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Mathematics LibreTexts

7.3 Double Angle Identities

Two special cases of the sum of angles identities arise often enough that we choose to state these identities separately.

Identities

The double angle identities

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image750.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image752.gif

These identities follow from the sum of angles identities.

Proof of Sine Double Angle Identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image754.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image756.gif                                       Apply the sum of angles identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image758.gif         Simplify

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image760.gif                               

Establishing the identity

Try it Now: 1

Show File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image762.gif by using the sum of angles identity for cosine.

For the cosine double angle identity, there are three forms of the identity stated because the basic form, File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image762.gif, can be rewritten using the Pythagorean Identity.   Rearranging the Pythagorean Identity results in the equalityFile:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image764.gif, and by substituting this into the basic double angle identity, we obtain the second form of the double angle identity.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image762.gif             Substituting using the Pythagorean identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image766.gif        Simplifying

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image768.gif

 

Example 1

If File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image770.gif and θ is in the second quadrant, find exact values for File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image772.gif and File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image774.gif.

SOLUTION

To evaluateFile:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image774.gif, since we know the value for File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image776.gif, we can use the version of the double angle that only involves sine.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image778.gif

Since the double angle for sine involves both sine and cosine, we’ll need to first find File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image780.gif, which we can do using the Pythagorean Identity.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image105.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image783.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image785.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image787.gif                     

Since θ is in the second quadrant, we know that cos(θ) < 0, so

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image789.gif

Now we can evaluate the sine double angle

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image791.gif

Example 2

Simplify the expressions

a) File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image793.gif                  b) File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image795.gif

SOLUTION

a) Notice that the expression is in the same form as one version of the double angle identity for cosine:  File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image797.gif.  Using this,

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image799.gif

b) This expression looks similar to the result of the double angle identity for sine.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image795.gif                   Factoring a 4 out of the original expression

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image801.gif               Applying the double angle identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image803.gif

We can use the double angle identities to simplify expressions and prove identities.

Example 2b

Simplify File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image805.gif.

SOLUTION

With three choices for how to rewrite the double angle, we need to consider which will be the most useful. To simplify this expression, it would be great if the denominator would cancel with something in the numerator, which would require a factor of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image807.gif in the numerator, which is most likely to occur if we rewrite the numerator with a mix of sine and cosine.

 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image805.gif                                              Apply the double angle identity

=File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image810.gif                                       Factor the numerator

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image812.gif                Cancelling the common factor

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image814.gif                                           Resulting in the most simplified form

Example 3

Prove File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image816.gif.

SOLUTION

Since the right side seems a bit more complicated than the left side, we begin there.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image818.gif                                     Rewrite the secants in terms of cosine

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image820.gif                                

 

At this point, we could rewrite the bottom with common denominators, subtract the terms, invert and multiply, then simplify.  Alternatively, we can multiple both the top and bottom by File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image822.gif, the common denominator:

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image824.gif              Distribute on the bottom

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image826.gif                  Simplify

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image828.gif                                Rewrite the denominator as a double angle

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image830.gif                                        Rewrite as a secant

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image832.gif                                         Establishing the identity

Try it Now: 2

Use an identity to find the exact value of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image834.gif.

As with other identities, we can also use the double angle identities for solving equations.

Example 4

Solve File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image836.gif for all solutions with File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image050.gif.

SOLUTION

In general when solving trig equations, it makes things more complicated when we have a mix of sines and cosines and when we have a mix of functions with different periods.  In this case, we can use a double angle identity to rewrite the cos(2t).  When choosing which form of the double angle identity to use, we notice that we have a cosine on the right side of the equation. We try to limit our equation to one trig function, which we can do by choosing the version of the double angle formula for cosine that only involves cosine.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image836.gif                               Apply the double angle identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image839.gif                        This is quadratic in cosine, so make one side 0

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image841.gif                  Factor

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image843.gif              Break this apart to solve each part separately

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image845.jpg

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image847.gif         or         File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image849.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image851.gif             or         File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image853.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image855.gif or File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image857.gif    or         File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image859.gif

Looking at a graph of cos(2t) and cos(t) shown together, we can verify that these three solutions on [0, 2π) seem reasonable.

Example 5

A cannonball is fired with velocity of 100 meters per second.  If it is launched at an angle of θ, the vertical component of the velocity will be File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image861.gif and the horizontal component will be File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image863.gif.  Ignoring wind resistance, the height of the cannonball will follow the equation File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image865.gif and horizontal position will follow the equation File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image867.gif.   If you want to hit a target 900 meters away, at what angle should you aim the cannon?

 

To hit the target 900 meters away, we want File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image869.gifat the time when the cannonball hits the ground, when File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image871.gif.  To solve this problem, we will first solve for the time, t, when the cannonball hits the ground.  Our answer will depend upon the angleFile:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image873.gif.

