Mathematics

Trigonometric Identities and Equations

223 Questions

Solve a variety of questions based on trigonometric identities and mathematical equations. Key areas covered include double angle formulas, the law of sines, and quadrant angles. This material helps students preparing for advanced mathematics exams, UPSC, and state public service commissions.

Double angle formulasLaw of sinesTrigonometric quadrantsLaplace transformSecant functionTaylor series

Trigonometric Identities and Equations Questions

Multiple choice

Find the value of $x$ in the equation $\sin x = \frac{1}{2}$.

  1. $30\degree$
  2. $45\degree$
  3. $60\degree$
  4. $90\degree$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the inverse sine function, we get $x = \sin^{-1}\left(\frac{1}{2}\right) = 30\degree$.

Multiple choice

What is the double-angle identity for cosine?

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

  2. cos(2x) = 2cos^2(x) - 1

  3. cos(2x) = 1 - 2sin^2(x)

  4. cos(2x) = 2cos(x) - 1
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The double-angle identity for cosine is cos(2x) = cos^2(x) - sin^2(x).

Multiple choice

Which of the following is the half-angle identity for sine?

  1. sin(x/2) = sqrt((1 - cos(x))/2)

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

  3. sin(x/2) = (1 - cos(x))/2

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

The half-angle identity for sine is sin(x/2) = sqrt((1 - cos(x))/2).

Multiple choice

What is the sum-to-product identity for sine and cosine?

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

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

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

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

The sum-to-product identity for sine and cosine is sin(x) + cos(x) = 2sin((x+y)/2)cos((x-y)/2).

Multiple choice

Which of the following is the difference-to-product identity for sine and cosine?

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

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

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

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

The difference-to-product identity for sine and cosine is sin(x) - cos(x) = 2sin((x-y)/2)cos((x+y)/2).

Multiple choice

What is the power-reducing identity for sine?

  1. sin^2(x) = (1 - cos(2x))/2

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

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

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

The power-reducing identity for sine is sin^2(x) = (1 - cos(2x))/2.

Multiple choice

Which of the following is the power-reducing identity for cosine?

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

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

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

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

The power-reducing identity for cosine is cos^2(x) = (1 + cos(2x))/2.

Multiple choice

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

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

Brahmagupta's formula for cosine is derived from the half-angle formula for cosine.

Multiple choice

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

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

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

Multiple choice

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

  1. $\sin(A + B) = \sin A \cos B + \cos A \sin B$
  2. $\sin(A + B) = \sin A \sin B + \cos A \cos B$
  3. $\sin(A + B) = \tan A \sec B + \cot A \csc B$
  4. $\sin(A + B) = \cos A \sec B + \sin A \csc B$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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

Multiple choice

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

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

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

Multiple choice

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

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

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

Multiple choice

What is the value of the sine of 30 degrees according to Aryabhata?

  1. 0.5

  2. 0.866

  3. 1

  4. 1.5
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Aryabhata calculated the sine of 30 degrees to be 0.5, which is an accurate approximation.