Textbook Question
Solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. a = 30.4, c = 50.2
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2. Trigonometric Functions on Right Triangles
Solving Right Triangles
4:34 minutesProblem 31
Textbook Question
In Exercises 29–36, find the length x to the nearest whole unit.
Verified step by step guidance1
Identify the two right triangles in the diagram. The larger triangle has a base of 1600 units and an unknown height x. The angles given are 66° and 24°, which are complementary to the right angle in each triangle.
Use the tangent function, which relates the opposite side to the adjacent side in a right triangle. For the larger triangle with angle 66°, write the equation: \(\tan(66^\circ) = \frac{x}{1600}\).
Solve for x in the equation from step 2: \(x = 1600 \times \tan(66^\circ)\).
For the smaller triangle, note that the height is the difference between the two heights formed by the two triangles. Use the tangent function for the 24° angle: \(\tan(24^\circ) = \frac{h}{1600}\), where h is the height of the smaller triangle.
Calculate the height h from the smaller triangle: \(h = 1600 \times \tan(24^\circ)\). Then, subtract h from the height x of the larger triangle to find the length of the segment labeled x in the diagram.
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Video duration:
4mKey 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 known.
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Angle Sum Property of Triangles
The angle sum property states that the sum of the interior angles of any triangle is 180°. In this problem, knowing two angles (66° and 24°) helps confirm the triangle's structure and can assist in identifying complementary or supplementary angles needed for calculations.
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Using Trigonometric Ratios to Find Lengths
To find an unknown side length like x, use trigonometric ratios with the given angles and known side lengths. For example, tangent relates the opposite side to the adjacent side, allowing calculation of height x when the base and angle are known. This approach is key to solving the problem.
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