Textbook Question
In Exercises 29–51, find the exact value of each expression. Do not use a calculator. cos⁻¹ (− 1/2)
Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblem 35
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblem 35Chapter 2, Problem 35
In Exercises 29–36, find the length x to the nearest whole unit.
Verified step by step guidance1
Identify the right triangle and the given information: one angle is 22°, the other non-right angle is 54°, the base adjacent to the 22° angle is 200 units, and the height (opposite side to the 54° angle) is x, which we need to find.
Recall that in a right triangle, the sum of the two non-right angles is 90°, which matches the given angles 22° and 54°, confirming the triangle's angle measures.
Use the tangent function for the 54° angle, since tangent relates the opposite side to the adjacent side: \(\tan(54^\circ) = \frac{x}{200}\).
Rearrange the equation to solve for x: \(x = 200 \times \tan(54^\circ)\).
Calculate the value of \(\tan(54^\circ)\) using a calculator, then multiply by 200 to find the length x, rounding to the nearest whole unit.
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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, enabling the calculation of unknown side lengths or angles.
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45-45-90 Triangles
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 help find missing sides or angles when some measurements are known.
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5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Angle Sum and Complementary Angles in Triangles
The sum of angles in any triangle is 180°. In a right triangle, the two acute angles are complementary, summing to 90°. Understanding this helps identify which trigonometric ratios to use based on the given angles and sides.
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3:35Intro to Complementary & Supplementary Angles
Related Practice
Textbook Question
In Exercises 29–44, graph two periods of the given cosecant or secant function. y = sec x/3
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Textbook Question
In Exercises 35–42, determine the amplitude and period of each function. Then graph one period of the function. y = cos 2x
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Textbook Question
In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. tan⁻¹ (−20)
Textbook Question
In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. _ cos⁻¹ √5/7
Textbook Question
Graph two periods of the given cosecant or secant function.
y = 3 sec x
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