Math / Trigonometry

QuestionFind the exact values of $r$ and $y$ given $B=76^{\circ}$ , and $x=6$ . NOTE: You will need to use trigonometric functions in your answers. $r=$ $\square$ $B=76^{\circ}$ $y=$

Studdy Solution

STEP 1

What is this asking? We need to find the lengths of the hypotenuse $r$ and the opposite side $y$ of a right triangle, given that one angle is $76^\circ$ and the adjacent side is $6$ . Watch out! Don't forget to use the correct trigonometric ratios for the right triangle, and make sure your calculator is in degree mode!

STEP 2

1. Use cosine to find $r$
2. Use sine to find $y$

STEP 3

Alright, let's start by using the cosine function!
Remember, cosine relates the adjacent side to the hypotenuse in a right triangle.
The formula is:
$\cos(B) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r}$

STEP 4

We know $B = 76^\circ$ and $x = 6$ .
Plug these into the formula:
$\cos(76^\circ) = \frac{6}{r}$

STEP 5

Now, let's solve for $r$ .
Multiply both sides by $r$ to get rid of the fraction:
$r \cdot \cos(76^\circ) = 6$

STEP 6

Next, divide both sides by $\cos(76^\circ)$ to isolate $r$ :
$r = \frac{6}{\cos(76^\circ)}$

STEP 7

Now, let's find $y$ using the sine function!
Sine relates the opposite side to the hypotenuse.
The formula is:
$\sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r}$

STEP 8

We already found $r$ in the previous step.
So, plug in $B = 76^\circ$ and the expression for $r$ :
$\sin(76^\circ) = \frac{y}{\frac{6}{\cos(76^\circ)}}$

STEP 9

Simplify the equation by multiplying both sides by $\frac{6}{\cos(76^\circ)}$ :
$y = \sin(76^\circ) \cdot \frac{6}{\cos(76^\circ)}$

STEP 10

The exact values are: - $r = \frac{6}{\cos(76^\circ)}$ - $y = \sin(76^\circ) \cdot \frac{6}{\cos(76^\circ)}$

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QuestionFind the exact values of $r$ and $y$ given $B=76^{\circ}$ , and $x=6$ . NOTE: You will need to use trigonometric functions in your answers. $r=$ $\square$ $B=76^{\circ}$ $y=$

Studdy Solution

STEP 1

STEP 2

1. Use cosine to find $r$
2. Use sine to find $y$

STEP 3

Alright, let's start by using the cosine function!
Remember, cosine relates the adjacent side to the hypotenuse in a right triangle.
The formula is:
$\cos(B) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r}$

STEP 4

We know $B = 76^\circ$ and $x = 6$ .
Plug these into the formula:
$\cos(76^\circ) = \frac{6}{r}$

STEP 5

Now, let's solve for $r$ .
Multiply both sides by $r$ to get rid of the fraction:
$r \cdot \cos(76^\circ) = 6$

STEP 6

Next, divide both sides by $\cos(76^\circ)$ to isolate $r$ :
$r = \frac{6}{\cos(76^\circ)}$

STEP 7

Now, let's find $y$ using the sine function!
Sine relates the opposite side to the hypotenuse.
The formula is:
$\sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r}$

STEP 8

We already found $r$ in the previous step.
So, plug in $B = 76^\circ$ and the expression for $r$ :
$\sin(76^\circ) = \frac{y}{\frac{6}{\cos(76^\circ)}}$

STEP 9

Simplify the equation by multiplying both sides by $\frac{6}{\cos(76^\circ)}$ :
$y = \sin(76^\circ) \cdot \frac{6}{\cos(76^\circ)}$

STEP 10

The exact values are: - $r = \frac{6}{\cos(76^\circ)}$ - $y = \sin(76^\circ) \cdot \frac{6}{\cos(76^\circ)}$

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QuestionFind the exact values of rrr and yyy given B=76∘B=76^{\circ}B=76∘, and x=6x=6x=6. NOTE: You will need to use trigonometric functions in your answers. r=r=r= □\square□ B=76∘B=76^{\circ}B=76∘ y=y=y=

