Trigonometric Identities. Sum and Differences

Transcription

1 Trigonometric Identities Sum and Differences

2 WARNING: While viewing this pdf, the viewer may experience the following: 1.) Shock.) Confusion.) Denial 4.) Disbelief 5.) I never learned this 6.) Fear 7.) Rage 8.) Terror 9.) Indifference 10.) Kanye

3 Identities FORMULA sin(α + β) = sin(α)cos(β) + cos(α)sin(β) sin(α β) = sin(α)cos(β) cos(α)sin(β) cos(α + β) = cos(α)cos(β) sin(α)sin(β) cos(α β) = cos(α)cos(β) + sin(α)sin(β) tan(a) + tan(b) tan(α + β) = 1 tan(a)tan(b) tan(a) tan(b) tan(α β) = 1 + tan(a)tan(b) Notice: Only for tan and cos does the operation sign change to it s opposite

4 You would need to frequently refer back to the Unit Circle while solving Trigonometric Identities/ Equations

5 cos75 Cos75 Cos(10-15) This slide will explain how to solve the equation using the formula : cos(α β) First find two degrees from the Unit Circle that when subtracted equals the degree give in the problem. For this problem the degree is 75. We re going to using the degrees 10 and 15 cos10cos15+sin10sin15 Next plug in the degrees/numbers in the extended formula: cos(α)cos(β) + sin(α)sin(β) α = 10 β= 15 ( ) + (- 1 ) ANSWER: ( 4 ) OR 6 4 Now, refer back to the Unit Circle and find the numbers that correspond with degree. Don t forget to change the sign to its opposite. Multiply from denominator by dominator and numerator to numerator. Side Note Sin is the y coordinate Cos is the X coordinate

6 Cos75 Solve cos75 We re first going to solve this equation using the formula: cos(α + β) cos(45+0) cos45cos0-sin45sin0 ( ) ( 1 ) 6 or First, find two numbers that when added together, equals the number in the problem. For this problem the number is 75. We re going to use the numbers 45 and 5 degrees. Next step is plug in the numbers in the extended formula of cos(α + β) : Cosαsinβ- SinαCosβ Next refer back to the Unit Circle and find number that applies for the section for example: Cos45 = Sin45 = cos0 = Sin0 = 1 Multiply the Dominator by Dominator and Numerator to Numerator for each section and you got your answer *Don t forget to change the operation from addition to subtraction

7 Sin75 Sin75 sin(0 + 45) For this problem we ll being using the formula: sin(α + β) = sin(α)cos(β) + cos(α)sin(β) Similar to the problems before it, pick two numbers that when added together is 75. For this example, it ll be 0 and 45 sin0cos45 + cos0sin ANSWER Plug in the degrees into the extended portion of the formula Refer back to the Unit circle and match the coordinates/numbers with the it corresponding place. For example: SIN0: 1 COS45: Multiply denominator to denominator, and numerator to numerator SIN AND COS DIFFER BY FORMULA BUT ARE SOLVED VERY SIMILARLY

8 SIN75 sin75 sin FIND TWO NUMBERS THAT WHEN SUBTRACTED FROM EACH OTHER EQUALS THE GIVEN NUMBER OF THE PROBLEM. For this example, we are going to use 10 and 15 which when subtracted equals 75 sin10cos15 cos10sin15 (- 1 ) 4 ( 6 4 ) Answer: Plug in the degrees into the formula with the 10 being α being 10 and 15 being β. Next, refer back to the Unit Circle and place the numbers that correspond to what s being asked for in the formula. For example: Sin10 = 1 SIMPLIFY. cos15 =

9 WHY DON T YOU TAKE A 5 MINUTE BREAK?

10 tan75 TAN75 tan 45+ tan 0 1 tan 45 tan = + = = = + ANSWER: + For this equation we are first going to use the formula: tan(α + β) Find two degree that when added together equal the given degree in the problem For this it would 75, and, we re using 45 & 0 Find the tan of the degree (refer to the breakdown section). Next plug in the number into the formula and find a common denominator. For this one the common denominator is. So multiply 1 by. The denominator will cancel & out leaving you with + Next multiply the denominator by it s conjugate. Add together like terms Next do you basic math solving (PEMDAS). So, divide 6 in to 1 and 6. tan(a) + tan(b) 1 tan(a)tan(b) Breakdown TAN0 : 1 = 1 TAN45: = 1 SIDE NOTE tan = SIN COS 1 can also bee written as

11 tan75 We ll be using the formula: tan(α β) Find the tan of the degree (refer to the breakdown section). Next plug in the number into the formula and find a common denominator. For this one the common denominator is. So multiply 1 by. The denominator will cancel & out leaving you with + Multiply the whole problem by the denominator conjugate. Then, add together like terms Next do you basic math solving (PEMDAS). So, divide 6 in to 1 and 6. Breakdown Tan10: 1 = Tan15: = 1

12 WHEN YOU THOUGHT YOU RE DONE WITH THE PDF ONLY TO FIND OUT THAT YOU HAVE TO DO THE PRACTICE PROBLEMS AT THE END OF IT

13 Practice Problems 1.) Solve sin105 using sin(α + β) or sin(α - β).) Solve sin195.) Solve cos85 4.) Solve cos-15 using cos(α β) 5.) Solve tan105 using the formula, tan(α β)

14 SIN105 sin sin60cos45 + cos60sin40 ( ) + ( 1 ) or sin sin150cos45 cos150sin45 ( 1 ) (- ) Answer: SIN195 sin sin15cos60 + cos15sin60 sin ( 1 ) + ( ) or sin40cos45 - cos40sin45 (- 1 ) - (- ) Answer: COS85 cos 15 0 cos15cos0 sin15sin0 + 1 Or solve it in this way cos cos40cos45 sin40sin45 ( - 1 ) (- ) Answer: 6 4

15 Cos-15 cos(0-45) cos0cos45 sin0sin45 ( ) + (1 ) Answer: tan105 Tan(15 0) tan15 tan0 1 + (tan15)(tan0) ( ) + + Tan15: = 1 Tan0: 1 = 1 or +( ) Answer:

16 For further explanations (explanation video) ference.xml For further explanation for tan75 : On page 556 in the Algebra and Trigonometry: Fifth Element book