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

What is the addition formula for sine?

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

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

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

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

The addition formula for sine states that the sine of the sum of two angles is equal to the sine of the first angle multiplied by the cosine of the second angle plus the cosine of the first angle multiplied by the sine of the second angle.

Multiple choice

What is the addition formula for cosine?

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

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

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

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

The addition formula for cosine states that the cosine of the sum of two angles is equal to the cosine of the first angle multiplied by the cosine of the second angle minus the sine of the first angle multiplied by the sine of the second angle.

Multiple choice

What is the subtraction formula for sine?

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

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

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

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

The subtraction formula for sine states that the sine of the difference of two angles is equal to the sine of the first angle multiplied by the cosine of the second angle minus the cosine of the first angle multiplied by the sine of the second angle.

Multiple choice

What is the subtraction formula for cosine?

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

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

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

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

The subtraction formula for cosine states that the cosine of the difference of two angles is equal to the cosine of the first angle multiplied by the cosine of the second angle plus the sine of the first angle multiplied by the sine of the second angle.

Multiple choice

What is the double-angle formula for sine?

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

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

  3. sin(2x) = sin(x) - sin(x)

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

The double-angle formula for sine states that the sine of twice an angle is equal to twice the sine of the angle multiplied by the cosine of the angle.

Multiple choice

What is the double-angle formula 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 formula for cosine states that the cosine of twice an angle is equal to the square of the cosine of the angle minus the square of the sine of the angle.

Multiple choice

What was Brahmagupta's formula for the sine of an angle?

  1. $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
  2. $\sin \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
  3. $\sin \theta = \frac{\text{opposite}}{\text{adjacent}}$
  4. $\sin \theta = \frac{\text{hypotenuse}}{\text{adjacent}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Brahmagupta's formula for the sine of an angle is $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$.

Multiple choice

What was Brahmagupta's formula for the cosine of an angle?

  1. $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
  2. $\cos \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
  3. $\cos \theta = \frac{\text{opposite}}{\text{adjacent}}$
  4. $\cos \theta = \frac{\text{hypotenuse}}{\text{adjacent}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Brahmagupta's formula for the cosine of an angle is $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$.

Multiple choice

What is the general formula for the sine series?

  1. $sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$
  2. $sin(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \cdots$
  3. $sin(x) = x - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$
  4. $sin(x) = x + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \cdots$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The general formula for the sine series is $sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$.

Multiple choice

What is the general formula for the cosine series?

  1. $cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$
  2. $cos(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \cdots$
  3. $cos(x) = 1 - \frac{x}{2} + \frac{x^2}{4} - \frac{x^3}{6} + \cdots$
  4. $cos(x) = 1 + \frac{x}{2} + \frac{x^2}{4} + \frac{x^3}{6} + \cdots$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The general formula for the cosine series is $cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$.

Multiple choice

What is the value of $sin(\frac{\pi}{2})$ using the sine series?

  1. 0

  2. 1

  3. -1

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

Using the sine series, we have $sin(\frac{\pi}{2}) = \frac{\pi}{2} - \frac{(\frac{\pi}{2})^3}{3!} + \frac{(\frac{\pi}{2})^5}{5!} - \frac{(\frac{\pi}{2})^7}{7!} + \cdots = 1$.

Multiple choice

What is the value of $cos(\frac{\pi}{2})$ using the cosine series?

  1. 0

  2. 1

  3. -1

  4. \frac{1}{2}
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the cosine series, we have $cos(\frac{\pi}{2}) = 1 - \frac{(\frac{\pi}{2})^2}{2!} + \frac{(\frac{\pi}{2})^4}{4!} - \frac{(\frac{\pi}{2})^6}{6!} + \cdots = 0$.

Multiple choice

What is the value of $sin(\frac{\pi}{3})$ using the sine series?

  1. 0

  2. 1

  3. -1

  4. \frac{1}{2}
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

Using the sine series, we have $sin(\frac{\pi}{3}) = \frac{\pi}{3} - \frac{(\frac{\pi}{3})^3}{3!} + \frac{(\frac{\pi}{3})^5}{5!} - \frac{(\frac{\pi}{3})^7}{7!} + \cdots = \frac{\sqrt{3}}{2}$.

Multiple choice

What is the value of $cos(\frac{\pi}{3})$ using the cosine series?

  1. 0

  2. 1

  3. -1

  4. \frac{1}{2}
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Using the cosine series, we have $cos(\frac{\pi}{3}) = 1 - \frac{(\frac{\pi}{3})^2}{2!} + \frac{(\frac{\pi}{3})^4}{4!} - \frac{(\frac{\pi}{3})^6}{6!} + \cdots = \frac{1}{2}$.

Multiple choice

What is the value of $sin(\frac{\pi}{4})$ using the sine series?

  1. 0

  2. 1

  3. -1

  4. \frac{1}{2}
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

Using the sine series, we have $sin(\frac{\pi}{4}) = \frac{\pi}{4} - \frac{(\frac{\pi}{4})^3}{3!} + \frac{(\frac{\pi}{4})^5}{5!} - \frac{(\frac{\pi}{4})^7}{7!} + \cdots = \frac{1}{\sqrt{2}}$.