Trigonometric Ratios of Complementary Angles

Trigonometry ratios of complementary angles are the ratios related to trigonometry when their angles are complementary. Complementary angles are angles when the sum of angles is 90 degrees. Suppose we have two angles ‘a’ and ‘b’ that are complementary, then if sin a = x then sin b = y now cos a = y and cos b = x.

In this article, we will learn about Trigonometric Ratios, Complementary Angles, Trigonometric Ratios of Complementary Angles, Examples, and others in detail.

What are Trigonometric Ratios?

Trigonometric ratios are essential mathematical functions that describe the relationships between the angles and sides of a right-angled triangle. These ratios are derived from the lengths of the sides of a triangle and the measurements of its angles, especially in the context of a right triangle, which has one angle measuring 90 degrees.

In the right triangle, there are three sides:

• Hypotenuse
• Side Adjacent to Specific Angle
• Side Opposite that Angle

The relationships between these sides and angles form the basis of trigonometric ratios. The ratios are derived by comparing the lengths of these sides and involve fundamental functions like sine, cosine, and tangent.

• Sine (sin): Sine of an angle is defined by the ratio of the length of the side opposite an acute angle to the length of the hypotenuse.

• Cosine (cos): Cosine of an angle is defined by the ratio of the length of the side adjacent to an acute angle to the length of the hypotenuse.

• Tangent (tan): Tangent of an angle is defined by the ratio of the length of the side opposite an acute angle to the length of the side adjacent to that angle.

• Cosecant (cosec): Cosecant of an angle is defined by the ratio of the length of the hypotenuse and the side opposite the angle.

• Secant (sec): Secant of an angle is defined by the ratio of the length of the hypotenuse and the side and the side adjacent to the angle.

• Cotangent (cot): Cotangent of an angle is defined by the ratio of the length of sides that are adjacent to the angle and the side that is opposite to the angle.

What are Complementary Angles?

Complementary angles are pairs of angles that, when added together, result in a sum of 90 degrees. In essence, they complement each other to form a right angle. The definition is fundamental in geometry, providing a basis for understanding the relationships between angles within geometric figures.

∠x + ∠y = 90

For instance, if one angle measures 30 degrees, its complementary angle would measure 60 degrees, as their sum equals 90 degrees. This concept is pivotal in solving various problems involving angles, aiding in the analysis of geometric shapes and measurements.

Types of Complementary Angles

There are two types of complementary angles that are,

• Adjacent Complementary Angles
• Non- adjacent Complementary Angles

Adjacent Complementary Angles

“Adjacent Complementary Angles” refers to a pair of angles that share a vertex and arm.

Non- adjacent Complementary Angles:

Non-adjacent complementary angles are a pair of two complementary angles that are not next to one another.

Trigonometric Ratios of Complementary Angles Formula

In a right-angled triangle, if two angles are complementary (their sum is 90 degrees), the trigonometric ratios of one angle are directly related to the complementary angle’s ratios.

• sin A = BC/AC

• cos A = AB/AC

• tan A = BC/AB

• cosec A = 1/sin A = AC/BC

• sec A = 1/cos A = AC/AB

• cot A = 1/tan A = AB/BC

Trigonometric Ratios of Complementary Angles Derivation

The trigonometric ratios of the complement of angle ∠A (90° – A) in the same triangle are:

• sin (90° – A) = AB/AC

• cos (90° – A) = BC/AC

• tan (90° – A) = AB/BC

• cosec (90° – A) = 1/sin (90° – A) = AC/AB

• sec (90° – A) = 1/cos (90° – A) = AC/BC

• cot (90° – A) = 1/tan (90° – A) = BC/AB

These trigonometric ratios when compared to the ratios of angle ∠A exhibit some intriguing relationships:

• sin (90° – A) = cos A ⇔ cos (90° – A) = sin A

• tan (90° – A) = cot A ⇔ cot (90° – A) = tan A

• sec (90° – A) = cosec A ⇔ cosec (90° – A) = sec A

These relations hold true for all values of A lying between 0° and 90°.

