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Solving Trigonometric Equations using Sum and Difference Formulas

Solve sine, cosine, and tangent of angles that are added or subtracted.

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Solving Trig Equations using Sum and Difference Formulas

As Agent Trigonometry, you are now given another piece of the puzzle: sin(π2x)=1 . What is the value of x ?

Guidance

Lastly, we can use the sum and difference formulas to solve trigonometric equations. For this concept, we will only find solutions in the interval 0x<2π .

Example A

Solve cos(xπ)=22 .

Solution: Use the formula to simplify the left-hand side and then solve for x .

cos(xπ)cosxcosπ+sinxsinπcosxcosx=22=22=22=22

The cosine negative in the 2nd and 3rd quadrants. x=3π4 and 5π4 .

Example B

Solve sin(x+π4)+1=sin(π4x) .

Solution:

sin(x+π4)+1sinxcosπ4+cosxsinπ4+1sinx22+cosx22+12sinxsinx=sin(π4x)=sinπ4cosxcosπ4sinx=22.cosx22sinx=1=12=22

In the interval, x=5π4 and 7π4 .

Example C

Solve 2sin(x+π3)=tanπ3 .

Solution:

2sin(x+π3)2(sinxcosπ3+cosxsinπ3)2sinx12+2cosx32sinx+3cosxsinxsin2x1cos2x00=tanπ3=3=3=3=3(1cosx)=3(12cosx+cos2x)square both sides=36cosx+3cos2x  substitute sin2x=1cos2x=4cos2x6cosx+2=2cos2x3cosx+1

At this point, we can factor the equation to be (2cosx1)(cosx1)=0 . cosx=12 , and 1, so x=0,π3,5π3 . Be careful with these answers. When we check these solutions it turns out that 5π3 does not work.

2sin(5π3+π3)2sin2π0=tanπ3=33

Therefore, 5π3 is an extraneous solution.

Concept Problem Revisit

In the previous lesson you solved the expression sin(π2x) as:

sin(π2x)=sinπ2cosxcosπ2sinx=1cosx0sinx=cosx

So what you're now looking for is the value of x where cosx=1 .

The cosine of 180 is equal to 1 .

Guided Practice

Solve the following equations in the interval 0x<2π .

1. cos(2πx)=12

2. sin(π6x)+1=sin(x+π6)

3. cos(π2+x)=tanπ4

Answers

1.

cos(2πx)cos2πcosx+sin2πsinxcosxx=12=12=12=π3 and 5π3

2.

sin(π6x)+1sinπ6cosxcosπ6sinx+112cosx32sinx+1113x=sin(x+π6)=sinxcosπ6+cosxsinπ6=32sinx+12cosx=3sinx=sinx=sin1(13)=0.6155 and 2.5261 rad

3.

cos(π2+x)cosπ2cosxsinπ2sinxsinxsinxx=tanπ4=1=1=1=3π2

Explore More

Solve the following trig equations in the interval 0x<2π .

  1. sin(xπ)=22
  2. cos(2π+x)=1
  3. tan(x+π4)=1
  4. sin(π2x)=12
  5. sin(x+3π4)+sin(x3π4)=1
  6. sin(x+π6)=sin(xπ6)
  7. cos(x+π6)=cos(xπ6)+1
  8. cos(x+π3)+cos(xπ3)=1
  9. tan(x+π)+2sin(x+π)=0
  10. tan(x+π)+cos(x+π2)=0
  11. tan(x+π4)=tan(xπ4)
  12. sin(x5π3)sin(x2π3)=0
  13. 4sin(x+π)2=2cos(x+π2)
  14. 1+2cos(xπ)+cosx=0
  15. Real Life Application The height, h (in feet), of two people in different seats on a Ferris wheel can be modeled by h1=50cos3t+46 and h2=50cos3(t3π4)+46 where t is the time (in minutes). When are the two people at the same height?

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