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Introduction
In this problem, we are given a right triangle with certain measurements and are asked to find the value of mZQ. To solve this, we need to use the properties of right triangles and apply trigonometric concepts. The given measurements are R = √30 and √62 = Q. We will use these values to find the value of mZQ.
Understanding the Problem
The problem involves a right triangle with two known measurements: R = √30 and √62 = Q. We need to find the value of mZQ, which is the measure of the angle opposite to the side with length Q. To solve this, we can use the trigonometric ratios in a right triangle.
Using Trigonometric Ratios
In a right triangle, the trigonometric ratios are defined as follows:
- Sine (sin): The ratio of the length of the side opposite to the angle to the length of the hypotenuse.
- Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
- Tangent (tan): The ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle.
We can use these ratios to find the value of mZQ.
Finding mZQ
To find mZQ, we can use the tangent ratio:
tan(mZQ) = Q / R
We are given that Q = √62 and R = √30. Substituting these values into the equation, we get:
tan(mZQ) = √62 / √30
Simplifying the Expression
To simplify the expression, we can rationalize the denominator by multiplying both the numerator and the denominator by √30:
tan(mZQ) = (√62 × √30) / (√30 × √30) = (√1860) / 30 = √1860 / 30
Evaluating the Expression
To evaluate the expression, we can simplify the square root:
√1860 = √(30 × 62) = √(30 × 2 × 31) = √(60 × 31) = √1860
Now, we can substitute this value back into the expression:
tan(mZQ) = √1860 / 30
Finding the Value of mZQ
To find the value of mZQ, we can use the inverse tangent function (arctangent):
mZQ = arctan(√1860 / 30)
Evaluating the Inverse Tangent
To evaluate the inverse tangent, we can use a calculator or a trigonometric table. The value of arctan(√1860 / 30) is approximately 1.1 radians.
Converting to Degrees
To convert the value from radians to degrees, we can multiply by (180 / π):
mZQ ≈ 1.1 × (180 / π) ≈ 63.4°
Conclusion
In this problem, we used the properties of right triangles and trigonometric ratios to find the value of mZQ. We started with the given measurements R = √30 and Q = √62, and used the tangent ratio to find the value of mZQ. We then simplified the expression and evaluated the inverse tangent to find the value of mZQ in radians. Finally, we converted the value from radians to degrees.
The final answer is: 63.4
Introduction
In our previous article, we solved the problem of finding the value of mZQ in a right triangle with given measurements R = √30 and Q = √62. We used the properties of right triangles and trigonometric ratios to find the value of mZQ. In this article, we will answer some frequently asked questions related to the problem.
Q: What is the significance of the measurements R = √30 and Q = √62?
A: The measurements R = √30 and Q = √62 are the lengths of the sides of the right triangle. R is the length of the side adjacent to the angle mZQ, and Q is the length of the side opposite to the angle mZQ.
Q: How did you simplify the expression tan(mZQ) = √62 / √30?
A: We simplified the expression by rationalizing the denominator. We multiplied both the numerator and the denominator by √30 to get rid of the square root in the denominator.
Q: What is the inverse tangent function (arctangent)?
A: The inverse tangent function (arctangent) is a mathematical function that returns the angle whose tangent is a given value. In this problem, we used the arctangent function to find the value of mZQ.
Q: Why did you convert the value of mZQ from radians to degrees?
A: We converted the value of mZQ from radians to degrees because the problem statement asked for the value of mZQ in degrees. Radians are a unit of angle measurement, and degrees are a more common unit of angle measurement.
Q: Can you explain the trigonometric ratios in a right triangle?
A: In a right triangle, the trigonometric ratios are defined as follows:
- Sine (sin): The ratio of the length of the side opposite to the angle to the length of the hypotenuse.
- Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
- Tangent (tan): The ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle.
Q: How did you find the value of mZQ using the tangent ratio?
A: We found the value of mZQ using the tangent ratio by substituting the given values of Q and R into the equation tan(mZQ) = Q / R. We then simplified the expression and evaluated the inverse tangent to find the value of mZQ.
Q: What is the final answer to the problem?
A: The final answer to the problem is 63.4 degrees.
Q: Can you provide a summary of the problem and the solution?
A: Here is a summary of the problem and the solution:
- The problem involves finding the value of mZQ in a right triangle with given measurements R = √30 and Q = √62.
- We used the properties of right triangles and trigonometric ratios to find the value of mZQ.
- We simplified the expression tan(mZQ) = √62 / √30 and evaluated the inverse tangent to find the value of mZQ.
- We converted the value of mZQ from radians to degrees and found the final answer to be 63.4 degrees.
Conclusion
In this article, we answered some frequently asked questions related to the problem of finding the value of mZQ in a right triangle. We provided explanations and examples to help clarify the concepts and procedures involved in solving the problem.