In Exercises 5–18, the unit circle has been divided into twelve e...

Question

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋.6 3 2 3 6 6 3 2 3 6Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.
In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.sec 5𝜋/3

Unit circle with coordinates and angles for trigonometric functions in trigonometry course.

In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.sec 5𝜋/3

Accepted Answer

Hello, everyone. We are given a unit circle and asked to determine the value of the trigonometric function at the given real number T using the X Y coordinates in the diagram or specify that the expression is undefined here. We're looking for the value of the sea can of seven pi divided by six, we are given four answer choices. So first when I look at the sea camp of seven pi divided by six, I can identify T in this case is seven pi divided by six. So going to my unit circle finding seven pi divided by six. I see that that matches with the coordinates negative radical three divided by two, a negative one half from there. I recall that my X value is the cosine of T. So the cosine of seven pi divided by six and my Y value is the sign of T. So the sign of seven pi divided by six. Unfortunately, I'm not looking for sign or cosine, I'm looking for CCA but I know that the sea can't of any value is the same as one divided by the cosine of that value. So if we are looking for the sea can of seven pi divided by six. We can work with one divided by the cosine of seven pi divided by six which is negative radical three divided by two. I want to take care of my complex fraction. So I'm gonna rewrite it as two fractions. So one divided by negative radical 3/2. Recall my rules of fractions. I know I'm multiplying by the reciprocal. So I have one times negative two divided by radical three. And this will give me negative two divided by radical three. We're almost there, but we don't want to leave a radical in our denominator. So when I have negative two divided by radical three, I'm gonna rationalize that denominator by multiplying top and bottom of the fraction by radical three. And this will leave me negative two. Radical three is my numerator. In my, my denominator radical three multiplied by radical three will give us a three. So the C can of seven pi divided by six is negative two radical three divided by three which is answer choice. A have a nice day.

Key Concepts

Unit Circle

Trigonometric Functions

Radians and Angle Measurement

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