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Given that sin π₯ cos 60 equals one-quarter, find the value of π₯, where π₯ is greater than zero and less than 90 degrees.
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Now to answer this question, we might begin by spotting that cos of 60 is one of the trigonometric values that we should know by heart.
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And whilst we can derive these, we certainly donβt want to be doing that every time.
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So thereβs a handy table that can help.
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This table contains values for sin, cos, and tan of zero, 30, 45, 60, and 90 degrees.
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We fill this table in in the following manner.
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We begin by writing zero, one, two, three, and four.
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Then we reverse this for cosine: four, three, two, one, and zero.
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We add a denominator of two to every single one of these values.
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Then we take the square root of the numerator only.
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And this allows us to simplify a number of these expressions.
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The square root of zero is zero, so the square root of zero over two is also zero, meaning that sin of zero and cos of 90 are both zero.
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The square root of one is simply one, so sin of 30 and cos of 60 are one-half.
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Finally, the square root of four is two.
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So sin of 90 and cos of zero are two over two, which is simply equal to one.
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So those are our values for sin of π and cos of π.
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But what about tan of π?
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Well, we divide any value for sin of π by any value of cos of π to get the corresponding value of tan of π.
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tan of zero then is zero divided by one, which is zero.
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tan of 30 is one-half divided by root three over two.
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In fact, though, since the denominators are the same, we can simply divide the numerators.
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So tan of 30 is one over root three, which is equivalent to root three over three.
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tan of 45 is root two over two divided by root two over two, which is one.
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tan of 60 is root three divided by one, which is root three.
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And then when we try to find tan of 90, we get one divided by zero, which is undefined.
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So tan of 90 is undefined.
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And this is really useful because we can now rewrite the left-hand side of our equation by finding the value of cos 60 in our table.
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cos of 60 is equal to one-half.
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So our equation sin π₯ cos 60 equals a quarter becomes sin π₯ times one-half equals a quarter.
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And in fact, it will be helpful to isolate sin of π₯ because then we can find the relevant values in our table.
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To do so, weβre going to divide by one-half.
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And actually, thatβs the same as multiplying by two.
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So sin of π₯ is a quarter times two, which is simply equal to one-half.
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So we now need to ask ourselves what value of π₯ in our table gives us sin π₯ equals one-half.
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Well, we might notice that sin of 30 equals one-half.
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So for the equation sin π₯ equals one-half to hold for values of π₯ between zero and 90 degrees, π₯ must be equal to 30.
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So given that sin π₯ cos 60 equals a quarter, our value of π₯ is 30 degrees.