Trigonometric Ratios of Complementary Angles

The trigonometric ratios of complementary angles refer to the relationships between trigonometric functions (sine, cosine, tangent, etc.) of two angles that add up to 90 degrees (a right angle). Understanding these relationships is crucial for simplifying trigonometric expressions and solving various problems in trigonometry and geometry.

Formulae

Let $A$ be an angle. Then the complement of $A$ is $90^\circ – A$. The key relationships are:

$\sin(90^\circ – A) = \cos A$
$\cos(90^\circ – A) = \sin A$
$\tan(90^\circ – A) = \cot A$
$\cot(90^\circ – A) = \tan A$
$\sec(90^\circ – A) = \csc A$
$\csc(90^\circ – A) = \sec A$

Examples

Example-1: Simplify $\sin(60^\circ)$.

We can rewrite $60^\circ$ as $90^\circ – 30^\circ$. Therefore:

$\sin(60^\circ) = \sin(90^\circ – 30^\circ) = \cos(30^\circ) = \frac{\sqrt{3}}{2}$

Example-2: Find the value of $x$ if $\cos(2x + 10^\circ) = \sin(3x – 20^\circ)$.

Using the complementary angle identity, we know that $\cos(A) = \sin(90^\circ – A)$. Therefore:

$\sin(90^\circ – (2x + 10^\circ)) = \sin(3x – 20^\circ)$

Since the sines are equal, their arguments must be equal (or differ by multiples of 360 degrees, which we can consider for this problem):

$90^\circ – (2x + 10^\circ) = 3x – 20^\circ$

$80^\circ – 2x = 3x – 20^\circ$

$100^\circ = 5x$

$x = 20^\circ$

Theorem with Proof

Theorem: In a right-angled triangle, the sine of an acute angle is equal to the cosine of its complementary angle.

Proof:

Consider a right-angled triangle $ABC$, where $\angle C = 90^\circ$. Let $\angle A = A$. Then, $\angle B = 90^\circ – A$ (since the sum of angles in a triangle is $180^\circ$).

Let $a$ be the side opposite angle $A$, $b$ be the side adjacent to angle $A$, and $c$ be the hypotenuse.

By definition:

$\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c}$
$\cos B = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{c}$ (since the side adjacent to angle $B$ is $a$)

Therefore, $\sin A = \cos B$. Since $B = 90^\circ – A$, we have $\sin A = \cos(90^\circ – A)$. This proves the theorem.

Common mistakes by students

Incorrectly identifying the complementary angle: Students often struggle to identify which angle is the complement of the given angle. Remember that the two angles must add up to 90 degrees.
Mixing up the trigonometric functions: Confusing sine with cosine, tangent with cotangent, and secant with cosecant. Practice is key!
Applying the rules incorrectly: For instance, writing $\sin(90^\circ – A) = -\sin A$ instead of $\sin(90^\circ – A) = \cos A$.

Real Life Application

These trigonometric identities are fundamental in various fields:

Navigation: Used in calculating distances and bearings.
Engineering: Applied in structural analysis and design.
Physics: Used in calculations involving angles and forces (e.g., projectile motion).
Surveying: Used for determining heights, distances, and angles of objects.

Fun Fact

The relationship between sine and cosine (and other cofunctions) can be visualized using the unit circle. As you move around the circle, the x and y coordinates (which represent cosine and sine, respectively) switch roles when you move 90 degrees.