Math / Trigonometry

QuestionUse an addition or subtraction formula to write the expression as a trigonometric function of one number: $\cos \frac{3 \pi}{7} \cos \frac{2 \pi}{21}+\sin \frac{3 \pi}{7} \sin \frac{2 \pi}{21}=\cos \frac{\pi}{A}=\frac{B}{2}$ . $B=$

Studdy Solution

STEP 1

What is this asking? We're asked to rewrite a trigonometric expression using an addition or subtraction formula and then find the values of $A$ and $B$ . Watch out! Remember those angle addition and subtraction formulas!
Also, be careful with those fractions – common denominators are key!

STEP 2

1. Rewrite using a formula
2. Simplify the expression
3. Find A
4. Find B

STEP 3

The given expression looks a lot like the cosine subtraction formula: $\cos(x - y) = \cos x \cos y + \sin x \sin y$ .
Let's use this!

STEP 4

In our case, $x = \frac{3\pi}{7}$ and $y = \frac{2\pi}{21}$ , so we can rewrite the expression as: $\cos\left(\frac{3\pi}{7} - \frac{2\pi}{21}\right).$

STEP 5

To subtract the fractions inside the cosine, we need a common denominator.
The common denominator for 7 and 21 is 21.

STEP 6

We can rewrite $\frac{3\pi}{7}$ as $\frac{9\pi}{21}$ by multiplying the top and bottom by 3.
So, our expression becomes: $\cos\left(\frac{9\pi}{21} - \frac{2\pi}{21}\right).$

STEP 7

Now, we can subtract the fractions inside: $\cos\left(\frac{9\pi}{21} - \frac{2\pi}{21}\right) = \cos\left(\frac{7\pi}{21}\right).$

STEP 8

We can simplify $\frac{7\pi}{21}$ to $\frac{\pi}{3}$ by dividing the top and bottom by 7.
So, our simplified expression is: $\cos\left(\frac{\pi}{3}\right).$

STEP 9

The problem states that our simplified expression equals $\cos\frac{\pi}{A}$ .
We found that our simplified expression is $\cos\frac{\pi}{3}$ .

STEP 10

By comparing $\cos\frac{\pi}{3}$ with $\cos\frac{\pi}{A}$ , we can see that $A = \mathbf{3}$ .

STEP 11

We know that $\cos\frac{\pi}{3} = \frac{1}{2}$ .

STEP 12

The problem also states that $\cos\frac{\pi}{A} = \frac{B}{2}$ .
We found that $\cos\frac{\pi}{A} = \cos\frac{\pi}{3} = \frac{1}{2}$ .

STEP 13

By comparing $\frac{1}{2}$ with $\frac{B}{2}$ , we can see that $B = \mathbf{1}$ .

STEP 14

$A = 3$ and $B = 1$ .

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QuestionUse an addition or subtraction formula to write the expression as a trigonometric function of one number: $\cos \frac{3 \pi}{7} \cos \frac{2 \pi}{21}+\sin \frac{3 \pi}{7} \sin \frac{2 \pi}{21}=\cos \frac{\pi}{A}=\frac{B}{2}$ . $B=$

Studdy Solution

STEP 1

What is this asking? We're asked to rewrite a trigonometric expression using an addition or subtraction formula and then find the values of $A$ and $B$ . Watch out! Remember those angle addition and subtraction formulas!
Also, be careful with those fractions – common denominators are key!

STEP 2

1. Rewrite using a formula
2. Simplify the expression
3. Find A
4. Find B

STEP 3

The given expression looks a lot like the cosine subtraction formula: $\cos(x - y) = \cos x \cos y + \sin x \sin y$ .
Let's use this!

STEP 4

In our case, $x = \frac{3\pi}{7}$ and $y = \frac{2\pi}{21}$ , so we can rewrite the expression as: $\cos\left(\frac{3\pi}{7} - \frac{2\pi}{21}\right).$

STEP 5

To subtract the fractions inside the cosine, we need a common denominator.
The common denominator for 7 and 21 is 21.

STEP 6

We can rewrite $\frac{3\pi}{7}$ as $\frac{9\pi}{21}$ by multiplying the top and bottom by 3.
So, our expression becomes: $\cos\left(\frac{9\pi}{21} - \frac{2\pi}{21}\right).$

STEP 7

Now, we can subtract the fractions inside: $\cos\left(\frac{9\pi}{21} - \frac{2\pi}{21}\right) = \cos\left(\frac{7\pi}{21}\right).$

STEP 8

We can simplify $\frac{7\pi}{21}$ to $\frac{\pi}{3}$ by dividing the top and bottom by 7.
So, our simplified expression is: $\cos\left(\frac{\pi}{3}\right).$

STEP 9

The problem states that our simplified expression equals $\cos\frac{\pi}{A}$ .
We found that our simplified expression is $\cos\frac{\pi}{3}$ .

STEP 10

By comparing $\cos\frac{\pi}{3}$ with $\cos\frac{\pi}{A}$ , we can see that $A = \mathbf{3}$ .

STEP 11

We know that $\cos\frac{\pi}{3} = \frac{1}{2}$ .

STEP 12

The problem also states that $\cos\frac{\pi}{A} = \frac{B}{2}$ .
We found that $\cos\frac{\pi}{A} = \cos\frac{\pi}{3} = \frac{1}{2}$ .

STEP 13

By comparing $\frac{1}{2}$ with $\frac{B}{2}$ , we can see that $B = \mathbf{1}$ .

