Video Transcript
Simplify sin of 90 degrees minus 𝜃
divided by cos of 90 degrees minus 𝜃 multiplied by tan 𝜃. Is it (A) negative one, (B) one,
(C) tan 𝜃, (D) negative tan 𝜃, or (E) cot 𝜃?
To answer this question, we begin
by recalling the properties of complementary angles. Complementary angles sum to 90
degrees. These are useful in trigonometry
because if 𝜃 is one angle in a right triangle, then the other angle is the
complementary angle, 90 degrees minus 𝜃. We can use this to find a set of
identities called the cofunction identities.
First, we draw 𝜃 in standard
position as follows. We can include the complementary
angle in our diagram as shown. Since 𝜃 plus 90 degrees minus 𝜃
equals 90 degrees, we can construct the following congruent triangle. This is the angle 90 degrees minus
𝜃 in standard position, so the coordinates of the point of intersection give us the
sine and cosine values of this angle. Since these triangles are
congruent, we can equate the corresponding sides to get cos 𝜃 equals sin of 90
degrees minus 𝜃 and sin 𝜃 equals cos of 90 degrees minus 𝜃. These are the cofunction
identities, and they are true for any angle 𝜃 measured in degrees.
Returning to our expression, we can
rewrite sin of 90 degrees minus 𝜃 over cos of 90 degrees minus 𝜃 as cos 𝜃 over
sin 𝜃. Next, we recall that tan 𝜃 is
equal to sin 𝜃 over cos 𝜃. So our expression simplifies to cos
𝜃 over sin 𝜃 multiplied by sin 𝜃 over cos 𝜃. Multiplying the numerators and
denominators separately gives us sin 𝜃 cos 𝜃 divided by sin 𝜃 cos 𝜃, which is
equal to one. We can therefore conclude that sin
of 90 degrees minus 𝜃 divided by cos of 90 degrees minus 𝜃 multiplied by tan 𝜃 is
equal to one. And the correct answer is option
(B).
