In Exercises 29–36, find the length x to the nearest whole unit.

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Identify the right triangle and the given information: the height (opposite side) is 800 units, and the angles adjacent to the base are 37° and 72°.

Recognize that the height of 800 is opposite to the 72° angle in the smaller right triangle formed inside the larger triangle.

Use the tangent function for the smaller triangle: \(\tan(72^\circ) = \frac{800}{\text{adjacent side}}\). Solve for the adjacent side, which is the segment of the base adjacent to the 72° angle.

Use the tangent function for the larger triangle: \(\tan(37^\circ) = \frac{800}{x + \text{adjacent side}}\). Here, \(x + \text{adjacent side}\) is the total base length of the larger triangle.

Solve the two tangent equations step-by-step to find the length \(x\), which is the segment of the base adjacent to the 37° angle.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Right Triangle Trigonometry

Right triangle trigonometry involves relationships between the angles and sides of a right triangle. The primary trigonometric ratios—sine, cosine, and tangent—relate an angle to the ratios of two sides. These ratios are essential for finding unknown side lengths or angles when some measurements are given.

Trigonometric Ratios (Sine, Cosine, Tangent)

Sine, cosine, and tangent are ratios defined for an acute angle in a right triangle: sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent. These ratios allow calculation of unknown sides or angles when at least one side length and one angle are known.

Angle Sum in a Triangle and Complementary Angles

The sum of angles in any triangle is 180°. In a right triangle, the two non-right angles are complementary, meaning they add up to 90°. This property helps identify missing angles and apply the correct trigonometric ratios to solve for unknown sides.

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