STEP 14

$A = 3$ and $B = 1$ .

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Studdy Solution

STEP 1

What is this asking? We're asked to rewrite a trigonometric expression using an addition or subtraction formula and then find the values of AAA and BBB. Watch out! Remember those angle addition and subtraction formulas!Also, be careful with those fractions – common denominators are key!

STEP 2

1. Rewrite using a formula2. Simplify the expression3. Find A4. Find B

STEP 3

The given expression looks a lot like the cosine subtraction formula: cos⁡(x−y)=cos⁡xcos⁡y+sin⁡xsin⁡y\cos(x - y) = \cos x \cos y + \sin x \sin ycos(x−y)=cosxcosy+sinxsiny.Let's **use this**!

STEP 4

In our case, x=3π7x = \frac{3\pi}{7}x=73π​ and y=2π21y = \frac{2\pi}{21}y=212π​, so we can rewrite the expression as: cos⁡(3π7−2π21). \cos\left(\frac{3\pi}{7} - \frac{2\pi}{21}\right). cos(73π​−212π​).

STEP 5

To subtract the fractions inside the cosine, we need a common denominator.The **common denominator** for 7 and 21 is 21.

STEP 6

We can rewrite 3π7\frac{3\pi}{7}73π​ as 9π21\frac{9\pi}{21}219π​ by multiplying the top and bottom by 3.So, our expression becomes: cos⁡(9π21−2π21). \cos\left(\frac{9\pi}{21} - \frac{2\pi}{21}\right). cos(219π​−212π​).

STEP 7

Now, we can subtract the fractions inside: cos⁡(9π21−2π21)=cos⁡(7π21). \cos\left(\frac{9\pi}{21} - \frac{2\pi}{21}\right) = \cos\left(\frac{7\pi}{21}\right). cos(219π​−212π​)=cos(217π​).

STEP 8

We can simplify 7π21\frac{7\pi}{21}217π​ to π3\frac{\pi}{3}3π​ by dividing the top and bottom by 7.So, our **simplified expression** is: cos⁡(π3). \cos\left(\frac{\pi}{3}\right). cos(3π​).

STEP 9

The problem states that our simplified expression equals cos⁡πA\cos\frac{\pi}{A}cosAπ​.We found that our simplified expression is cos⁡π3\cos\frac{\pi}{3}cos3π​.

STEP 10

By comparing cos⁡π3\cos\frac{\pi}{3}cos3π​ with cos⁡πA\cos\frac{\pi}{A}cosAπ​, we can see that A=3A = \mathbf{3}A=3.

STEP 11

We know that cos⁡π3=12\cos\frac{\pi}{3} = \frac{1}{2}cos3π​=21​.

STEP 12

The problem also states that cos⁡πA=B2\cos\frac{\pi}{A} = \frac{B}{2}cosAπ​=2B​.We found that cos⁡πA=cos⁡π3=12\cos\frac{\pi}{A} = \cos\frac{\pi}{3} = \frac{1}{2}cosAπ​=cos3π​=21​.

STEP 13

By comparing 12\frac{1}{2}21​ with B2\frac{B}{2}2B​, we can see that B=1B = \mathbf{1}B=1.

STEP 14

A=3A = 3A=3 and B=1B = 1B=1.

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What is this asking? We're asked to rewrite a trigonometric expression using an addition or subtraction formula and then find the values of $A$ and $B$ . Watch out! Remember those angle addition and subtraction formulas!
Also, be careful with those fractions – common denominators are key!

1. Rewrite using a formula
2. Simplify the expression
3. Find A
4. Find B

The given expression looks a lot like the cosine subtraction formula: $\cos(x - y) = \cos x \cos y + \sin x \sin y$ .
Let's use this!

In our case, $x = \frac{3\pi}{7}$ and $y = \frac{2\pi}{21}$ , so we can rewrite the expression as: $\cos\left(\frac{3\pi}{7} - \frac{2\pi}{21}\right).$

To subtract the fractions inside the cosine, we need a common denominator.
The common denominator for 7 and 21 is 21.

We can rewrite $\frac{3\pi}{7}$ as $\frac{9\pi}{21}$ by multiplying the top and bottom by 3.
So, our expression becomes: $\cos\left(\frac{9\pi}{21} - \frac{2\pi}{21}\right).$

Now, we can subtract the fractions inside: $\cos\left(\frac{9\pi}{21} - \frac{2\pi}{21}\right) = \cos\left(\frac{7\pi}{21}\right).$

We can simplify $\frac{7\pi}{21}$ to $\frac{\pi}{3}$ by dividing the top and bottom by 7.
So, our simplified expression is: $\cos\left(\frac{\pi}{3}\right).$

The problem states that our simplified expression equals $\cos\frac{\pi}{A}$ .
We found that our simplified expression is $\cos\frac{\pi}{3}$ .

By comparing $\cos\frac{\pi}{3}$ with $\cos\frac{\pi}{A}$ , we can see that $A = \mathbf{3}$ .

We know that $\cos\frac{\pi}{3} = \frac{1}{2}$ .

The problem also states that $\cos\frac{\pi}{A} = \frac{B}{2}$ .
We found that $\cos\frac{\pi}{A} = \cos\frac{\pi}{3} = \frac{1}{2}$ .

By comparing $\frac{1}{2}$ with $\frac{B}{2}$ , we can see that $B = \mathbf{1}$ .

$A = 3$ and $B = 1$ .