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Find the value of one plus sin two π΄ divided by one plus cos two π΄ given the tan of π΄ is five twenty-sixths where π΄ is greater than zero and less than π over three radians without using a calculator.
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Before trying to calculate the value of the expression, letβs consider the information we are given.
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We are told that the tan of angle π΄ is five twenty-sixths and that π΄ lies between zero and π by three radians.
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Using the CAST diagram, we know that if an angle lies between zero and π by two, the sine of the angle, the cosine of the angle, and the tangent of the angle will all be positive.
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This is true in this case as the angle π΄ lies between zero and π by three radians.
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Using our knowledge of right-angle trigonometry, we know that the sin of angle π is equal to the opposite over the hypotenuse, the cos of angle π is equal to the adjacent over the hypotenuse, and the tan of angle π is equal to the opposite over the adjacent.
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In this question, we are told the tan of angle π΄ is equal to five twenty-sixths.
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We can calculate the length of the hypotenuse using the Pythagorean theorem.
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β squared is equal to 26 squared plus five squared.
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26 squared plus five squared is equal to 701.
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We can then square root both sides of our equation such that β is equal to the square root of 701.
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The sin of angle π΄ is, therefore, equal to five over the square root of 701 and the cos of angle π΄ is equal to 26 over the square root of 701.
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Letβs now consider the expression one plus the sin of two π΄ divided by one plus the cos of two π΄.
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We know that the sin of two π is equal two sin π cos π and that the cos of two π is equal to two multiplied by cos squared π minus one.
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The expression in this question can be rewritten as one plus two sin π΄ cos π΄ divided by one plus two cos squared π΄ minus one.
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On the denominator, one plus one is equal to zero.
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We can now substitute the values of sin π΄ and cos π΄ into this expression.
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On the numerator, we have one plus two multiplied by five over the square root of 701 multiplied by 26 over the square root of 701.
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On the denominator, we have two multiplied by 26 over the square root of 701 squared.
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The numerator simplifies to 961 over 701 and the denominator simplifies to 1352 over 701.
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Multiplying the numerator and denominator by 701 gives us a final answer of 961 over 1352.
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This is the value of one plus sin two π΄ divided by one plus cos two π΄.