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How to Calculate tan B for a Right Triangle with Given Side Lengths
Mathematics
Grade 10 (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 the tangent trigonometric function 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: Verify that the fraction is in simplest form: √31 is an irrational number with no perfect square factors, so √31/12 cannot be simplified further.
Knowledge Points Involved
1
Tangent Function in Right Triangles
For an acute angle θ in a right triangle, tan(θ) 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 primary trigonometric ratios, used to relate angle measures to side lengths in right triangles.
2
Identifying Opposite and Adjacent Sides
In a right triangle, for a given acute angle: the opposite side is the side that does not form the angle, the adjacent side is the side that forms the angle along with the hypotenuse, and the hypotenuse is the longest side opposite the right angle. This distinction is critical for applying trigonometric ratios correctly.
3
Simplifying Radical Fractions
A fraction with a radical in the numerator 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 prime number, √31 cannot be simplified, so √31/12 is already in simplest form.
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