Sin(t) - cos(t) can be rewritten as?

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In summary, the trigonometric identity for Sin(t) - cos(t) is tan(t - π/4). Sin(t) - cos(t) can also be rewritten as √2 sin(t - π/4) or expressed as √2 sin(t + π/4) or √2 cos(t - π/4) using trigonometric identities. The graph of Sin(t) - cos(t) has the same amplitude and period as sin(t) and cos(t), but is shifted by π/4 to the right. This identity can be useful in calculating the difference between two periodic quantities that are out of phase with each other, such as sound waves or alternating currents.
  • #1
spj1
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The problem:

sin(t) - cos(t) can be rewritten as ? - The answer is sqrt(2)cos(3PI/4 - t).

there is an equation Re^(xt)cos(bt - angle).
R = sqrt(A^2+B^2)

So i get the following answer:

sqrt(2)(cos(t - 3PI/4)).


Why does the correct answer have the 3PI/4 switched with the t??
 
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  • #2
If you remember the graph of cosine, the property should come to mind that:
cos(-x)=cos(x), which is what's being used here.
 

1. What is the trigonometric identity for Sin(t) - cos(t)?

The trigonometric identity for Sin(t) - cos(t) is tan(t - π/4).

2. How do you rewrite Sin(t) - cos(t) in terms of other trigonometric functions?

Sin(t) - cos(t) can be rewritten as √2 sin(t - π/4), which is equivalent to √2 (sin(t)cos(π/4) - cos(t)sin(π/4)).

3. Can Sin(t) - cos(t) be expressed in terms of sine or cosine alone?

Yes, Sin(t) - cos(t) can be expressed as √2 sin(t + π/4) or √2 cos(t - π/4), using the trigonometric identities sin(A + B) = sin(A)cos(B) + cos(A)sin(B) and cos(A - B) = cos(A)cos(B) + sin(A)sin(B).

4. How does the graph of Sin(t) - cos(t) compare to that of sin(t) and cos(t)?

The graph of Sin(t) - cos(t) has the same amplitude and period as both sin(t) and cos(t), but is shifted by π/4 to the right. This means that the peaks and valleys of Sin(t) - cos(t) occur at different points on the x-axis compared to sin(t) and cos(t).

5. In what real-world situations would the trigonometric identity Sin(t) - cos(t) be useful?

The trigonometric identity Sin(t) - cos(t) can be useful in calculating the difference between two periodic quantities that are out of phase with each other. For example, it could be used to find the difference in amplitude between two sound waves with different frequencies or the difference in voltage between two alternating currents with different phases.

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