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How to Calculate tan B in a Right Triangle with Sides √31 and 12
Mathematics
Grade 10 (Junior High School)
Question Content
In right triangle DCB with right angle at C, side DC = √31, side CB = 12. Express tan B as a fraction in simplest terms.
Correct Answer
√31/12
Detailed Solution Steps
1
Step 1: Recall the definition of tangent for an acute angle in a right triangle: tan(θ) = length of opposite side / length of adjacent side.
2
Step 2: Identify the sides relative to angle B. The opposite side to angle B is DC = √31, and the adjacent side (that is not the hypotenuse) is CB = 12.
3
Step 3: Substitute the values into the tangent formula: tan B = opposite/adjacent = DC/CB = √31/12.
4
Step 4: Confirm that √31 is a prime number, so the fraction √31/12 cannot be simplified further.
Knowledge Points Involved
1
Tangent Ratio in Right Triangles
For any acute angle in a right triangle, the tangent of the angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Formula: tan(θ) = opposite/adjacent. This is one of the three basic trigonometric ratios used to solve right triangle problems.
2
Identifying Opposite and Adjacent Sides
In a right triangle, relative to a given acute angle, the opposite side is the side that does not form the angle, while the adjacent side is one of the two sides that form the angle (excluding the hypotenuse, which is the longest side opposite the right angle).
3
Simplifying Radical Fractions
A fraction with a radical in the numerator is in simplest form if the radical cannot be simplified (i.e., the number under the radical has no perfect square factors other than 1) and there are no radicals in the denominator. Since 31 is a prime number, √31 cannot be simplified, so √31/12 is already in simplest terms.
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