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Find the Missing Side of a Triangle, Which Has One Angle of 40°, a Right Angle, and Adjacent Side Length 12.

Answer – The missing opposite side in the given right-angled triangle can be calculated by employing the trigonometric ratio tangent for the 40° angle. The opposite side is found to be ≅ 10 cm.

Explanation:

We begin by drawing a triangle and labeling all its angles and sides based on the given information. We know that the side of length 12 is adjacent to the 40° angle. Therefore, it must be between this angle and the right angle as shown below. Let’s call the unknown opposite side x.

A right-angled triangle with a 40-degree angle and adjacent side 12

Since we know the adjacent side of the triangle, the most suitable trigonometric ratio for calculating x is tangent. The formula for this is:

Tan θ = Opposite side / Adjacent side

On providing the value of θ as 40 and adjacent side length as 12, we get:

Tan 40 = x / 12

Finally, to calculate the unknown opposite side x, we cross-multiply and arrive at the following equation:

x = Tan 40 × 12

x = 0.8390 × 12

x = 10.06 ≅ 10

Therefore, the length of the missing side of the given triangle is 10.

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Precalculus Right Triangles Trigonometric Ratios Trigonometry

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