Textbook Question
The graph of a cotangent function is given. Select the equation for each graph from the following options: y = cot(x + π/2), y = cot(x + π), y = −cot x, y= −cot(x − π/2).
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2. Trigonometric Functions on Right Triangles
Solving Right Triangles
2:40 minutesProblem 49
Textbook Question
In Exercises 49–54, find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round answers to the nearest whole number.
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Verified step by step guidance1
Identify the sides of the right triangle relative to the given angle of 21° at vertex P. The side PR = 413 m is adjacent to angle P, and side QR is opposite to angle P. The hypotenuse is PQ.
To find the length of the side opposite to angle P (which is QR), use the sine function, since sine relates the opposite side to the hypotenuse: \(\sin(21^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{QR}{PQ}\).
To find the length of the hypotenuse PQ, use the cosine function, since cosine relates the adjacent side to the hypotenuse: \(\cos(21^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{PR}{PQ} = \frac{413}{PQ}\).
Rearrange the cosine equation to solve for the hypotenuse PQ: \(PQ = \frac{413}{\cos(21^\circ)}\).
Once you find PQ, substitute it back into the sine equation to solve for QR: \(QR = PQ \times \sin(21^\circ)\). This will give you the length of the side opposite the 21° angle.
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2mKey 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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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 one side length and one angle are known.
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Using Given Angle and Side to Find Unknown Side
Given an angle and one side length in a right triangle, you can use trigonometric ratios to find the length of another side. For example, with angle 21° and adjacent side 413 m, cosine can find the hypotenuse, or tangent can find the opposite side. Rounding to the nearest whole number is often required for practical answers.
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