Question

Find the complement of each of the following angles:

Answer 1

Hint: Draw a line perpendicular to the base line and assume the complementary angle to be $x$. Subtract the given angle from ${{90}^{\circ }}$ to find the value of $x$. The value of $x$ obtained will be the answer.

Complete step-by-step answer:
Complementary angles are angle pairs whose measures sum to one right angle or ${{90}^{\circ }}$. If two complementary angles are adjacent their non-shared side forms a right angle. In Euclidean geometry, the two acute angles in a right angle triangle are complementary, because the sum of internal angles of a triangle is 180 degrees, and the right angle itself accounts for ninety degrees. The adjective complementary is from Latin complementum, associated with the verb complete, “to fill up”. An acute angle is filled up by its complement to form a right angle. The difference between an angle and a right angle is termed as the complement of the angle.
If two A and B angles are complementary, the following relationships hold:
$\begin{align}
  & {{\sin }^{2}}A+{{\sin }^{2}}B=1 \\
 & \tan A=\cot B \\
 & {{\cos }^{2}}A+{{\cos }^{2}}B=1 \\
 & \sec A=\cos ecB \\
\end{align}$
The tangent of an angle equals the cotangent of its complement and its secant equals the cosecant of its complement. The prefix “co-“ in the names of some trigonometric ratios refers to the word “complementary”.
Now, we come to the question. Let us assume that the complementary angle of \[{{57}^{\circ }}\] is $x$. Then, $\begin{align}
  & x+{{57}^{\circ }}={{90}^{\circ }} \\
 & x={{90}^{\circ }}-{{57}^{\circ }} \\
 & x={{33}^{\circ }} \\
\end{align}$
Hence, the complementary angle is ${{33}^{\circ }}$.

Note: We don’t have to equate the sum equal to ${{180}^{\circ }}$ because that will be the pair of supplementary angles. So, for finding the complementary angle we have to put the sum of all given angles equal to ${{90}^{\circ }}$.Complementary angles are angle pairs whose measures sum to one right angle or ${{90}^{\circ }}$.Supplementary angles are angle pairs whose measures sum to one straight angle or ${{180}^{\circ }}$.