Question 187426: #1) If tan of theta = (6/5) and cos of theta < 0, use the fundamental identities to evaluate the other five trigonometric functions of theta.
#2) use the fundamental identities to simplify csc squared if beta (1-cos squared of beta)
#3) factor and simplify. (sec "to the power of 4" x- tan "to the power of 4" x/ sec "squared" x + tan "squared" x)
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! #1) If tan of theta = (6/5) and cos of theta < 0, use the fundamental identities to evaluate the other five trigonometric functions of theta.
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If tan(theta) = 6/5 and cos(theta) < 0 then y = -6 and x = -5
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Solve for "r": r = sqrt(6^2+5^2) = sqrt(61)
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sin(theta) = y/r = -6/sqrt(61)
cos(theta) = x/r = -5/sqrt(61)
tan(theta) = y/x = 6/5
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The csc, sec, and cot are the inverse of sin, cos, tan
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#2) use the fundamental identities to simplify csc squared of beta (1-cos squared of beta)
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csc^2(1 - cos^2)
= csc^2 - (cos^2/sin^2)
= (1/sin^2) - (cos^2/sin^2)
= (1 - cos^2)/sin^2
= sin^2/sin^2
= 1
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#3) factor and simplify. (sec "to the power of 4" x- tan "to the power of 4" x/ sec "squared" x + tan "squared" x)
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[sec^4(x) - tan^4(x)] / [sec^2(x) + tan^2(x)]
= [(sec^2+tan^2)(sec^2-tan^2)] / [sec^2 + tan^2]
= sec^2(x) - tan^2(x)
= 1
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Cheers,
Stan H.
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