Trigonometric Ratios of Complementary Angles Explained
Trigonometric Ratios of Complementary Angles describe the relationship between two angles whose sum is 90°. The sine of one angle equals the cosine of its complement, while tangent and cotangent, and secant and cosecant, form reciprocal pairs. This page covers the identities, formulas, examples and shortcuts used to solve aptitude questions.
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What Are Trigonometric Ratios of Complementary Angles?
The identities are 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, and cosec(90° − θ) = sec θ. In a right triangle, the two acute angles are complementary, so the opposite side for one angle becomes the adjacent side for the other.
Trigonometric Ratios of Complementary Angles Video Lessons
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Complementary Angle Trigonometric Identities
Learn complementary-angle identities such as sin θ = cos(90° − θ), and understand how related trigonometric ratios transform when angles add up to 90 degrees.
Trigonometric Ratios of Complementary Angles Formula & Tricks
Important Formulas
Sine and cosine interchange when an angle is replaced by its complement.
Tangent and cotangent interchange for complementary angles.
Secant and cosecant interchange for complementary angles.
The complement is found by subtracting the given angle from 90°.
Quick Tricks
When 90° − θ appears inside a ratio, replace the ratio with its paired cofunction: sin with cos, tan with cot, and sec with cosec.
Before applying an identity, confirm that the two angles add to 90°. The rule does not apply to angles whose sum is not 90°.
The pairs tan-cot, sec-cosec are reciprocal ratios, but the complementary-angle identity changes the function directly. Do not write tan(90° − θ) as 1/tan(90° − θ); write it as cot θ.
Trigonometric Ratios of Complementary Angles Concepts
Complementary-Angle Cofunction Identities
For any angle θ for which the ratios are defined, sin(90° − θ) = cos θ and cos(90° − θ) = sin θ. Similarly, tan(90° − θ) = cot θ, cot(90° − θ) = tan θ, sec(90° − θ) = cosec θ, and cosec(90° − θ) = sec θ.
Applying the Identity to Numerical Angles
For example, 58° = 90° − 32°. Therefore, sin 58° = sin(90° − 32°) = cos 32°, and cot 58° = cot(90° − 32°) = tan 32°.
Using Complementary Ratios in a Right Triangle
Let the acute angles be A and B, with A + B = 90°. If the sides opposite A, opposite B and the hypotenuse are respectively a, b and c, then sin A = a/c and cos B = b/c. Since the side opposite A is the side adjacent to B, sin A = cos B.
Simplifying Expressions with Complementary Angles
The substitution can turn an expression into a standard identity. For instance, sin(90° − θ) + cos θ becomes cos θ + cos θ = 2 cos θ. The same replacement applies to reciprocal ratios such as sec and cosec.
Trigonometric Ratios of Complementary Angles Visual Guide & Data
Use the table and chart below to understand the data better.
Cofunction Identity Table
Each trigonometric ratio of 90° − θ is paired with a cofunction of θ.
| Ratio involving the complement | Equivalent ratio |
|---|---|
| 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 |
Note: The angle θ is generally an acute angle when these identities are used in right-triangle problems.
Complementary Acute Angles in a Right Triangle
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Trigonometric Ratios of Complementary Angles: Revision Points
Use these identities and rules for quick simplification and calculation.
- Complement of θ is 90° − θ.
- sin(90° − θ) = cos θ and cos(90° − θ) = sin θ.
- tan(90° − θ) = cot θ and cot(90° − θ) = tan θ.
- sec(90° − θ) = cosec θ and cosec(90° − θ) = sec θ.
- The two acute angles in a right triangle are complementary.
- Confirm that the angles add to 90° before applying a complementary-angle identity.
- For a numerical angle α, write α as 90° − (90° − α) to identify its cofunction.
Trigonometric Ratios of Complementary Angles FAQs
What is the formula for sin(90° − θ)?
sin(90° − θ) = cos θ. For example, sin 70° = cos 20°.
What is the complement of 37°?
The complement is 90° − 37° = 53°.
How are tan and cot related for complementary angles?
tan(90° − θ) = cot θ and cot(90° − θ) = tan θ. Thus, tan 65° = cot 25°.
If sin θ = 3/5, what is cos(90° − θ)?
cos(90° − θ) = sin θ = 3/5.
Are 35° and 55° complementary angles?
Yes. Their sum is 35° + 55° = 90°.
What is sec(90° − θ) equal to?
sec(90° − θ) = cosec θ. For example, sec 60° = cosec 30° = 2.