Trigonometric Ratios of Complementary Angles

Complementary angles are a pair of angles whose sum is 90°. In simple terms, they “complete” each other to form a right angle.

∠x + ∠y = 90

In trigonometry, complementary angles show special relationships between the trigonometric ratios. These relationships allow one trigonometric function to be expressed in terms of another.

For an angle θ, the following identities hold:

In Simple Words

• Sine ↔ Cosine interchange
• Tangent ↔ Cotangent interchange
• Secant ↔ Cosecant interchange

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(H)
• Side Adjacent to Specific Angle or Base(b)
• Side Opposite that Angle or perpendicular(p)

The relationships between these sides and angles form the basis of trigonometric 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

For instance, if one angle measures 30 degrees, its complementary angle would measure 60 degrees, as their sum equals 90 degrees.

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 for all values of A lying between 0° and 90°.

Related Articles:

• Trigonometric Identities
• Sin and Cos Formulas
• Trigonometry Table
• Pythagoras Theorem

Solved Examples

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

Using trigonometric relation: sin² θ + cos² θ = 1

Given, cos θ = 4/5

sin² θ = 1 - cos² θ

⇒ sin² θ = 1 - (4/5)²

⇒ sin² θ = 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).

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).

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).

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).

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.