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Find the set of values satisfying the equation five cos squared π equals four, where zero degrees is less than or equal to π which is less than 360 degrees.
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Give the answer to the nearest minute.
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Weβre being asked to solve the trigonometric equation five cos squared π equals four for π, where π is greater than or equal to zero and less than 360 degrees.
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So, where do we start?
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Well, we begin by solving just like any other equation.
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Weβre going to divide both sides by five such that cos squared π is four-fifths or 0.8.
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Next, we take the square root of both sides of our equation, remembering to take both the positive and negative square root of four-fifths.
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The square root of four-fifths can be written as two root five over five.
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So, we have two equations that we need to solve, cos of π equals two root five over five or negative two root five over five.
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We solve for π by taking the inverse cos of both sides of our equation.
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When we take the inverse cos of two root five over five, making sure our calculator is in degree mode, we get 26.5650 and so on.
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And the inverse cos of negative two root five over five is 153.4349 and so on.
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Now, weβre going to leave those values unrounded for now, but weβre not quite finished.
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We wanted to find the set of values for π is greater than or equal to zero and less than 360 degrees.
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And so, there are a couple of ways we can find the other solutions.
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One way is to think about the shape of the graph π¦ equals cos of π₯.
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In the interval from π is greater than or equal to zero to less than 360, it looks a little like this with maximums at one and minimums at negative one.
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The line π¦ equals two root five over five looks a little something like this.
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We know it has one solution at 26.5 and so on degrees.
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Now, we can see that the graph has reflection or symmetry about the line π₯ equals 180 degrees, so we find the other value of π by subtracting 26.5 and so on from 360 degrees.
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That gives us 333.4349 and so on.
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Similarly, if we sketch the line, π¦ equals negative two root five over five, it looks a bit like this.
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This time, one solution lies at 153.4 degrees and so one.
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To find the fourth solution, we subtract this from 360.
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That gives us 206.5650 and so on.
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Now, we want to round our answers to the nearest minute.
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We could multiply the decimal part by 60 to achieve this.
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Alternatively, thereβs a button on most calculators that will do this for us.
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And it looks a little something like this.
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When we press this button for our first value, we get 26 degrees and 34 minutes correct to the nearest minute.
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We get 153 degrees and 26 minutes for our next solution.
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And our other two solutions are 260 degrees and 34 minutes and 333 degrees and 26 minutes.
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And so, the set of values that satisfy our solution are shown.
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Now, there was another way we could have found these values, and that was to use the CAST diagram.
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The CAST diagram looks like this.
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We put the letters C-A-S-T, and this shows us where our values for cos π, sin π, tan π, or all three are positive.
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Now, our first solution to cos π equals two root five over five was 26.56 and so on degrees.
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The other solution will be here in the fourth quadrant where cos π is positive.
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We find this value of π by subtracting 26.56 from 360.
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And that gives us, to the nearest minute, 333 degrees and 26 minutes.
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Our other solution was π equals 153.43 and so on.
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Cos π is also negative in this third quadrant, so we subtract 153.43 and so on from 360 degrees.
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And once again, we get 206 degrees and 34 minutes correct to the nearest minute.