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 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 $sin(\frac{\pi}{4}) = \frac{\pi}{4} - \frac{(\frac{\pi}{4})^3}{3!} + \frac{(\frac{\pi}{4})^5}{5!} - \frac{(\frac{\pi}{4})^7}{7!} + \cdots = \frac{1}{\sqrt{2}}$.

Multiple choice

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

  1. 0

  2. 1

  3. -1

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

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

Multiple choice

What is the value of $sin(\frac{\pi}{6})$ using the sine 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 sine series, we have $sin(\frac{\pi}{6}) = \frac{\pi}{6} - \frac{(\frac{\pi}{6})^3}{3!} + \frac{(\frac{\pi}{6})^5}{5!} - \frac{(\frac{\pi}{6})^7}{7!} + \cdots = \frac{1}{2}$.

Multiple choice

What is the value of $cos(\frac{\pi}{6})$ using the cosine series?

  1. 0

  2. 1

  3. -1

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

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

Multiple choice

What is the Laplace transform of the cosine function $\cos(at)$?

  1. $\frac{s}{s^2+a^2}$
  2. $\frac{a}{s^2+a^2}$
  3. $\frac{s}{s^2-a^2}$
  4. $\frac{a}{s^2-a^2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Laplace transform of the cosine function $\cos(at)$ is $\frac{s}{s^2+a^2}$.

Multiple choice

What does the symbol (\sin) represent?

  1. Sine

  2. Cosine

  3. Tangent

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

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

Multiple choice

What does the symbol (\cos) represent?

  1. Sine

  2. Cosine

  3. Tangent

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

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