### Video Transcript

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.

Before trying to calculate the
value of the expression, letβs consider the information we are given. We are told that the tan of angle
π΄ is five twenty-sixths and that π΄ lies between zero and π by three radians. 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. This is true in this case as the
angle π΄ lies between zero and π by three radians.

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. In this question, we are told the
tan of angle π΄ is equal to five twenty-sixths. We can calculate the length of the
hypotenuse using the Pythagorean theorem. β squared is equal to 26 squared
plus five squared. 26 squared plus five squared is
equal to 701. We can then square root both sides
of our equation such that β is equal to the square root of 701. 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.

Letβs now consider the expression
one plus the sin of two π΄ divided by one plus the cos of two π΄. 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. 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. On the denominator, one plus one is
equal to zero.

We can now substitute the values of
sin π΄ and cos π΄ into this expression. 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. On the denominator, we have two
multiplied by 26 over the square root of 701 squared. The numerator simplifies to 961
over 701 and the denominator simplifies to 1352 over 701. Multiplying the numerator and
denominator by 701 gives us a final answer of 961 over 1352. This is the value of one plus sin
two π΄ divided by one plus cos two π΄.