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Express tan B as a Simplest Fraction in a Right Triangle
Mathematics
High School
Question Content
Express tan B as a fraction in simplest terms. The diagram shows a right triangle DCB with a right angle at C, where CD = √31 and CB = 12.
Correct Answer
√31 / 12
Detailed Solution Steps
1
1. Recall the definition of tangent in a right triangle: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
2
2. Identify the sides relative to \( \angle B \):
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- The side **opposite** \( \angle B \) is \( CD \) (length \( \sqrt{31} \)).
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- The side **adjacent** to \( \angle B \) is \( CB \) (length \( 12 \)).
5
3. Apply the tangent ratio: \( \tan B = \frac{\text{opposite}}{\text{adjacent}} = \frac{CD}{CB} = \frac{\sqrt{31}}{12} \).
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4. Simplify the fraction: \( \sqrt{31} \) and \( 12 \) have no common factors, so \( \frac{\sqrt{31}}{12} \) is in simplest form.
Knowledge Points Involved
1
Tangent Ratio in Right Triangles
In a right triangle, the tangent of an acute angle \( \theta \) is defined as the ratio of the length of the side **opposite** \( \theta \) to the length of the side **adjacent** to \( \theta \): \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). This ratio helps relate the angles of a right triangle to the lengths of its sides.
2
Simplifying Fractions with Radicals
To simplify a fraction with a radical in the numerator (e.g., \( \frac{\sqrt{a}}{b} \)), check if the radical and the denominator have common factors. If the radical’s radicand (e.g., \( a \)) is a prime number or has no square factors, and the denominator is an integer with no common factors with the radicand, the fraction is already in simplest form.
3
Right Triangle Side Identification
In a right triangle, the legs are the two sides forming the right angle, and the hypotenuse is the side opposite the right angle. For an acute angle, the 'opposite' side is across from the angle, and the 'adjacent' side is next to the angle (forming the angle with the hypotenuse).
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