Pythagorean Relationship among Trigonometric Ratios

In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Thus, for any angle A within a right triangle:

sin² A + cos² A = 1

This forms the foundation for many trigonometric identities and relationships.

Learn more about, Pythagoras Theorem

Reciprocal and Quotient Identities

Trigonometric ratios have reciprocal relationships, where the reciprocal of one ratio is equal to another ratio involving the same angle. For instance:

• sin A = 1/cosec A

• cos A = 1/sec A

• tan A = 1/cot A

Additionally, there are quotient identities derived from these ratios:

• tan A = sin A / cos A

• cot A = cos A / sin A

Trigonometric Ratios of Complementary Angles – Conclusion

In conclusion, trigonometric ratios are fundamental tools in mathematics, providing essential relationships between angles and sides within right triangles. Their practical applications extend across numerous fields, emphasizing their significance in problem-solving and design where spatial relationships are critical.

Related Resources,

• Trigonometric Identities
• Sin and Cos Formulas
• Trigonometry Table

Trigonometric Ratios of Complementary Angles – Examples

Some examples of complementary angles in trigonometric ratio are,

Example 1: Given, cos θ = 4/5 and θ is an acute angle. Find the value of sin (90° – θ).

Solution:

Using trigonometric relation: sin² θ + cos² θ = 1

Given, cos θ = 4/5

sin² θ = 1 – cos² θ

= 1 – (4/5)²

= 1 – 16/25 = 9/25

sin θ = ±3/5 (as θ is acute, sin θ = 3/5)

Now, sin (90° – θ) = cos θ = 4/5

Therefore, sin (90° – θ) = 4/5.

Example 2: If cot A = 5/12, find sec (90° – A).

Solution:

Given, cot A = 5/12

Since cot A = 1/tan A, we can find tan A = 12/5

Using the trigonometric identity: 1 + tan² A = sec² A

1 + (12/5)² = sec² A

1 + 144/25 = sec² A

169/25 = sec² A

sec A = ±13/5

Now, sec (90° – A) = sec A = 13/5.

Example 3: If sin α = 7/25, find cos (90° – α).

Solution:

Given, sin α = 7/25

Using the trigonometric identity: sin² α + cos² α = 1

cos² α = 1 – sin² α

cos² α = 1 – (7/25)²

cos² α = 1 – 49/625

cos² α = 576/625

cos α = ±24/25

Now, cos (90° – α) = sin α = 7/25.

Example 4: If sec β = 29/21, find tan (90° – β).

Solution:

Given, sec β = 29/21

Since sec β = 1/cos β, we can find cos β = 21/29

Using the trigonometric identity: tan² β + 1 = sec² β

tan² β + 1 = (29/21)²

tan² β = (29/21)² – 1

tan² β = 841/441 – 1

tan² β = 2

tan β = √2

Now, tan (90° – β) = cot β = 1/√2.

Example 5: If tan θ = 3/4, find cosec (90° – θ).

Solution:

Given, tan θ = 3/4

Since tan θ = 1/cot θ, we can find cot θ = 4/3

Using the trigonometric identity: cosec² θ = 1 + cot² θ

cosec² θ = 1 + (4/3)²

= 1 + 16/9 = 25/9

cosec θ = ±5/3

Now, cosec (90° – θ) = cosec θ = 5/3.

FAQs on Trigonometric Ratios of Complementary Angles

What are Complementary Angles in Trigonometric Ratios?

Complementary angles are the set of two angles where their sum is equal to 90°. For example, 40° and 50° are complementary to each and their sum is equal to 90°.

What is Significance of Trigonometric Ratios?

Trigonometric ratios play a crucial role in various fields such as physics, engineering, architecture, and astronomy. They help solve problems related to distances, angles, and forces.

What are Complementary Angles?

Complementary angles are the pair of angles whose sum is equal to 90 degrees.

What is Complementary Angle Theorem?

Complementary Angles theorem states that, “If two angles are complements of the same angle (or congruent angles), then the two angles are congruent in itself.”