Determine whether each statement is true or false. See Example 4....

Question

Determine whether each statement is true or false. See Example 4.

cot 30° < tan 40°

Accepted Answer

Hello. Today we're going to be determining whether the given statement is true or false. So we want to determine that cotangent of degrees is less than tangent of degrees. Well, in order to answer this question, let's see if we can rewrite the cotangent function. In terms of tangent, we have a co function formula that states that cotangent of theta is equal to tangent of 90 degrees minus the. So if we wanted to rewrite cotangent in terms of tangent, we're going to need to use this co function equation. So for cotangent, the given angle theta is equal to 58 degrees. And if we plug in this value of data into the co function formula, that would give us that cotangent of 58 degrees is equal to tangent of 90 degrees minus degrees, 90 minus 58 will give us the value of 32. And that means that cotangent of 58 degrees can be rewritten as tangent of 32 degrees. So using this value, we can rewrite the statement as tangent of 32 degrees is less than tangent of 69 degrees. Now, in order to answer this question, we're going to need to take a look at the first period of the graph of tangent. The first period of the graph of tangent has two asymptotes. The first Asymptote is located at negative pi divided by two which is equivalent to negative 90 degrees. And the second Asymptote is located at pi divided by two which is equivalent to positive 90 degrees. And the graph of the tangent function has a rough snakelike shape roughly like this. Now, since we want to compare tangent of 32 degrees to tangent of 69 degrees, we want to pay attention to the to the function as it approaches 90 degrees. Let's just roughly say that tangent of 32 degrees is located here and tangent of 69 degrees is located here. Now, as the graph of tangent is approaching 90 degrees, the the graph of the tangent is going to increase over time. What this means is that the value of tangent of 32 degrees is going to be less than tangent of 69 degrees. And with that being said, that means that the statement is going to be true and the overall answer is going to be a. 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.

Determine whether each statement is true or false. See Example 4. cot 30° < tan 40°

Key Concepts

Cotangent and Tangent Functions

Angle Measurement in Trigonometry

Comparison of Trigonometric Values

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