Textbook Question
Decide whether each statement is true or false. If false, explain why.
The tangent and secant functions are undefined for the same values.
847
views
5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 11
Textbook Question
Find the exact value of each real number y. Do not use a calculator.
y = sec⁻¹ (―2)
Verified step by step guidance1
Recall that the function \( y = \sec^{-1}(x) \) is the inverse secant function, which gives an angle \( y \) such that \( \sec y = x \). Here, we want to find \( y \) such that \( \sec y = -2 \).
Use the identity relating secant and cosine: \( \sec y = \frac{1}{\cos y} \). So, \( \sec y = -2 \) implies \( \frac{1}{\cos y} = -2 \), which means \( \cos y = -\frac{1}{2} \).
Determine the range of \( y = \sec^{-1}(x) \). By definition, \( y \) lies in \( [0, \pi] \) excluding \( \frac{\pi}{2} \), because secant is not defined at \( \frac{\pi}{2} \).
Find all angles \( y \) in the interval \( [0, \pi] \) where \( \cos y = -\frac{1}{2} \). Recall the unit circle values where cosine equals \( -\frac{1}{2} \).
Select the angle(s) from the previous step that satisfy the domain of \( \sec^{-1} \) and write the exact value(s) of \( y \) accordingly.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Secant Function (sec⁻¹ or arcsec)
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 helps find the angle y such that sec(y) = -2.
Recommended video:
6:22
Graphs of Secant and Cosecant Functions
Relationship Between Secant and Cosine
Secant is the reciprocal of cosine, so sec(y) = 1/cos(y). To find y when sec(y) = -2, we rewrite it as cos(y) = -1/2. This relationship allows us to use cosine values to determine the angle.
Recommended video:
6:22Graphs of Secant and Cosecant Functions
Exact Values of Cosine for Special Angles
Certain angles have well-known cosine values, such as cos(120°) = cos(2π/3) = -1/2. Recognizing these exact values enables finding the precise angle y without a calculator when given sec(y) = -2.
Recommended video:
6:04Example 1
Related Videos
Related Practice