Question

How do you find the complement and supplement of $\dfrac{3\pi }{4}$ ?

Answer 1

We have to find the complementary and supplementary angles of $\dfrac{3\pi }{4}$. As we know that the sum of two complementary angles is $\dfrac{\pi }{2}$ and the sum of two supplementary angles is $\pi$. So we will use the concept to get the desired answer.

Complete step by step solution:
We have been given a measure of angle $\dfrac{3\pi }{4}$.
We have to find the complement and supplement of a given angle.
Now, we know that sum of two complementary angles is $\dfrac{\pi }{2}$ so to find the complement of the given angle we need to subtract $\dfrac{3\pi }{4}$ from $\dfrac{\pi }{2}$. Then we will get
$\Rightarrow \dfrac{\pi }{2}-\dfrac{3\pi }{4}$
Now, solving the above obtained equation we will get
$\begin{aligned} & \Rightarrow \dfrac{2\pi -3\pi }{4} \\\ & \Rightarrow \dfrac{-\pi }{4} \\\ \end{aligned}$
Now, we know that the sum of two supplementary angles is $\pi$. So to find the supplement of the given angle we need to subtract $\dfrac{3\pi }{4}$from $\pi$. Then we will get
$\Rightarrow \pi -\dfrac{3\pi }{4}$
Now, solving the above obtained equation we will get
$\begin{aligned} & \Rightarrow \dfrac{4\pi -3\pi }{4} \\\ & \Rightarrow \dfrac{\pi }{4} \\\ \end{aligned}$
Hence we get the complement of $\dfrac{3\pi }{4}$ as $-\dfrac{\pi }{4}$ and supplement of $\dfrac{3\pi }{4}$ as $\dfrac{\pi }{4}$.

Note: Alternatively students can assume the complementary angle as $\theta$ and then equate the sum of $\dfrac{3\pi }{4}$ and $\theta$ to $\dfrac{\pi }{2}$. Then by solving the obtained equation we get the value of $\theta$ i.e. value of complementary angle. Similarly we can assume the supplementary angle as $\alpha$ and then equate the sum of $\dfrac{3\pi }{4}$ and $\alpha$to $\pi$. Then by solving the obtained equation we get the value of $\alpha$i.e. value of supplementary angle.