Find the value of sin 𝜃 given cos
of 𝜃 is equal to negative 21 over 29, where 𝜃 is greater than 90 degrees and less
than 180 degrees.
In order to answer this question,
we’ll use the Pythagorean identity sin squared 𝜃 plus cos squared 𝜃 is equal to
one. We are told that 𝜃 lies between 90
and 180 degrees, so it is worth considering our CAST diagram. The angle lies in the second
quadrant, which means that the sin of angle 𝜃 must be positive. The cos of angle 𝜃 and tan of
angle 𝜃 must be negative. This ties in with the fact that we
are told that the cos of 𝜃 is equal to negative 21 over 29. It also helps us in that we know
the answer for sin 𝜃 must be positive.
We can substitute the value of cos
𝜃 into the Pythagorean identity. Squaring negative 21 over 29 gives
us 441 over 841. We can then subtract this from both
sides of our equation such that sin squared 𝜃 is equal to one minus 441 over
841. The right-hand side simplifies to
400 over 841. We can then square root both sides
of our equation so that sin of 𝜃 is equal to positive or negative the square root
of 400 over 841.
Square rooting the numerator gives
us 20 and the denominator 29. As sin of 𝜃 must be positive, if
the cos of 𝜃 is negative 21 over 29 and 𝜃 lies between 90 and 180 degrees, then
the sin of 𝜃 is equal to 20 over 29.