Math / Trigonometry

QuestionQuestion 8, 5.4.41
Write the expression as a function of $x$ , with no angle measure involved. $\sin \left(\frac{2 \pi}{3}+x\right)$ $\square$

Studdy Solution

STEP 1

What is this asking? Rewrite the expression $\sin(\frac{2\pi}{3} + x)$ so it only uses $x$ and no angles besides $x$ itself. Watch out! Remember your angle addition formulas, and don't mix up sine and cosine!

STEP 2

1. Expand the sine of the sum
2. Evaluate known values
3. Combine and simplify

STEP 3

We're given the expression $\sin(\frac{2\pi}{3} + x)$ , and we want to get rid of that $\frac{2\pi}{3}$ .
The angle addition formula for sine is perfect for this!
It says: $\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$ In our case, $a = \frac{2\pi}{3}$ and $b = x$ , so let's plug those in: $\sin(\frac{2\pi}{3} + x) = \sin(\frac{2\pi}{3})\cos(x) + \cos(\frac{2\pi}{3})\sin(x)$

STEP 4

We know $\sin(\frac{2\pi}{3})$ and $\cos(\frac{2\pi}{3})$ from the unit circle! $\frac{2\pi}{3}$ radians is 120 degrees.
Remember, sine is the y-coordinate and cosine is the x-coordinate.
So: $\sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2}$ $\cos(\frac{2\pi}{3}) = -\frac{1}{2}$

STEP 5

Let's plug these values back into our expanded expression: $\sin(\frac{2\pi}{3} + x) = (\frac{\sqrt{3}}{2})\cos(x) + (-\frac{1}{2})\sin(x)$

STEP 6

Now, we just need to clean things up a bit.
That negative sign can go right in front of the second term: $\sin(\frac{2\pi}{3} + x) = \frac{\sqrt{3}}{2}\cos(x) - \frac{1}{2}\sin(x)$ And there we go!

STEP 7

Our final expression, with no angle measure involved except $x$ , is: $\frac{\sqrt{3}}{2}\cos(x) - \frac{1}{2}\sin(x)$

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QuestionQuestion 8, 5.4.41
Write the expression as a function of $x$ , with no angle measure involved. $\sin \left(\frac{2 \pi}{3}+x\right)$ $\square$

Studdy Solution

STEP 1

What is this asking? Rewrite the expression $\sin(\frac{2\pi}{3} + x)$ so it only uses $x$ and no angles besides $x$ itself. Watch out! Remember your angle addition formulas, and don't mix up sine and cosine!

STEP 2

1. Expand the sine of the sum
2. Evaluate known values
3. Combine and simplify

STEP 3

STEP 4

We know $\sin(\frac{2\pi}{3})$ and $\cos(\frac{2\pi}{3})$ from the unit circle! $\frac{2\pi}{3}$ radians is 120 degrees.
Remember, sine is the y-coordinate and cosine is the x-coordinate.
So: $\sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2}$ $\cos(\frac{2\pi}{3}) = -\frac{1}{2}$

STEP 5

Let's plug these values back into our expanded expression: $\sin(\frac{2\pi}{3} + x) = (\frac{\sqrt{3}}{2})\cos(x) + (-\frac{1}{2})\sin(x)$

STEP 6

Now, we just need to clean things up a bit.
That negative sign can go right in front of the second term: $\sin(\frac{2\pi}{3} + x) = \frac{\sqrt{3}}{2}\cos(x) - \frac{1}{2}\sin(x)$ And there we go!

STEP 7

Our final expression, with no angle measure involved except $x$ , is: $\frac{\sqrt{3}}{2}\cos(x) - \frac{1}{2}\sin(x)$

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QuestionQuestion 8, 5.4.41Write the expression as a function of xxx, with no angle measure involved. sin⁡(2π3+x)\sin \left(\frac{2 \pi}{3}+x\right)sin(32π​+x) □\square□

Studdy Solution

STEP 1

What is this asking? Rewrite the expression sin⁡(2π3+x)\sin(\frac{2\pi}{3} + x)sin(32π​+x) so it only uses xxx and no angles besides xxx itself. Watch out! Remember your angle addition formulas, and don't mix up sine and cosine!

STEP 2

1. Expand the sine of the sum2. Evaluate known values3. Combine and simplify

STEP 3

STEP 4

STEP 5

Let's **plug these values** back into our expanded expression: sin⁡(2π3+x)=(32)cos⁡(x)+(−12)sin⁡(x) \sin(\frac{2\pi}{3} + x) = (\frac{\sqrt{3}}{2})\cos(x) + (-\frac{1}{2})\sin(x) sin(32π​+x)=(23​​)cos(x)+(−21​)sin(x)

STEP 6

Now, we just need to **clean things up** a bit.That negative sign can go right in front of the second term: sin⁡(2π3+x)=32cos⁡(x)−12sin⁡(x) \sin(\frac{2\pi}{3} + x) = \frac{\sqrt{3}}{2}\cos(x) - \frac{1}{2}\sin(x) sin(32π​+x)=23​​cos(x)−21​sin(x) And there we go!

STEP 7

Our final expression, with no angle measure involved except xxx, is: 32cos⁡(x)−12sin⁡(x) \frac{\sqrt{3}}{2}\cos(x) - \frac{1}{2}\sin(x) 23​​cos(x)−21​sin(x)

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QuestionQuestion 8, 5.4.41
Write the expression as a function of $x$ , with no angle measure involved. $\sin \left(\frac{2 \pi}{3}+x\right)$ $\square$

What is this asking? Rewrite the expression $\sin(\frac{2\pi}{3} + x)$ so it only uses $x$ and no angles besides $x$ itself. Watch out! Remember your angle addition formulas, and don't mix up sine and cosine!

1. Expand the sine of the sum
2. Evaluate known values
3. Combine and simplify

Let's plug these values back into our expanded expression: $\sin(\frac{2\pi}{3} + x) = (\frac{\sqrt{3}}{2})\cos(x) + (-\frac{1}{2})\sin(x)$

Now, we just need to clean things up a bit.
That negative sign can go right in front of the second term: $\sin(\frac{2\pi}{3} + x) = \frac{\sqrt{3}}{2}\cos(x) - \frac{1}{2}\sin(x)$ And there we go!

Our final expression, with no angle measure involved except $x$ , is: $\frac{\sqrt{3}}{2}\cos(x) - \frac{1}{2}\sin(x)$