Given that cot of 𝜃 equals
negative three-halves, where 𝜋 over two is less than 𝜃 which is less than 𝜋,
evaluate sec squared 𝜃 without using a calculator.
Before we do anything else, it will
be helpful to identify the place where this 𝜃 must fall. It’s falling between 𝜋 over two
and 𝜋. So we sketch a coordinate grid and
label it with the radian measures. If 𝜃 falls between 𝜋 over two and
𝜋, the 𝜃 will fall somewhere in the second quadrant. If we sketch a line and our angle
𝜃, we could then make a sketch of a right-angled triangle.
From there, we’ll need to remember
our trig relationships. Since we know that cot of our angle
is negative three over two, we know that the three represents the adjacent side
length and the two represents the opposite side length. We can use that information to
label this sketch. Our goal is to find sec squared
𝜃. To do that, we’ll multiply sec of
𝜃 by sec of 𝜃.
That secant relationship is the
hypotenuse over the adjacent side length. But in order to solve this problem,
we’ll need to figure out what the hypotenuse is. We can use the Pythagorean theorem
to do this. Two squared plus three squared
equals the hypotenuse squared. That will be four plus nine, 13,
equals the hypotenuse squared, which means the hypotenuse equals the square root of
But we need to be really careful
here. We have to consider, because our
angle falls in this second quadrant, if the secant is going to be positive or
negative. To do that, we can use a CAST
diagram. We already know we’re interested in
the second quadrant. And because there’s an S here, it
tells us that the sine relationship is positive, but the cosine and tangent
relationships will be negative.
secant is the inverse of
cosine. And that means if the cosine value
is negative, the secant value will be negative. Since the secant is the hypotenuse
over the adjacent side length, we have the square root of 13 over three, but we know
that this value must be negative. And that makes the sec of 𝜃 the
negative square root of 13 over three. sec squared will be negative square root of
13 over three times negative square root of 13 over three. The two negatives become
positive. The square root of 13 multiplied by
the square root of 13 equals 13. And three times three is nine. Under these conditions, sec squared
𝜃 is 13 over nine.