Textbook Question
Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1.
1/csc(90°-51°)
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1. Measuring Angles
Angles in Standard Position
1:44 minutesProblem 2.3.36
Textbook Question
Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.
sec θ = 1.1606249
Verified step by step guidance1
Recall the definition of secant in terms of cosine: \(\sec \theta = \frac{1}{\cos \theta}\). This means that \(\cos \theta = \frac{1}{\sec \theta}\).
Substitute the given value of \(\sec \theta\) into the equation: \(\cos \theta = \frac{1}{1.1606249}\).
Calculate the value of \(\cos \theta\) using the reciprocal of the given secant value (do not compute the final number here, just set up the expression).
Use the inverse cosine function to find \(\theta\): \(\theta = \cos^{-1}(\text{value from step 3})\). This will give \(\theta\) in radians or degrees depending on your calculator settings.
Since the problem asks for \(\theta\) in degrees within the interval \([0^\circ, 90^\circ)\), ensure your calculator is set to degrees and that the angle you find lies within this interval.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of Secant Function
The secant function, sec θ, is the reciprocal of the cosine function, defined as sec θ = 1/cos θ. Understanding this relationship allows you to convert the given secant value into a cosine value, which is often easier to work with when solving for θ.
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Graphs of Secant and Cosecant Functions
Inverse Trigonometric Functions
To find the angle θ from a trigonometric value, you use inverse functions such as arccos or arcsec. Since sec θ = 1.1606249, you first find cos θ = 1/1.1606249, then apply the inverse cosine function to determine θ within the specified interval.
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Domain and Range Restrictions
The problem restricts θ to the interval [0°, 90°), which corresponds to the first quadrant where cosine values are positive. This restriction ensures a unique solution for θ and guides the selection of the correct angle from the inverse function's output.
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