Mathematics

Trigonometric Identities

87 Questions

Trigonometric identities focus on solving equations using tangent, sine, and cosine properties. Questions involve half angle formulas and slope calculations. This topic is a staple in the quantitative aptitude section of major competitive examinations.

Tangent propertiesAngle formulasTrigonometric equationsHalf angle identitiesSlope calculations

Trigonometric Identities Questions

Multiple choice

What is the value of (tan 45^\circ)?

  1. \(1\)
  2. \(\sqrt{2}\)
  3. \(\sqrt{3}\)
  4. \(2\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We know that (tan 45^\circ = 1).

Multiple choice

What is the double angle formula for tangent?

  1. \(tan 2A = \frac{2 tan A}{1 - tan^2 A}\)
  2. \(tan 2A = tan A + tan A\)
  3. \(tan 2A = cos A + cos A\)
  4. \(tan 2A = 2 tan A sin A\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The double angle formula for tangent states that (tan 2A = \frac{2 tan A}{1 - tan^2 A}).

Multiple choice

What is the half angle formula for tangent?

  1. \(tan \frac{A}{2} = \pm \sqrt{\frac{1 - cos A}{1 + cos A}}\)
  2. \(tan \frac{A}{2} = tan A + tan A\)
  3. \(tan \frac{A}{2} = cos A + cos A\)
  4. \(tan \frac{A}{2} = 2 tan A sin A\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The half angle formula for tangent states that (tan \frac{A}{2} = \pm \sqrt{\frac{1 - cos A}{1 + cos A}}).

Multiple choice

Which of the following is an identity?

  1. sin^2(x) + cos^2(x) = 1

  2. sin(x) + cos(x) = 1

  3. sin(x) - cos(x) = 1

  4. sin(x) / cos(x) = tan(x)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The identity sin^2(x) + cos^2(x) = 1 is known as the Pythagorean identity. It is true for all values of x.

Multiple choice

Which of the following is the quotient identity for tangent?

  1. tan(x) = sin(x) / cos(x)

  2. tan(x) = cos(x) / sin(x)

  3. tan(x) = sin(x) + cos(x)

  4. tan(x) = cos(x) - sin(x)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The quotient identity for tangent is tan(x) = sin(x) / cos(x).

Multiple choice

What is the sum-to-product identity for tangent?

  1. tan(x) + tan(y) = (sin(x+y))/(cos(x)cos(y))

  2. tan(x) + tan(y) = (sin(x-y))/(cos(x)cos(y))

  3. tan(x) + tan(y) = (cos(x+y))/(sin(x)sin(y))

  4. tan(x) + tan(y) = (cos(x-y))/(sin(x)sin(y))
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sum-to-product identity for tangent is tan(x) + tan(y) = (sin(x+y))/(cos(x)cos(y)).

Multiple choice

Which of the following is the difference-to-product identity for tangent?

  1. tan(x) - tan(y) = (sin(x-y))/(cos(x)cos(y))

  2. tan(x) - tan(y) = (sin(x+y))/(cos(x)cos(y))

  3. tan(x) - tan(y) = (cos(x-y))/(sin(x)sin(y))

  4. tan(x) - tan(y) = (cos(x+y))/(sin(x)sin(y))
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The difference-to-product identity for tangent is tan(x) - tan(y) = (sin(x-y))/(cos(x)cos(y)).

Multiple choice

What is the half-angle identity for tangent?

  1. tan(x/2) = (1 - cos(x))/(1 + cos(x))

  2. tan(x/2) = (1 + cos(x))/(1 - cos(x))

  3. tan(x/2) = (sin(x))/(cos(x))

  4. tan(x/2) = (cos(x))/(sin(x))
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The half-angle identity for tangent is tan(x/2) = (1 - cos(x))/(1 + cos(x)).

Multiple choice

Which of the following is the double-angle identity for tangent?

  1. tan(2x) = (2tan(x))/(1 - tan^2(x))

  2. tan(2x) = (2tan(x))/(1 + tan^2(x))

  3. tan(2x) = (tan(x))/(1 - tan^2(x))

  4. tan(2x) = (tan(x))/(1 + tan^2(x))
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The double-angle identity for tangent is tan(2x) = (2tan(x))/(1 - tan^2(x)).

Multiple choice

Brahmagupta's formula for calculating the tangent of an angle is given by:

  1. $\tan A = \frac{\sin A}{\cos A}$
  2. $\tan A = \frac{\cos A}{\sin A}$
  3. $\tan A = \frac{\sin A}{\sec A}$
  4. $\tan A = \frac{\cos A}{\csc A}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Brahmagupta's formula for tangent is derived from the definition of tangent.

Multiple choice

Brahmagupta's formula for calculating the tangent of the sum of two angles is given by:

  1. $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
  2. $\tan(A + B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$
  3. $\tan(A + B) = \frac{\sin A + \sin B}{\cos A + \cos B}$
  4. $\tan(A + B) = \frac{\cos A + \cos B}{\sin A + \sin B}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Brahmagupta's formula for the tangent of the sum of two angles is derived from the angle addition formula for tangent.

Multiple choice

Brahmagupta's formula for calculating the tangent of the difference of two angles is given by:

  1. $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$
  2. $\tan(A - B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
  3. $\tan(A - B) = \frac{\sin A - \sin B}{\cos A - \cos B}$
  4. $\tan(A - B) = \frac{\cos A - \cos B}{\sin A - \sin B}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Brahmagupta's formula for the tangent of the difference of two angles is derived from the angle difference formula for tangent.