Math / Trigonometry

QuestionUse an addition or subtraction formula to write the expression as a trigonometric functi $\frac{\tan 73^{\circ}-\tan 13^{\circ}}{1+\tan 73^{\circ} \tan 13^{\circ}}=\tan A^{\circ}=\sqrt{B}$ A = $B=$ Submit Question

Studdy Solution

STEP 1

What is this asking? We're asked to rewrite a tricky-looking expression involving tangents as a single tangent function using an addition or subtraction formula, then find the angle $A$ and the value of $B$ such that $\tan A^\circ = \sqrt{B}$ . Watch out! It's super easy to mix up the tangent addition and subtraction formulas!
Make sure you pick the right one and watch out for the signs.

STEP 2

1. Identify the Formula
2. Apply the Formula
3. Calculate the Tangent
4. Find A and B

STEP 3

Alright, let's decode this mathematical puzzle!
The given expression looks a whole lot like the tangent subtraction formula.
Remember, the tangent subtraction formula is: $\tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \cdot \tan y}$ This is exactly the form we have!

STEP 4

Let's match the parts of our expression to the formula.
We can see that $x = 73^\circ$ and $y = 13^\circ$ .
So, we can rewrite our expression as: $\frac{\tan 73^\circ - \tan 13^\circ}{1 + \tan 73^\circ \cdot \tan 13^\circ} = \tan(73^\circ - 13^\circ)$

STEP 5

Now, let's simplify the angle inside the tangent function: $\tan(73^\circ - 13^\circ) = \tan(60^\circ)$

STEP 6

We know that $\tan(60^\circ)$ is a special value!
It's equal to $\sqrt{3}$ : $\tan(60^\circ) = \sqrt{3}$

STEP 7

The problem tells us that this whole thing is equal to $\tan A^\circ = \sqrt{B}$ .
We've found that our expression equals $\tan(60^\circ)$ , which is also equal to $\sqrt{3}$ .
So, we can match these up: $\tan A^\circ = \tan 60^\circ$ $\sqrt{B} = \sqrt{3}$

STEP 8

From the first equation, we can see that $A = \mathbf{60}$ .

STEP 9

From the second equation, we can see that $B = \mathbf{3}$ .

STEP 10

$A = 60$ $B = 3$

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QuestionUse an addition or subtraction formula to write the expression as a trigonometric functi $\frac{\tan 73^{\circ}-\tan 13^{\circ}}{1+\tan 73^{\circ} \tan 13^{\circ}}=\tan A^{\circ}=\sqrt{B}$ A = $B=$ Submit Question

Studdy Solution

STEP 1

STEP 2

1. Identify the Formula
2. Apply the Formula
3. Calculate the Tangent
4. Find A and B

STEP 3

Alright, let's decode this mathematical puzzle!
The given expression looks a whole lot like the tangent subtraction formula.
Remember, the tangent subtraction formula is: $\tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \cdot \tan y}$ This is exactly the form we have!

STEP 4

Let's match the parts of our expression to the formula.
We can see that $x = 73^\circ$ and $y = 13^\circ$ .
So, we can rewrite our expression as: $\frac{\tan 73^\circ - \tan 13^\circ}{1 + \tan 73^\circ \cdot \tan 13^\circ} = \tan(73^\circ - 13^\circ)$

STEP 5

Now, let's simplify the angle inside the tangent function: $\tan(73^\circ - 13^\circ) = \tan(60^\circ)$

STEP 6

We know that $\tan(60^\circ)$ is a special value!
It's equal to $\sqrt{3}$ : $\tan(60^\circ) = \sqrt{3}$

STEP 7

The problem tells us that this whole thing is equal to $\tan A^\circ = \sqrt{B}$ .
We've found that our expression equals $\tan(60^\circ)$ , which is also equal to $\sqrt{3}$ .
So, we can match these up: $\tan A^\circ = \tan 60^\circ$ $\sqrt{B} = \sqrt{3}$

STEP 8

From the first equation, we can see that $A = \mathbf{60}$ .

STEP 9

From the second equation, we can see that $B = \mathbf{3}$ .

STEP 10

$A = 60$ $B = 3$

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Studdy Solution

STEP 1

STEP 2

1. Identify the Formula2. Apply the Formula3. Calculate the Tangent4. Find A and B

STEP 3

STEP 4

STEP 5

Now, let's **simplify** the angle inside the tangent function: tan⁡(73∘−13∘)=tan⁡(60∘) \tan(73^\circ - 13^\circ) = \tan(60^\circ) tan(73∘−13∘)=tan(60∘)

STEP 6

We know that tan⁡(60∘)\tan(60^\circ)tan(60∘) is a special value!It's equal to 3\sqrt{3}3​: tan⁡(60∘)=3 \tan(60^\circ) = \sqrt{3} tan(60∘)=3​

STEP 7

STEP 8

From the first equation, we can see that A=60A = \mathbf{60}A=60.

STEP 9

From the second equation, we can see that B=3B = \mathbf{3}B=3.

STEP 10

A=60A = 60A=60 B=3B = 3B=3

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1. Identify the Formula
2. Apply the Formula
3. Calculate the Tangent
4. Find A and B

Now, let's simplify the angle inside the tangent function: $\tan(73^\circ - 13^\circ) = \tan(60^\circ)$

We know that $\tan(60^\circ)$ is a special value!
It's equal to $\sqrt{3}$ : $\tan(60^\circ) = \sqrt{3}$

From the first equation, we can see that $A = \mathbf{60}$ .

From the second equation, we can see that $B = \mathbf{3}$ .

$A = 60$ $B = 3$