Math / Geometry

Question8. Find the value of the hypotenuse $60.25$
9. Find the value of the angle $B$ (in the lower ri

Studdy Solution

STEP 1

1. We have a right triangle, triangle ABC.
2. Side AC is 4 inches, side CB is 7 inches, and angle C is a right angle.
3. We need to find the value of the hypotenuse (side AB).
4. We need to find the value of angle B.

STEP 2

1. Use the Pythagorean Theorem to find the hypotenuse.
2. Use trigonometric ratios to find the value of angle B.

STEP 3

To find the hypotenuse, we use the Pythagorean Theorem, which states:
$a^2 + b^2 = c^2$
where $a$ and $b$ are the legs of the triangle, and $c$ is the hypotenuse.

STEP 4

Substitute the given values into the Pythagorean Theorem:
$4^2 + 7^2 = c^2$
Calculate:
$16 + 49 = c^2$

STEP 5

Simplify and solve for $c$ :
$65 = c^2$
Take the square root of both sides:
$c = \sqrt{65}$

STEP 6

To find angle B, use the tangent function, which is defined as:
$\tan(B) = \frac{\text{opposite}}{\text{adjacent}}$
For angle B, the opposite side is AC (4 inches) and the adjacent side is CB (7 inches).

STEP 7

Calculate the tangent of angle B:
$\tan(B) = \frac{4}{7}$

STEP 8

Find angle B using the inverse tangent function:
$B = \tan^{-1}\left(\frac{4}{7}\right)$
Use a calculator to find the value of angle B in degrees.
The value of the hypotenuse is $\sqrt{65}$ inches, and the value of angle B is approximately $\tan^{-1}\left(\frac{4}{7}\right)$ degrees.

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Question8. Find the value of the hypotenuse $60.25$
9. Find the value of the angle $B$ (in the lower ri

Studdy Solution

STEP 1

1. We have a right triangle, triangle ABC.
2. Side AC is 4 inches, side CB is 7 inches, and angle C is a right angle.
3. We need to find the value of the hypotenuse (side AB).
4. We need to find the value of angle B.

STEP 2

1. Use the Pythagorean Theorem to find the hypotenuse.
2. Use trigonometric ratios to find the value of angle B.

STEP 3

To find the hypotenuse, we use the Pythagorean Theorem, which states:
$a^2 + b^2 = c^2$
where $a$ and $b$ are the legs of the triangle, and $c$ is the hypotenuse.

STEP 4

Substitute the given values into the Pythagorean Theorem:
$4^2 + 7^2 = c^2$
Calculate:
$16 + 49 = c^2$

STEP 5

Simplify and solve for $c$ :
$65 = c^2$
Take the square root of both sides:
$c = \sqrt{65}$

STEP 6

To find angle B, use the tangent function, which is defined as:
$\tan(B) = \frac{\text{opposite}}{\text{adjacent}}$
For angle B, the opposite side is AC (4 inches) and the adjacent side is CB (7 inches).

STEP 7

Calculate the tangent of angle B:
$\tan(B) = \frac{4}{7}$

STEP 8

Find angle B using the inverse tangent function:
$B = \tan^{-1}\left(\frac{4}{7}\right)$
Use a calculator to find the value of angle B in degrees.
The value of the hypotenuse is $\sqrt{65}$ inches, and the value of angle B is approximately $\tan^{-1}\left(\frac{4}{7}\right)$ degrees.

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Question8. Find the value of the hypotenuse 60.2560.2560.259. Find the value of the angle BBB (in the lower ri

Studdy Solution

STEP 1

1. We have a right triangle, triangle ABC.2. Side AC is 4 inches, side CB is 7 inches, and angle C is a right angle.3. We need to find the value of the hypotenuse (side AB).4. We need to find the value of angle B.

STEP 2

1. Use the Pythagorean Theorem to find the hypotenuse.2. Use trigonometric ratios to find the value of angle B.

STEP 3

To find the hypotenuse, we use the Pythagorean Theorem, which states:a2+b2=c2 a^2 + b^2 = c^2 a2+b2=c2where a a a and b b b are the legs of the triangle, and c c c is the hypotenuse.

STEP 4

Substitute the given values into the Pythagorean Theorem:42+72=c2 4^2 + 7^2 = c^2 42+72=c2Calculate:16+49=c2 16 + 49 = c^2 16+49=c2

STEP 5

Simplify and solve for c c c:65=c2 65 = c^2 65=c2Take the square root of both sides:c=65 c = \sqrt{65} c=65​

STEP 6

To find angle B, use the tangent function, which is defined as:tan⁡(B)=oppositeadjacent \tan(B) = \frac{\text{opposite}}{\text{adjacent}} tan(B)=adjacentopposite​For angle B, the opposite side is AC (4 inches) and the adjacent side is CB (7 inches).

STEP 7

Calculate the tangent of angle B:tan⁡(B)=47 \tan(B) = \frac{4}{7} tan(B)=74​

STEP 8

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Question8. Find the value of the hypotenuse $60.25$
9. Find the value of the angle $B$ (in the lower ri

1. We have a right triangle, triangle ABC.
2. Side AC is 4 inches, side CB is 7 inches, and angle C is a right angle.
3. We need to find the value of the hypotenuse (side AB).
4. We need to find the value of angle B.

1. Use the Pythagorean Theorem to find the hypotenuse.
2. Use trigonometric ratios to find the value of angle B.

To find the hypotenuse, we use the Pythagorean Theorem, which states:
$a^2 + b^2 = c^2$
where $a$ and $b$ are the legs of the triangle, and $c$ is the hypotenuse.

Substitute the given values into the Pythagorean Theorem:
$4^2 + 7^2 = c^2$
Calculate:
$16 + 49 = c^2$

Simplify and solve for $c$ :
$65 = c^2$
Take the square root of both sides:
$c = \sqrt{65}$

To find angle B, use the tangent function, which is defined as:
$\tan(B) = \frac{\text{opposite}}{\text{adjacent}}$
For angle B, the opposite side is AC (4 inches) and the adjacent side is CB (7 inches).

Calculate the tangent of angle B:
$\tan(B) = \frac{4}{7}$