Math / Trigonometry

QuestionGiven a right angled triangle $X Y Z$ with angles $Y=\frac{\pi}{2}$ and $Z=0.02$ side $X=3.7 \mathrm{~mm}$ find side $Z$ . Give your answer in mm to 2 decimal places.

Studdy Solution

STEP 1

1. Triangle $XYZ$ is a right-angled triangle with $\angle Y = \frac{\pi}{2}$ .
2. $\angle Z = 0.02$ radians.
3. Side $X$ is opposite $\angle Z$ and has a length of $3.7 \, \text{mm}$ .
4. We need to find the length of side $Z$ , which is adjacent to $\angle Z$ .

STEP 2

1. Recall the trigonometric relationship for a right-angled triangle.
2. Use the tangent function to relate the sides.
3. Solve for side $Z$ .
4. Round the answer to two decimal places.

STEP 3

Recall the trigonometric relationship for a right-angled triangle:
The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side:
$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

STEP 4

Use the tangent function to relate the sides:
For $\angle Z = 0.02$ , the opposite side is $X = 3.7 \, \text{mm}$ , and the adjacent side is $Z$ :
$\tan(0.02) = \frac{3.7}{Z}$

STEP 5

Solve for side $Z$ :
Rearrange the equation to solve for $Z$ :
$Z = \frac{3.7}{\tan(0.02)}$
Calculate $\tan(0.02)$ and then find $Z$ :
$\tan(0.02) \approx 0.02$ (using small angle approximation)
$Z \approx \frac{3.7}{0.02}$
$Z \approx 185$

STEP 6

Round the answer to two decimal places:
Since $Z \approx 185$ , the answer is already to two decimal places:
$Z = 185.00 \, \text{mm}$
The length of side $Z$ is:
$\boxed{185.00 \, \text{mm}}$

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QuestionGiven a right angled triangle $X Y Z$ with angles $Y=\frac{\pi}{2}$ and $Z=0.02$ side $X=3.7 \mathrm{~mm}$ find side $Z$ . Give your answer in mm to 2 decimal places.

Studdy Solution

STEP 1

1. Triangle $XYZ$ is a right-angled triangle with $\angle Y = \frac{\pi}{2}$ .
2. $\angle Z = 0.02$ radians.
3. Side $X$ is opposite $\angle Z$ and has a length of $3.7 \, \text{mm}$ .
4. We need to find the length of side $Z$ , which is adjacent to $\angle Z$ .

STEP 2

1. Recall the trigonometric relationship for a right-angled triangle.
2. Use the tangent function to relate the sides.
3. Solve for side $Z$ .
4. Round the answer to two decimal places.

STEP 3

Recall the trigonometric relationship for a right-angled triangle:
The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side:
$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

STEP 4

Use the tangent function to relate the sides:
For $\angle Z = 0.02$ , the opposite side is $X = 3.7 \, \text{mm}$ , and the adjacent side is $Z$ :
$\tan(0.02) = \frac{3.7}{Z}$

STEP 5

Solve for side $Z$ :
Rearrange the equation to solve for $Z$ :
$Z = \frac{3.7}{\tan(0.02)}$
Calculate $\tan(0.02)$ and then find $Z$ :
$\tan(0.02) \approx 0.02$ (using small angle approximation)
$Z \approx \frac{3.7}{0.02}$
$Z \approx 185$

STEP 6

Round the answer to two decimal places:
Since $Z \approx 185$ , the answer is already to two decimal places:
$Z = 185.00 \, \text{mm}$
The length of side $Z$ is:
$\boxed{185.00 \, \text{mm}}$

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QuestionGiven a right angled triangle XYZX Y ZXYZ with angles Y=π2Y=\frac{\pi}{2}Y=2π​ and Z=0.02Z=0.02Z=0.02 side X=3.7 mmX=3.7 \mathrm{~mm}X=3.7 mm find side ZZZ. Give your answer in mm to 2 decimal places.

Studdy Solution

STEP 1

STEP 2

1. Recall the trigonometric relationship for a right-angled triangle.2. Use the tangent function to relate the sides.3. Solve for side ZZZ.4. Round the answer to two decimal places.

STEP 3

Recall the trigonometric relationship for a right-angled triangle:The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side:tan⁡(θ)=OppositeAdjacent \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} tan(θ)=AdjacentOpposite​

STEP 4

Use the tangent function to relate the sides:For ∠Z=0.02\angle Z = 0.02∠Z=0.02, the opposite side is X=3.7 mmX = 3.7 \, \text{mm}X=3.7mm, and the adjacent side is ZZZ:tan⁡(0.02)=3.7Z \tan(0.02) = \frac{3.7}{Z} tan(0.02)=Z3.7​

STEP 5

STEP 6

Round the answer to two decimal places:Since Z≈185Z \approx 185Z≈185, the answer is already to two decimal places:Z=185.00 mm Z = 185.00 \, \text{mm} Z=185.00mm The length of side ZZZ is:185.00 mm \boxed{185.00 \, \text{mm}} 185.00mm​

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionGiven a right angled triangle $X Y Z$ with angles $Y=\frac{\pi}{2}$ and $Z=0.02$ side $X=3.7 \mathrm{~mm}$ find side $Z$ . Give your answer in mm to 2 decimal places.

1. Recall the trigonometric relationship for a right-angled triangle.
2. Use the tangent function to relate the sides.
3. Solve for side $Z$ .
4. Round the answer to two decimal places.

Recall the trigonometric relationship for a right-angled triangle:
The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side:
$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

Use the tangent function to relate the sides:
For $\angle Z = 0.02$ , the opposite side is $X = 3.7 \, \text{mm}$ , and the adjacent side is $Z$ :
$\tan(0.02) = \frac{3.7}{Z}$

Round the answer to two decimal places:
Since $Z \approx 185$ , the answer is already to two decimal places:
$Z = 185.00 \, \text{mm}$
The length of side $Z$ is:
$\boxed{185.00 \, \text{mm}}$