In Exercises 1–26, find the exact value of each expression.
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Question

In Exercises 1–26, find the exact value of each expression.
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sec⁻¹ (−√2)

Accepted Answer

Hello, today we're gonna be evaluating the following trigonometric expression. So what we are given is seek an inverse of negative two. Now, what we wanna do in order to evaluate this trigonometric expression is we want to set this expression equal to an arbitrary constant which in this case, we can use X. So we're going to create an equation which is going to be seeking inverse of negative two is equal to X. Then what we wanna do is we want to use trigonometric identities in order to reduce or simplify this equation. So we have a property for trigonometric functions that state if you multiply a trigonometric function by its inverse or you multiply the inverse of a trigonometric function with the respective trigonometric function. What is going to happen is that the product of the function and its inverse are going to cancel each other out leaving us with the inside angle which in this case is X. So in order to simplify this equation, we're going to multiply both sides of the equation by second doing so eliminates the inverse on the left hand side leaving us with the inside angle which is negative two and on the right hand side, what we are left with is sequent of X. Now recall that the identity of seeking of X is the inverse of cosine, which is one over cosine of X. We can use this property to rewrite the right hand side of the equation. And the equation becomes negative two is equal to one over cosine of X. Now, what we wanna do is you want to get cosine of X by itself in order to do this, we're gonna multiply both sides of the equation by cosine of X. Doing this allows us to reduce the denominator on the right hand side, leaving us with negative two multiplied by cosine of X is equal to one. Finally, in order to get cosine X by itself, we're going to divide both sides of the equation by negative two. And that is going to give us cosine of X is equal to negative 1/2. Now, the question here is what is an angle of data that we can plug in a cosine that will give us the value of negative 1/2. Now to answer this, we need to be familiar with our unit circle, but we also need to be careful when choosing data because keep in mind there are multiple values of eight of data. We can plug in a cosine to give us negative one half. But the original problem stated that we are working with C N inverse So we're going to need to use the domain of seek and inverse to answer data. The domain of seek and inverse is zero to pi which is the first half of the unit circle. So we only want to use values of data that exist in the 1st and 2nd quadrants of the unit circle. So what is the value of data that we can plug in a cosine from the domain of sequent? That'll give us negative 1/2. Well, that value of data is going to be two pi over three. Just to just to verify this, we can check this value on the unit circle. So we're going to draw a unit circle and keep in mind the domain we are working with is the first half of the unit circle between zero and pi two pi over three exists in the second quadrant of the unit circle which is within the domain we are working with. And the terminal point at this angle is going to be negative 1/2 comma square root 3/2. The cosine value of this of this angle is going to be negative 1/2. So what this means is that the original expression which is seeking inverse of negative two is going to equal to two pi over three, that is going to be the exact value of the given expression. And with that being said, the answer to this problem is going to be D So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 1–26, find the exact value of each expression. _ sec⁻¹ (−√2)

Key Concepts

Inverse Secant Function

Understanding Negative Values

Exact Values of Trigonometric Functions

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