Studdy Solution

STEP 1

STEP 2

1. Use cosine to find r r r2. Use sine to find y y y

STEP 3

Alright, let's start by using the cosine function!Remember, cosine relates the adjacent side to the hypotenuse in a right triangle.The formula is:cos⁡(B)=adjacenthypotenuse=xr\cos(B) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r}cos(B)=hypotenuseadjacent​=rx​

STEP 4

We know B=76∘ B = 76^\circ B=76∘ and x=6 x = 6 x=6.Plug these into the formula:cos⁡(76∘)=6r\cos(76^\circ) = \frac{6}{r}cos(76∘)=r6​

STEP 5

Now, let's solve for r r r.Multiply both sides by r r r to get rid of the fraction:r⋅cos⁡(76∘)=6r \cdot \cos(76^\circ) = 6r⋅cos(76∘)=6

STEP 6

Next, divide both sides by cos⁡(76∘)\cos(76^\circ)cos(76∘) to isolate r r r:r=6cos⁡(76∘)r = \frac{6}{\cos(76^\circ)}r=cos(76∘)6​

STEP 7

Now, let's find y y y using the sine function!Sine relates the opposite side to the hypotenuse.The formula is:sin⁡(B)=oppositehypotenuse=yr\sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r}sin(B)=hypotenuseopposite​=ry​

STEP 8

We already found r r r in the previous step.So, plug in B=76∘ B = 76^\circ B=76∘ and the expression for r r r:sin⁡(76∘)=y6cos⁡(76∘)\sin(76^\circ) = \frac{y}{\frac{6}{\cos(76^\circ)}}sin(76∘)=cos(76∘)6​y​

STEP 9

Simplify the equation by multiplying both sides by 6cos⁡(76∘)\frac{6}{\cos(76^\circ)}cos(76∘)6​:y=sin⁡(76∘)⋅6cos⁡(76∘)y = \sin(76^\circ) \cdot \frac{6}{\cos(76^\circ)}y=sin(76∘)⋅cos(76∘)6​

STEP 10

The exact values are: - r=6cos⁡(76∘) r = \frac{6}{\cos(76^\circ)} r=cos(76∘)6​ - y=sin⁡(76∘)⋅6cos⁡(76∘) y = \sin(76^\circ) \cdot \frac{6}{\cos(76^\circ)} y=sin(76∘)⋅cos(76∘)6​

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QuestionFind the exact values of $r$ and $y$ given $B=76^{\circ}$ , and $x=6$ . NOTE: You will need to use trigonometric functions in your answers. $r=$ $\square$ $B=76^{\circ}$ $y=$

1. Use cosine to find $r$
2. Use sine to find $y$

Alright, let's start by using the cosine function!
Remember, cosine relates the adjacent side to the hypotenuse in a right triangle.
The formula is:
$\cos(B) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r}$

We know $B = 76^\circ$ and $x = 6$ .
Plug these into the formula:
$\cos(76^\circ) = \frac{6}{r}$

Now, let's solve for $r$ .
Multiply both sides by $r$ to get rid of the fraction:
$r \cdot \cos(76^\circ) = 6$

Next, divide both sides by $\cos(76^\circ)$ to isolate $r$ :
$r = \frac{6}{\cos(76^\circ)}$

Now, let's find $y$ using the sine function!
Sine relates the opposite side to the hypotenuse.
The formula is:
$\sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r}$

We already found $r$ in the previous step.
So, plug in $B = 76^\circ$ and the expression for $r$ :
$\sin(76^\circ) = \frac{y}{\frac{6}{\cos(76^\circ)}}$

Simplify the equation by multiplying both sides by $\frac{6}{\cos(76^\circ)}$ :
$y = \sin(76^\circ) \cdot \frac{6}{\cos(76^\circ)}$

The exact values are: - $r = \frac{6}{\cos(76^\circ)}$ - $y = \sin(76^\circ) \cdot \frac{6}{\cos(76^\circ)}$