WEBVTT
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Find the exact value of inverse tan of negative one in radians in the interval minus π is less than π which is less than or equal to two π over three.
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Before we even think about answering this question, itβs really important to notice that it asks us to give our answer as an exact solution.
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Now, if we are lucky enough to have a scientific calculator to hand, we can type inverse tan of negative one into this.
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And it should as long as your calculator is in radians give us an answer of negative π over four.
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If a calculator is not allowed however, we will need to recall the definition of an inverse function to help us.
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A function has an inverse function if and only if it is one to one, meaning that each π¦-value has no more than one corresponding π₯-value.
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We can see that the tan graph fails this test.
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For any π¦-value on the tan graph, there are a whole number of corresponding π₯ solutions.
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Instead, we restrict the domain of our tan function to between negative π over two and positive π over two.
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On our unit circle, the values of inverse tan will be located on the right half of the circle, not including negative π over two and positive π over two since the tangent function is undefined at these points.
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These are the reference angles we know by heart.
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We can then use the symmetry of the unit circle to find the corresponding values that lie in the fourth quadrant.
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Now, recall tan π is equal to opposite over adjacent.
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In unit circle terms, tan π is equal to π¦ over π₯.
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We need to find an ordered pair on our unit circle that is between π over two and negative π over two such that π¦ over π₯ is equal to minus one.
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When π equals negative π over four, tan π is equal to root two over two divided by negative root two over two.
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This is equal to negative one.
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Therefore, the inverse tan of negative one is equal to negative π over four.