Find the missing side YZ and angle. Round to the nearest tenth if necessary.
Understand the Problem
The question is asking to find the missing side YZ and angle ∠Y, using the measurement of the triangle given. This involves applying trigonometric principles, specifically the sine and cosine laws.
Answer
\( m∠Y ≈ 78° \) and \( x = YZ ≈ 4.6 \) units.
Answer for screen readers
The values calculated are:
- ( m∠Y ≈ 78° )
- ( x = YZ ≈ 4.6 ) units.
Steps to Solve
- Identify given information
In triangle XYZ, we know:
- Angle ∠Z = 63°
- Angle ∠X = 39°
- Side XZ = 6 units
- Calculate missing angle ∠Y
The sum of angles in a triangle is 180°. So we can find angle ∠Y using:
$$ \angle Y = 180° - \angle Z - \angle X $$
Substituting the known values:
$$ \angle Y = 180° - 63° - 39° $$
- Calculate the value of angle ∠Y
Now, calculating ∠Y:
$$ \angle Y = 180° - 102° = 78° $$
- Use the Law of Sines to find side YZ
The Law of Sines states:
$$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$
In this context, we can set up the equation to find side YZ:
$$ \frac{YZ}{\sin X} = \frac{XZ}{\sin Y} $$
Substituting known values (where (XZ = 6) units and (X = 39°)):
$$ \frac{YZ}{\sin(39°)} = \frac{6}{\sin(78°)} $$
- Solve for side YZ
Rearranging the equation gives:
$$ YZ = 6 \cdot \frac{\sin(39°)}{\sin(78°)} $$
Now we can calculate YZ using a calculator.
The values calculated are:
- ( m∠Y ≈ 78° )
- ( x = YZ ≈ 4.6 ) units.
More Information
In this triangle, using the Law of Sines enables us to solve for missing sides and angles effectively. The angles’ relationships allow us to derive a variety of side lengths based on known measures.
Tips
- Forgetting that the angles in a triangle sum to 180°, which could lead to incorrect angle calculations.
- Misapplying the Law of Sines or mixing up the corresponding sides and angles.
- Not using a calculator set to degrees when evaluating sine values, leading to calculation errors.
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