigonometric values of the combination of two angles. Let’s look at the identities: Sum: Difference: sin( ) sin cos cos sin cos( ) cos cos sin sin tan tan tan( ) 1 tan tan sin( ) sin cos cos sin cos( ) cos cos sin sin tan tan tan( ) 1 tan tan Alpha, α, and Beta, β, represent two different angles. Using these identities is as simple as plugging values in and doing some basic Algebra. Example 1: Given: sin 20 0.342 cos 20 0.940 and sin15 0.259 cos15 0.966 Here is the sum identity for sine: , find the value for the sine of 35◦ sin( ) sin cos cos sin This identity states that the sine value of the sum of these angles is equal to the sine of the first times the cosine of the second, plus the cosine of the first times the sine of the second. We simply plug in the values: 1 sin(35 ) sin(15 20 ) sin( ) sin cos cos sin sin(15 20 ) sin15 cos 20 cos15 sin 20 sin(15 20 ) (0.259)(0.940) (0.966)(0.342) sin(15 20 ) 0.24346 0.330372 sin(15 20 ) 0.574 sin(35 ) 0.574 The important thing to remember is to make sure you get the angles in the correct order. For the sum identities, recall the addition is commutative, so it doesn’t matter which angle is Alpha and which is Beta. For difference identities, it does make a difference: Example 2: Given: sin 20 0.342 cos 20 0.940 and sin15 0.259 cos15 0.966 , find the value for the sine of 5◦ Since 20 – 15 = 5, we can use the difference identity for sine. 20◦ must be Alpha and 15◦ must be Beta: sin(5 ) sin(20 15 ) sin( ) sin cos cos sin sin(20 15 ) sin 20 cos15 cos 20 sin15 sin(20 15 ) (0.342)(0.966) (0.940)(0.259) sin(20 15 ) 0.330372 0.24346 sin(20 15 ) 0.086912 sin(5 ) 0.0869 Using two of the sum identities, we found the values for sine of 35◦ and 5◦. You can verify those with a calculator. 2 Example 3: Given: sin 20 0.342 cos 20 0.940 and sin15 0.259 cos15 0.966 , find the value for the cosine of 35◦ and 5◦ cos(35 ) cos(15 20 ) cos( ) cos cos sin sin cos(15 20 ) cos15 cos 20 sin15 sin 20 cos(15 20 ) (0.966)(0.940) (0.259)(0.342) cos(15 20 ) 0.90804 0.088578 cos(15 20 ) 0.819462 cos(35 ) 0.819 cos(5 ) cos(20 15 ) cos( ) cos cos sin sin cos(20 15 ) cos 20 cos15 sin 20 sin15 cos(20 15 ) (0.940)(0.966) (0.342)(0.259) cos(20 15 ) 0.90804 0.088578 cos(20 15 ) 0.996618 cos(5 ) 0.9966 You won’t be required to memorize these identities, but you will be required to know how to use them. Double AngleThe other set of identities that you’ll be required to know how to use (and not memorize!) are the double angle identities. The double angle identities allow you to find the trigonometric values for an angle twice as large as an angle that you already know. They are also used to help rewrite and solve trigonometric equations: 3 sin 2 2sin cos cos 2 cos2 sin 2 cos 2 2 cos2 1 cos 2 1 2sin 2 2 tan tan 2 1 tan 2 Example 4: Say you know: sin 20 0.342 cos 20 0.940 We can use the double angle formulas to find the sine and cosine of 40 degrees. Remember for cosine, there are three different formulas and you select the one you want based on whether or not you know the sine, cosine or both (of the original angle). In this case, we know both, so we have our choice: cos 2 2 cos 2 1 sin 2 2sin cos sin(40 ) 2(sin 20 )(cos 20 ) sin(40 ) 2(0.342)(0.940) sin(40 ) 0.6430 cos(40 ) 2(cos 20 ) 2 1 cos(40 ) 2(0.940) 2 1 cos(40 ) 1.7672 1 cos(40 ) 0.7672 Verify those results with your calculator. Example 6: Given: sin16 0.276 cos16 0.961 , find sin 32◦ sin 2 2sin cos sin(32 ) 2sin(16 ) cos(16 ) sin(32 ) 2(0.276)(0.961) sin(32 ) 0.530 Example 7: Given: sin 37 0.602 , find cos 74◦ We must use this double angle cosine formula: since we only know the sine value of 37 degrees. cos 2 1 2sin 2 4 cos 2 1 2sin 2 cos(74 ) 1 2sin 2 (37 ) cos(74 ) 1 2(0.602) 2 cos(74 ) 0.275 Example 8: Given: tan101 = -5.145 , find tan 202◦ 2 tan tan 2 1 tan 2 2 tan(101 ) tan(202 ) 1 tan 2 (101 ) 2( 5.145) tan(202 ) 1 ( 5.145) 2 10.29 tan(202 ) 1 26.471025 10.29 tan(202 ) 25.471025 tan(202 ) 0.404 Identities you’ll just need to know of: For this section, just make sure you read through the descriptions. You won’t be required to memorize the actual identities or know how to use them. I just want you to be aware of them for future reference. SymmetryThe symmetry identities tell us how the sign of a trigonometric value of sine or cosine varies with the corresponding angles in the different quadrants. In other words, if the sine value is negative for an angle, will it be positive or negative 360 degrees later? What about 180 degrees? What if it’s reflected over the x-axis? What if it’s reflected over the y-axis? Here are the identities. We have them for sine and cosine since the four other trigonometric values can be derived from sine and cosine: (k is any integer) Case I: sin( A 360 k ) sin A cos( A 360 k ) cos A Case II: sin( A 180 (2k 1)) sin A cos( A 180 (2k 1)) cos A 5 Case III: sin(360 k A) sin A Case IV: cos(360 k A) cos A sin(180 (2k 1) A) sin A cos(180 (2k 1) A) cos A Case I represents the coterminal angle in the same quadrant – the signs of both sine and cosine stay the same for the new angle. Case II represents adding 180 degrees to the original angle. When than happens, the sine and cosine of the new angle both take on the opposite signs they had originally. Case III represents reflecting the angle across the x-axis. When this occurs, the sign of the sine value changes, the cosine sign does not change. Case IV represents reflecting the angle across the y-axis. When this occurs, the sign of the cosine value changes, but the sine sign does not change. Remember that the numerical values are all the same, these identities just tell us if the sign changes or not. Half AngleLike the double angle identities, the half angle identities are what they sound like. They allow you to find the trigonometric values for an angle half the value of one already known. These identities are also used like the double angle, to help rewrite and solve trigonometric equations: 1 cos cos 2 2 sin tan tan tan 2 2 2 2 1 cos 2 1 cos 1 cos sin 1 cos 1 cos sin 6 Sum ProductThis group of identities will be used when you need to rewrite a sum or difference of two trigonometric values as a product: cos cos 2 cos( ) cos( ) 2 2 cos cos 2sin( )sin( ) 2 2 sin sin 2sin( ) cos( ) 2 2 sin sin 2 cos( )sin( ) 2 2 Product SumThis group of identities will be used when you need to rewrite a trigonometric product of values as a sum or difference: cos( ) cos( ) 2 cos( ) cos( ) sin sin 2 sin( ) sin( ) cos sin 2 sin( ) sin( ) sin cos 2 cos cos 7