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Find sec of minus 30 degrees without using a calculator.
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Let us start by recording what sec actually means.
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We have that sec of ๐ is equal to one over cos ๐.
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So therefore, we can write sec of minus 30 is equal to one over cos of minus 30.
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Now we just need to find the value of cos of minus 30.
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Now since minus 30 is not a nice angle to work with, we can look at a cos graph to try and find another angle ๐ for which cos of ๐ is equal to cos of minus 30.
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So letโs draw a cos graph.
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Here we have our cos graph.
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And we can mark on the angle minus 30.
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The horizontal dashed line represents the line ๐ฆ is equal to cos of minus 30.
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Therefore, any other point at which this line intersects the line ๐ฆ is equal to cos ๐ will give us another value of ๐, for which cos of minus 30 is equal to cos ๐.
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So letโs find another value of ๐.
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We can see that it also intersects at a point between naught and 90 degrees.
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Since the graph of cos is symmetric about the ๐ฆ-axis, we know that this point is at 30 degrees.
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And from this we can write cos of minus 30 is equal to cos of 30.
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And now we can substitute this back into our original equation.
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And this gives us that sec of minus 30 is equal to one over cos of 30.
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So all that remains to do is to find the value of cos 30.
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In order to do this, letโs start by drawing an equilateral triangle with sides of length two.
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So here is our equilateral triangle.
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Now, we can split it in half down the middle.
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Now, if we consider the triangle on the left, we can see that we have a right triangle with angles of 30 degrees and sixty degrees.
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Letโs draw this triangle out separately.
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So this triangle has a hypotenuse of two and one of the sides of length one.
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And now we can use Pythagoras to find the other side.
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And this will give us a side of length root three.
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Next, we will use SOHCAHTOA in order to find the value of cos of 30 degrees.
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Now, since weโre trying to find cos of an angle, weโre interested in CAH.
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This tells us that cos of an angle ๐ is equal to the adjacent over the hypotenuse.
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Now, the angle weโre interested in is 30 degrees.
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So letโs call 30 degrees ๐.
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And we can label on the adjacent, opposite, and hypotenuse of the triangle.
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So the adjacent is the side of length root three.
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The opposite is the side of length one.
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And the hypotenuse is the side of length two.
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Now we can write that cos of 30 is equal to the adjacent over the hypotenuse.
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So thatโs simply root three over two.
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And now we can take this value of cos 30 and put it back into our equation in order to find a value of sec of minus 30.
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We get that sec of minus 30 is equal to one over root three over two.
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This can be simplified to give us the solution that sec of minus 30 is equal to two over root three.