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Which set of Angles has the same Trigonometric Ratio? 

Explore the set of angles where trigonometric ratios coincide for a deeper understanding of mathematical relationships.

by Maivizhi A

Updated Mar 06, 2024

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Which set of Angles has the same Trigonometric Ratio?

Which Set of Angles has the same Trigonometric Ratio? 

A. sin 45° and tan 45°

B. sin 30° and cos 60°

C. cos 30° and tan 45°

D. tan 60° and sin 45°

Answer: 

The only option where the trigonometric ratios have the same value is B. sin 30° and cos 60°.

The trigonometric ratios (sine, cosine, tangent) of an angle relate the lengths of the sides in a right triangle.

However, these ratios have different relationships with each other and don't necessarily have the same values for different angles.

Here's a breakdown of the options:

A. sin 45° and tan 45°

  • sin 45° = √2 / 2 (opposite side / hypotenuse)
  • tan 45° = 1 (opposite side / adjacent side)

While both involve the opposite side in a 45-45-90 triangle, their values and relationships with other sides are different. Not the same.

B. sin 30° and cos 60°

  • sin 30° = 1 / 2 (opposite side / hypotenuse)
  • cos 60° = 1 / 2 (adjacent side / hypotenuse)

These two ratios are the same. They represent different sides in different triangles (30-60-90 and 30-60-90), but their values coincide.

C. cos 30° and tan 45°

  • cos 30° = √3 / 2 (adjacent side / hypotenuse)
  • tan 45° = 1 (opposite side / adjacent side)

These ratios do not have the same value and have different relationships with the sides in their respective triangles.

D. tan 60° and sin 45°

  • tan 60° = √3 (opposite side / adjacent side)
  • sin 45° = √2 / 2 (opposite side / hypotenuse)

These ratios do not have the same value and have different relationships with the sides in their respective triangles.

Therefore, the only option where the trigonometric ratios have the same value is B. sin 30° and cos 60°. 

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What are Trigonometric Ratios?  

Trigonometric ratios are mathematical relationships that exist between the angles and sides of a right triangle. In a right triangle, which is a triangle with one angle measuring 90 degrees, there are three primary trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). These ratios are defined as follows:

  1. Sine (sin): The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse (the longest side of the triangle). Mathematically, it can be expressed as: sin(theta) = (Opposite side) / (Hypotenuse)
  2. Cosine (cos): The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side (the side adjacent to the angle, but not the hypotenuse) to the length of the hypotenuse. It can be expressed as: cos(theta) = (Adjacent side) / (Hypotenuse)
  3. Tangent (tan): The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side. It can be expressed as: tan(theta) = (Opposite side) / (Adjacent side)

These ratios are fundamental in trigonometry and find extensive applications in various fields such as engineering, physics, astronomy, and navigation. Additionally, reciprocal trigonometric ratios such as cosecant (csc), secant (sec), and cotangent (cot) can also be defined, which are the reciprocals of sine, cosine, and tangent respectively.

Which Set of Angles has the same Trigonometric Ratio - FAQs

1. What are Trigonometric Ratios?

Trigonometric ratios are mathematical relationships that exist between the angles and sides of a right triangle. They include sine (sin), cosine (cos), and tangent (tan), defined based on the lengths of the sides relative to the angles.

2. What is the Definition of Sine (sin)?

Sine (sin) of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

3. What is the Definition of Cosine (cos)?

Cosine (cos) of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse.

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