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Solve the inverse sin of π₯ is equal to π by four.
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Now we read this as the inverse sin of π₯.
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So what does that actually mean?
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Well, the inverse is essentially the opposite.
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Itβs the operation that undoes another operation.
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So the inverse sin of π₯ is the opposite operation to finding sin of something.
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Now, since sine and the inverse of sine are opposite operations of one another, if we take the sin of the inverse sin of π₯, we will just be left with π₯.
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And so weβre going to take the sin of both sides of this equation.
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As expected, that leaves us with π₯ on the left-hand side.
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And then on the right, we get sin of π by four.
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Now, we could use a calculator to perform this part of the calculation.
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However, this is one of the values that we should indeed know.
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We can use a table to help us remember the key trigonometric values we should know by heart.
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These are sin, cos, and tan of π by six, π by four, and π by three radians.
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Now, of course, π by six radians is equal to 30 degrees and π by four radians is 45 degrees.
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π by three radians is 60 degrees.
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And so this table of values also holds when weβre working with degrees.
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To fill in this table, we write one, two, and three in the first row and then we reverse that in the second row.
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We then make all of these numbers into a fraction with a denominator of two as shown.
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Then we find the square root of all of our numerators.
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But of course, the square root of one is simply one, so we donβt actually need to write that.
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And now we see that sin of π by six is one-half and so on.
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Now to find the corresponding tan values, we divide the sine value by the cosine value.
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Since the denominators of the fractions are the same, this just looks like dividing the numerator.
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So tan of π by six is one over root three.
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Then tan of π by four is root two divided by root two, which is just one.
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And tan of π by three is root three divided by one, which is just the square root of three.
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Now remember, we said that π₯ was equal to sin of π by four.
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sin is the first row and π by four is the middle column.
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And so sin of π by four must be equal to root two over two.
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And so that is our value for π₯.
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The solution to the equation inverse sin of π₯ is equal to π by four is π₯ equals root two over two.