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How to Calculate tan B in a Right Triangle with Given Side Lengths
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 side opposite angle B is DC = √31, and the side adjacent to angle B (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 is already in its simplest form.
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. It is one of the three basic trigonometric ratios, used to relate angle measures to side lengths in right triangles.
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 always the longest side opposite the right angle).
3
Simplifying Radical Fractions
A fraction containing a radical is in simplest form if the radical has no perfect square factors other than 1, and there is no radical in the denominator. Since √31 is a radical with a prime radicand, √31/12 cannot be simplified further.
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