Textbook Question
In Exercises 1–26, find the exact value of each expression. _ tan⁻¹ (−√3)
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
4:31 minutesProblem 25
Textbook Question
In Exercises 1–26, find the exact value of each expression. _ sec⁻¹ (−√2)
Verified step by step guidance1
Recall that the function \( \sec^{-1}(x) \) is the inverse secant function, which gives the angle \( \theta \) such that \( \sec(\theta) = x \).
Identify the value inside the inverse secant: \( x = -\sqrt{2} \). We want to find \( \theta \) such that \( \sec(\theta) = -\sqrt{2} \).
Recall the relationship between secant and cosine: \( \sec(\theta) = \frac{1}{\cos(\theta)} \). So, \( \frac{1}{\cos(\theta)} = -\sqrt{2} \) implies \( \cos(\theta) = -\frac{1}{\sqrt{2}} \).
Determine the angle(s) \( \theta \) in the principal range of \( \sec^{-1} \), which is usually \( [0, \pi] \) excluding \( \frac{\pi}{2} \), where \( \cos(\theta) = -\frac{1}{\sqrt{2}} \).
Identify the exact angle(s) where cosine equals \( -\frac{1}{\sqrt{2}} \), which correspond to known special angles, and select the one within the principal range of the inverse secant function.
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Video duration:
4mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Secant Function (sec⁻¹)
The inverse secant function, sec⁻¹(x), returns the angle whose secant is x. It is defined for |x| ≥ 1, and its range is typically [0, π] excluding π/2. Understanding this function helps find the angle corresponding to a given secant value.
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Graphs of Secant and Cosecant Functions
Secant and Cosine Relationship
Secant is the reciprocal of cosine, so sec(θ) = 1/cos(θ). To find sec⁻¹(x), you can think in terms of cosine: cos(θ) = 1/x. This relationship is crucial for converting between secant and cosine values when solving trigonometric equations.
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Evaluating Exact Values Using Special Angles
Exact values of trigonometric functions often come from special angles like π/4, π/3, and π/6. Recognizing that sec(3π/4) = -√2, for example, allows you to determine the exact angle for sec⁻¹(−√2) without a calculator.
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