Trigonometry of Complementary Angles

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of a triangle. One of the fundamental concepts in trigonometry is the trigonometric ratios of complementary angles. This concept is crucial in solving problems related to angles in triangles, physics, and engineering applications. Understanding how trigonometric ratios behave with complementary angles helps simplify expressions and solve equations efficiently.

What are Ratios of Complementary Angles?

Two angles are said to be complementary if their sum is 90°. In mathematical terms: θ + (90° − θ) = 90°. This relationship helps transform trigonometric expressions and solve problems involving right-angled triangles.

Trigonometric Ratios of Complementary Angles Formula

For any acute angle θ, the six trigonometric ratios are as follows:

• sin(90° − θ) = cos θ
• cos(90° − θ) = sin θ
• tan(90° − θ) = cot θ
• cot(90° − θ) = tan θ
• sec(90° − θ) = cosec θ
• cosec(90° − θ) = sec θ

These formulas show that the trigonometric ratios of an angle can be expressed in terms of the ratios of its complementary angle.

Relation Between Trigonometric Ratios of Complementary Angles

The complementary angle relationships between trigonometric ratios can be explained using a right-angled triangle. Consider a right triangle ABC with:

• Angle B = θ
• Angle A = 90° − θ
• Hypotenuse = c
• Opposite side to θ = b
• Adjacent side to θ = a

Using the definitions of trigonometric ratios:

• sin θ = AC / AB
• cos θ = BC / AB
• tan θ = AC / BC
• cot θ = BC / AC
• sec θ = AB / BC
• cosec θ = AB / AC

Now, considering the complementary angle (90° − θ):

• sin(90° − θ) = cos θ = BC / AB
• cos(90° − θ) = sin θ = AC / AB
• tan(90° − θ) = cot θ = BC / AC
• cot(90° − θ) = tan θ = AC / BC
• sec(90° − θ) = cosec θ = AB / AC
• cosec(90° − θ) = sec θ = AB / BC

This establishes a fundamental connection between trigonometric ratios of complementary angles.

Complementary Angles Corollary

A corollary is a direct consequence of a theorem or principle. From the above derivations, we can establish:

• The sine of one angle equals the cosine of its complement.
• The tangent of one angle equals the cotangent of its complement.
• The secant of one angle equals the cosecant of its complement.
• These identities are valid only for acute angles in a right triangle.

Some Solved Examples Related to Trigonometry of Complementary Angles

Q1. Find the value of sin 37° × cos 53° + cos 37° × sin 53°.

Solution:
sin 53° = cos(90° − 53°) = cos 37°
cos 53° = sin(90° − 53°) = sin 37°
So, sin² 37° + cos² 37° = 1

Q2. Evaluate:

(cos 58° / sin 32°) + (sin 22° / cos 68°) − (cos 38° × cosec 52°) / (tan 18° × tan 35° × tan 72° × tan 55°)

Solution:
cos 58° = sin(90° − 58°) = sin 32°
sin 22° = cos(90° − 22°) = cos 68°
So the first two terms = 1 + 1 = 2

cos 38° × cosec 52° = cos 38° / sin 52° = cos 38° / cos(90° − 52°) = 1
Final answer = 1 + 1 − 1 = 1

Q3. Find the value of the following expression:

tan 1° × tan 2° × tan 3° … tan 89°

Solution:
P = tan 1° × tan 2° × tan 3° … tan 89°
Pairing: tan k° × tan(90° − k°) = 1
There are 44 such pairs and the unpaired middle term is tan 45° = 1
So, final product = 1

Q4. If tan 2A = cot(A − 21°), where 2A is an acute angle, find A.

Solution:
tan 2A = tan(90° − (A − 21°)) = tan(111° − A)
⇒ 2A = 111° − A
⇒ A = 37°

Real-Life Applications of Trigonometry of Complementary Angles

• Architecture & Construction: Designing slopes and inclinations of buildings and bridges.
• Aviation: Calculating angles of ascent and descent.
• Surveying: Measuring land elevations and distances.
• Navigation & GPS: Determining direction and location.
• Optics & Light Reflection: Controlling light paths in mirrors and lenses.
• Robotics & Machine Vision: Enabling precise movement and object detection.
• Sports & Biomechanics: Analyzing body posture in high jumps, diving, etc.

Conclusion

The concept of trigonometric ratios of complementary angles is fundamental in trigonometry. Understanding these relationships helps simplify calculations and solve complex trigonometric problems efficiently. By using the formulas and derivations discussed, we can easily relate different trigonometric functions and apply them in various real-world applications like physics, engineering, and geometry.