Trig Question Dealing with Sum, Difference, Double Angle, and Half Angle Formulas

rojareina

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Instructions: Find the exact value of the expression giventhat sec(x)= 3/2 and csc(y)=3 where x and y are in the 1st quadrant.

From what I have been able to deduce, cos(x)= 2/3 andsin(y)= 1/3. Because of this, I solved the following problems based on thisinformation. If anyone can double check and explain to me if I got somethingwrong, I would really appreciate it! I’m very bad at trig so I’m drowning inprecal right now in college.

  1. Sin(x+y)
    1. Sin((2/3)+(1/3)) = sin(2/3)cos(1/3) +sin(1/3)cos(2/3)
    2. = (2/3)(1/3) + (1/3)(2/3)
    3. = (4/9)
  2. Tan(x-y)
    1. tan((2/3)-(1/3)) = (tan(2/3) – tan(1/3))/(1 +tan(2/3)tan(1/3))
    2. = ((2/3) – (1/3))/(1 + (2/3)(1/3))
    3. = (1/3)/(11/9)
    4. =(3/11)
  3. Cos(y/2)
    1. Cos((1/3)/2)
    2. v PvwDtkxkmPwTA AAAAABJRU5ErkJggg==

    3. ULuUw8UKSCNLRt 7e15xWrz1fzC gxYQtFga4XIAAAAASUVORK5CYII=

    4. nbDgjObdDCAAAAAElFTkSuQmCC

    5. VJ3wCb394AzL6BkSLjGpMAAAAAElFTkSuQmCC
      /3
  4. Sin(2x)
    1. Sin(2(2/3))
    2. 2sin(2/3)cos(2/3)
    3. 2(2/3)(2/3)
    4. (8/9)
 
Instructions: Find the exact value of the expression giventhat sec(x)= 3/2 and csc(y)=3 where x and y are in the 1st quadrant.

From what I have been able to deduce,

cos(x)= 2/3 → sin(x) = √(5/9)

sin(y) = 1/3 → cos(y) = √(8/9)


sin(y)= 1/3. Because of this, I solved the following problems based on thisinformation. If anyone can double check and explain to me if I got somethingwrong, I would really appreciate it! I’m very bad at trig so I’m drowning inprecal right now in college.


  1. Sin(x+y)
    1. Sin((2/3)+(1/3)) = sin(2/3)cos(1/3) +sin(1/3)cos(2/3)
    2. = (2/3)(1/3) + (1/3)(2/3)
    3. = (4/9)
  2. Tan(x-y)
    1. tan((2/3)-(1/3)) = (tan(2/3) – tan(1/3))/(1 +tan(2/3)tan(1/3))
    2. = ((2/3) – (1/3))/(1 + (2/3)(1/3))
    3. = (1/3)/(11/9)
    4. =(3/11)
  3. Cos(y/2)
    1. Cos((1/3)/2)
    2. v PvwDtkxkmPwTA AAAAABJRU5ErkJggg==
    3. ULuUw8UKSCNLRt 7e15xWrz1fzC gxYQtFga4XIAAAAASUVORK5CYII=
    4. nbDgjObdDCAAAAAElFTkSuQmCC
    5. VJ3wCb394AzL6BkSLjGpMAAAAAElFTkSuQmCC
      /3
  4. Sin(2x)
    1. Sin(2(2/3))
    2. 2sin(2/3)cos(2/3)
    3. 2(2/3)(2/3)
    4. (8/9)
1. Sin(x+y) = sin(x)*cos(y) + cos(x)*sin(y) = 1/3 *√5 * 1/3 * √8 + 2/3 * 1/3 = 1/9 *(2 + √40)

continue....
 
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