Math / Trigonometry

QuestionOne of the sides $x, y$ and $r$ of the triangle in the figure below is given. Find exact values of the other two sides.
Find the values of $r$ and $y$ in the figure, given that $B=12^{\circ}, x=\lambda$ . $r=\lambda / \cos 12^{\circ}$ and $y=\lambda / \tan 12^{\circ}$ $r=\lambda / \sin 12^{\circ}$ and $y=\lambda / \cos 12^{\circ}$ $r=\lambda / \cos 12^{\circ}$ and $y=\lambda / \cos 12^{\circ}$ $r=\lambda / \sin 12^{\circ}$ and $y=\lambda / \tan 12^{\circ}$

Studdy Solution

STEP 1

What is this asking? We're trying to find the exact lengths of the sides $r$ and $y$ of a triangle, given that one side $x$ is $\lambda$ and angle $B$ is $12^\circ$ . Watch out! Remember to use the correct trigonometric functions for each side.
Mixing up sine, cosine, and tangent is a common mistake!

STEP 2

1. Identify the trigonometric relationships
2. Solve for $r$
3. Solve for $y$

STEP 3

Alright, let's get our trigonometry hats on!
We know angle $B = 12^\circ$ and side $x = \lambda$ .
We're looking for sides $r$ and $y$ .
In a right triangle, the relationships between the sides and angles are given by sine, cosine, and tangent.

STEP 4

The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse.
So, for angle $B$ , we have: $\cos(12^\circ) = \frac{x}{r}$

STEP 5

The tangent of an angle is the ratio of the opposite side to the adjacent side.
So, for angle $B$ , we have: $\tan(12^\circ) = \frac{y}{x}$

STEP 6

From the cosine relationship, we have: $\cos(12^\circ) = \frac{\lambda}{r}$

STEP 7

To solve for $r$ , multiply both sides by $r$ and divide by $\cos(12^\circ)$ : $r = \frac{\lambda}{\cos(12^\circ)}$

STEP 8

From the tangent relationship, we have: $\tan(12^\circ) = \frac{y}{\lambda}$

STEP 9

To solve for $y$ , multiply both sides by $\lambda$ : $y = \lambda \cdot \tan(12^\circ)$

STEP 10

The exact values of the sides are: $r = \frac{\lambda}{\cos(12^\circ)}$ $y = \lambda \cdot \tan(12^\circ)$

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Studdy Solution

STEP 1

STEP 2

1. Identify the trigonometric relationships
2. Solve for $r$
3. Solve for $y$

STEP 3

Alright, let's get our trigonometry hats on!
We know angle $B = 12^\circ$ and side $x = \lambda$ .
We're looking for sides $r$ and $y$ .
In a right triangle, the relationships between the sides and angles are given by sine, cosine, and tangent.

STEP 4

The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse.
So, for angle $B$ , we have: $\cos(12^\circ) = \frac{x}{r}$

STEP 5

The tangent of an angle is the ratio of the opposite side to the adjacent side.
So, for angle $B$ , we have: $\tan(12^\circ) = \frac{y}{x}$

STEP 6

From the cosine relationship, we have: $\cos(12^\circ) = \frac{\lambda}{r}$

STEP 7

To solve for $r$ , multiply both sides by $r$ and divide by $\cos(12^\circ)$ : $r = \frac{\lambda}{\cos(12^\circ)}$

STEP 8

From the tangent relationship, we have: $\tan(12^\circ) = \frac{y}{\lambda}$

STEP 9

To solve for $y$ , multiply both sides by $\lambda$ : $y = \lambda \cdot \tan(12^\circ)$

STEP 10

The exact values of the sides are: $r = \frac{\lambda}{\cos(12^\circ)}$ $y = \lambda \cdot \tan(12^\circ)$

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Studdy Solution

STEP 1

STEP 2

1. Identify the trigonometric relationships2. Solve for r r r3. Solve for y y y

STEP 3

Alright, let's get our trigonometry hats on!We know angle B=12∘ B = 12^\circ B=12∘ and side x=λ x = \lambda x=λ.We're looking for sides r r r and y y y.In a right triangle, the relationships between the sides and angles are given by sine, cosine, and tangent.

STEP 4

The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse.So, for angle B B B, we have: cos⁡(12∘)=xr \cos(12^\circ) = \frac{x}{r} cos(12∘)=rx​

STEP 5

The tangent of an angle is the ratio of the opposite side to the adjacent side.So, for angle B B B, we have: tan⁡(12∘)=yx \tan(12^\circ) = \frac{y}{x} tan(12∘)=xy​

STEP 6

From the cosine relationship, we have: cos⁡(12∘)=λr \cos(12^\circ) = \frac{\lambda}{r} cos(12∘)=rλ​

STEP 7

To solve for r r r, multiply both sides by r r r and divide by cos⁡(12∘)\cos(12^\circ)cos(12∘): r=λcos⁡(12∘) r = \frac{\lambda}{\cos(12^\circ)} r=cos(12∘)λ​

STEP 8

From the tangent relationship, we have: tan⁡(12∘)=yλ \tan(12^\circ) = \frac{y}{\lambda} tan(12∘)=λy​

STEP 9

To solve for y y y, multiply both sides by λ\lambdaλ: y=λ⋅tan⁡(12∘) y = \lambda \cdot \tan(12^\circ) y=λ⋅tan(12∘)

STEP 10

The exact values of the sides are: r=λcos⁡(12∘) r = \frac{\lambda}{\cos(12^\circ)} r=cos(12∘)λ​ y=λ⋅tan⁡(12∘) y = \lambda \cdot \tan(12^\circ) y=λ⋅tan(12∘)

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1. Identify the trigonometric relationships
2. Solve for $r$
3. Solve for $y$

Alright, let's get our trigonometry hats on!
We know angle $B = 12^\circ$ and side $x = \lambda$ .
We're looking for sides $r$ and $y$ .
In a right triangle, the relationships between the sides and angles are given by sine, cosine, and tangent.

The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse.
So, for angle $B$ , we have: $\cos(12^\circ) = \frac{x}{r}$

The tangent of an angle is the ratio of the opposite side to the adjacent side.
So, for angle $B$ , we have: $\tan(12^\circ) = \frac{y}{x}$

From the cosine relationship, we have: $\cos(12^\circ) = \frac{\lambda}{r}$

To solve for $r$ , multiply both sides by $r$ and divide by $\cos(12^\circ)$ : $r = \frac{\lambda}{\cos(12^\circ)}$

From the tangent relationship, we have: $\tan(12^\circ) = \frac{y}{\lambda}$

To solve for $y$ , multiply both sides by $\lambda$ : $y = \lambda \cdot \tan(12^\circ)$

The exact values of the sides are: $r = \frac{\lambda}{\cos(12^\circ)}$ $y = \lambda \cdot \tan(12^\circ)$