Mathematics

Trigonometric Identities

82 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 does the symbol (\tan) represent?

  1. Sine

  2. Cosine

  3. Tangent

  4. Cosecant
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

(\tan) is a mathematical symbol that represents the tangent of an angle. It is used to find the ratio of the length of the opposite side to the length of the adjacent side of a right triangle.

Multiple choice

Which of the following is an example of a series expansion used by Madhava of Sangamagrama for the arctangent function?

  1. $$\tan^{-1} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$
  2. $$\tan^{-1} x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
  3. $$\tan^{-1} x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \cdots$$

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Madhava of Sangamagrama developed a series expansion for the arctangent function, which is given by $$\tan^{-1} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$. This series converges for (|x| \leq 1).

Multiple choice

Which of the following is an example of a series expansion used by Madhava of Sangamagrama for the tangent function?

  1. $$\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \cdots$$
  2. $$\tan x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
  3. $$\tan x = 1 + \frac{x^2}{2} + \frac{x^4}{4} + \frac{x^6}{6} + \cdots$$

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Madhava of Sangamagrama developed a series expansion for the tangent function, which is given by $$\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \cdots$$. This series converges for (|x| \leq \frac{\pi}{4}).

Multiple choice

What is the value of the expression tan(π/4)?

  1. 0

  2. 1

  3. √2

  4. 2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The expression tan(π/4) means the tangent of the angle π/4. Since the tangent of an angle is the ratio of the opposite side to the adjacent side of a right triangle, and the opposite side and the adjacent side of a right triangle with an angle of π/4 are equal, the value of the expression is 1.

Multiple choice

What is the value of the expression tan(45°)?

  1. 1

  2. √2

  3. √3

  4. 2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The value of the expression tan(45°) is equal to 1.

Multiple choice

The equation (\frac{d}{dx} \tan x = \sec^2 x) is known as:

  1. Chain rule

  2. Product rule

  3. Quotient rule

  4. Trigonometric rule
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The equation (\frac{d}{dx} \tan x = \sec^2 x) is known as the trigonometric rule, which is used to find the derivative of a trigonometric function.

Multiple choice

The equation (\frac{d}{dx} \cot x = -\csc^2 x) is known as:

  1. Chain rule

  2. Product rule

  3. Quotient rule

  4. Trigonometric rule
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The equation (\frac{d}{dx} \cot x = -\csc^2 x) is known as the trigonometric rule, which is used to find the derivative of a trigonometric function.

Multiple choice

The formula $\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ is known as the:

  1. Pythagorean identity

  2. Euler's formula

  3. Sum-to-product formula

  4. Product-to-sum formula
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The formula $\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ is a sum-to-product formula in trigonometry, which expresses the tangent of the sum of two angles in terms of the tangents of the individual angles.

Multiple choice

What is the Pythagorean identity that relates the sine, cosine, and tangent functions?

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

The Pythagorean identity $$sin^2(x) + cos^2(x) = 1$$ establishes a fundamental relationship between the sine and cosine functions, stating that the sum of the squares of the sine and cosine of an angle is always equal to 1.

Multiple choice

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

  1. 0

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

In Indian trigonometry, the value of (\tan 45^\circ) is 1.

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).