 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image871.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image875.gif                              Factor

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image877.gif                              Break this apart to find two solutions

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image859.gif or

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image880.gif                                 Solve for t

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image882.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image884.gif

This shows that the height is 0 twice, once at t = 0 when the cannonball is fired, and again when the cannonball hits the ground after flying through the air.  This second value of t gives the time when the ball hits the ground in terms of the angle File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image873.gif.  We want the horizontal distance x(t) to be 900 when the ball hits the ground, in other words when File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image884.gif.

Since the target is 900 m away we start with

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image869.gif                                        Use the formula for x(t)

 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image888.gif                             Substitute the desired time, t from above

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image890.gif             Simplify

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image892.gif                 Isolate the cosine and sine product

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image894.gif

The left side of this equation almost looks like the result of the double angle identity for sine: File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image896.gif.

By dividing both sides of the double angle identity by 2, we get

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image898.gif.  Applying this to the equation above,

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image900.gif                                   Multiply by 2

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image902.gif                                 Use the inverse sine

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image904.gif                Divide by 2

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image906.gif, or about 30.94 degrees

Power Reduction and Half Angle Identities

Another use of the cosine double angle identities is to use them in reverse to rewrite a squared sine or cosine in terms of the double angle.  Starting with one form of the cosine double angle identity:

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image908.gif                     Isolate the cosine squared term

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image910.gif                     Add 1

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image912.gif                       Divide by 2

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image914.gif                       This is called a power reduction identity

Try it Now: 3

Use another form of the cosine double angle identity to prove the identity File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image916.gif.

Example 6

Rewrite File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image918.gif without any powers.

 

Since File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image920.gif, we can use the formula we found above

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image920.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image923.gif                                          Square the numerator and denominator

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image925.gif                                           Expand the numerator

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image927.gif                           Split apart the fraction

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image929.gif                         Apply the formula above to File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image931.gif

                                                                         File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image933.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image935.gif                 Simplify

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image937.gif                    Combine the constants

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image939.gif

The cosine double angle identities can also be used in reverse for evaluating angles that are half of a common angle.  Building from our formula File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image914.gif, if we let File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image941.gif, then File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image943.gif this identity becomes File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image945.gif.  Taking the square root, we obtain

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image947.gif, where the sign is determined by the quadrant. 

This is called a half-angle identity.

Try it Now: 4

Use your results from the last Try it Now to prove the identity

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image949.gif.

Example 7

Find an exact value for File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image951.gif

SOLUTION

Since 15 degrees is half of 30 degrees, we can use our result from above:

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image953.gif                  

We can evaluate the cosine.  Since 15 degrees is in the first quadrant, we need the positive result.

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image955.gif   

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image957.gif

Identities

Half-Angle Identities

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image947.gif                   File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image949.gif

Power Reduction Identities

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image914.gif                     File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image916.gif

Since these identities are easy to derive from the double-angle identities, the power reduction and half-angle identities are not ones you should need to memorize separately.

Important Topics of This Section

  • Double angle identity
  • Power reduction identity
  • Half angle identity
  • Using identities
    • Simplify equations
    • Prove identities
    • Solve equations

 

Try it Now Answers

  1. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image961.gif

 

  1. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image963.gif

 

 

  1. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image965.gif

 

 

  1. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image967.gif

Section 7.3 Exercises

 

1. If File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image969.gif and x is in quadrant I, then find exact values for (without solving for x):

a. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image971.gif       b. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image973.gif      c. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image975.gif

 

2. If File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image977.gif and x is in quadrant I, then find exact values for (without solving for x):

a. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image971.gif       b. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image973.gif      c. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image975.gif

 

Simplify each expression.

3. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image982.gif                                4. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image984.gif

5. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image986.gif                                             6. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image988.gif

7. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image990.gif                                   8. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image992.gif

9. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image994.gif                                        10. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image996.gif

 

Solve for all solutions on the interval File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image216.gif.

11. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image999.gif                             12. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1001.gif

13. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1003.gif                         14. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1005.gif

15. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1007.gif                                       16. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1009.gif

17. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1011.gif                              18. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1013.gif

 

Use a double angle, half angle, or power reduction formula to rewrite without exponents.

19. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1015.gif                                                   20. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1017.gif              

21. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1019.gif                                                    22. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1021.gif

23. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1023.gif                                              24. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1025.gif

 

25. If File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1027.gif and File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1029.gif, then find exact values for (without solving for x):

  1. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1031.gif                       b. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1033.gif                  c. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1035.gif

 

26. If File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1037.gif and File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1029.gif, then find exact values for (without solving for x):

  1. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1031.gif                       b. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1033.gif                  c. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1035.gif

 

 

 

Prove the identity.

27. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1043.gif

28. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1045.gif

29. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1047.gif

30. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1049.gif

31. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1051.gif

32. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1053.gif

33. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1055.gif

34. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1057.gif

35. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1059.gif

36. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image1061.gif

Contributors

  • David Lippman (Pierce College)
  • Melonie Rasmussen (Pierce College)