Complementary and Supplementary Angles (radians and rotations)
Two angles that sum to $90^\circ$90° or, in radians, $\frac{\pi}{2}$π2, are said to be complementary. The complement of an angle is its difference from $90^\circ$90° or from $\frac{\pi}{2}$π2.
Two angles that sum to $180^\circ$180° or, in radians, $\pi$π, are said to be supplementary. The supplement of an angle is its difference from $180^\circ$180° or $\pi$π.
We introduced the idea of complementary and supplementary angles here.
In a right-angled triangle, it is clear that the sine of an angle is equal to the cosine of its complement, and vice-versa. From the unit circle definition of the trigonometric functions, we see that the sine of an angle is equal to the sine of its supplement. Similarly, the cosine of an angle is the negative of the cosine of its supplement.
We may also wish to think of the difference between an angle and a full rotation.
Examples
Example 1
At $12:15$12:15 on a clock face, what is the acute angle between the hour and minute hands? What is the large angle? (An angle between $180^\circ$180° and $360^\circ$360° is called a reflex angle.)
The hour hand has moved $\frac{1}{4}$14 of the distance between '$12$12' and '$1$1'. So, it has moved $\left(\frac{1}{4}\div12\right)\times360^\circ$(14÷12)×360°. That is, $7.5^\circ$7.5°. The minute hand has moved $\frac{1}{4}$14 of a full turn or $\frac{1}{4}\times360^\circ=90^\circ$14×360°=90°. So, the angle between the hands is $90-7.5=82.5^\circ$90−7.5=82.5°.
The large or reflex angle is $360-82.5=277.5^\circ$360−82.5=277.5°.
Example 2
Angles $x$x and $y$y are complementary. If we are given coefficients $a$a and $b$b such that $a\times x=b\times y$a×x=b×y, express $x$x and $y$y separately in terms of $a$a and $b$b.
We write $x+y=\frac{\pi}{2}$x+y=π2 and $ax-by=0$ax−by=0 and solve these equations simultaneously. We have
| $x+y$x+y | $=$= | $\frac{\pi}{2}\ \ (1)$π2 (1) |
| and $ax-by$ax−by | $=$= | $0\ \ \ (2)$0 (2) |
| From $(1)$(1), $bx+by$bx+by | $=$= | $\frac{b\pi}{2}\ (3)$bπ2 (3) |
| Adding $(2)$(2) and $(3)$(3), $x\left(a+b\right)$x(a+b) | $=$= | $\frac{b\pi}{2}$bπ2 |
| $\therefore x$∴x | $=$= | $\frac{\pi}{2}\frac{b}{\left(a+b\right)}$π2b(a+b) |
| Substituting for $x$x in $(1)$(1), $y$y | $=$= | $\frac{\pi}{2}\frac{a}{\left(a+b\right)}$π2a(a+b) |
Worked Examples
Question 1
What is:
The complement of $23$23°
The complement of $80$80°
Half the complement of $46$46°
The complement of $90$90°
Question 2
Find the supplement of $\frac{\pi}{11}$π11 radians. You do not need to include the radian symbol in your answer.
Question 3
What is the smaller angle formed by the hour and minute hands of the clock? State your answer in degrees, not radians.
