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Find the set of values satisfying two sin 𝜃 plus cos 𝜃 sec 𝜃 equals zero, where 𝜃 is greater than or equal to zero and less than 360 degrees.
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In order to solve this equation, we begin by recalling that sec 𝜃 is equal to one over cos 𝜃.
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It is the reciprocal of cos 𝜃.
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We can, therefore, rewrite the equation as two sin 𝜃 plus cos 𝜃 multiplied by one over cos 𝜃 is equal to zero.
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The cos 𝜃’s cancel such that two sin 𝜃 plus one is equal to zero.
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We can then subtract one from both sides of this equation.
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Finally, dividing by two gives us sin 𝜃 is equal to negative one-half.
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We recall that sin of 30 degrees is one of our special angles and is equal to one-half.
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By drawing our CAST diagram, we can see that the solutions for sin 𝜃 equals negative one-half will be between 180 and 270 degrees and also between 270 and 360 degrees.
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The angles these lines make with the horizontal will be equal to 30 degrees.
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One of our solutions will be equal to 180 plus 30.
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This is equal to 210.
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The second solution will be equal to 360 minus 30.
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This is equal to 330.
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The values of 𝜃 that satisfy the equation are 210 degrees and 330 degrees.
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This can be written using set notation as shown.
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The set of values that satisfy two sin 𝜃 plus cos 𝜃 sec 𝜃 is equal to zero are 210 degrees and 330 degrees.