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 does the symbol (\sec) represent?

  1. Sine

  2. Cosine

  3. Tangent

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

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

Multiple choice

What does the symbol (\cot) represent?

  1. Sine

  2. Cosine

  3. Tangent

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

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

Multiple choice

What is the formula for the power series expansion of the sine function?

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

Multiple choice

What is the value of sine of 30 degrees according to Indian mathematicians?

  1. 1/2

  2. √3/2

  3. 1/√2

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

Indian mathematicians, including Aryabhata and Brahmagupta, calculated the value of sine of 30 degrees to be 1/2.

Multiple choice

What is the sine table?

  1. A table of values for the sine function

  2. A table of values for the cosine function

  3. A table of values for the tangent function

  4. A table of values for the cotangent function
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sine table is a table of values for the sine function, which gives the value of the sine of an angle for different values of the angle.

Multiple choice

How was the sine table used?

  1. To calculate the values of the sine function

  2. To calculate the values of the cosine function

  3. To calculate the values of the tangent function

  4. To calculate the values of the cotangent function
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sine table was used to calculate the values of the sine function for different angles.

Multiple choice

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

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

  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 sine function, which is given by $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$. This series converges for all (x).

Multiple choice

What is the Ramanujan modular equation?

  1. An equation that relates the Ramanujan theta function to the Dedekind eta function

  2. An equation that relates the Ramanujan theta function to the Jacobi theta function

  3. An equation that relates the Ramanujan theta function to the Weierstrass elliptic function

  4. An equation that relates the Ramanujan theta function to the Riemann zeta function
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Ramanujan modular equation is an equation that relates the Ramanujan theta function to the Dedekind eta function. It is a very important equation in number theory and has many applications.

Multiple choice

What is the general formula for the Madhava Series expansion of the sine function?

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

The general formula for the Madhava Series expansion of the sine function is $sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ...$

Multiple choice

What is the general formula for the Half-Angle Formula for sine?

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

The general formula for the Half-Angle Formula for sine is $sin(\frac{x}{2}) = \sqrt{\frac{1 - cos(x)}{2}}$.

Multiple choice

What is the general formula for the Half-Angle Formula for cosine?

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

The general formula for the Half-Angle Formula for cosine is $cos(\frac{x}{2}) = \sqrt{\frac{1 + cos(x)}{2}}$.

Multiple choice

What is the value of the expression sin(π/2)?

  1. 0

  2. 1

  3. √2/2

  4. 1/2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The expression sin(π/2) means the sine of the angle π/2. Since the sine of an angle is the ratio of the opposite side to the hypotenuse of a right triangle, and the opposite side of a right triangle with an angle of π/2 is equal to the hypotenuse, the value of the expression is 1.

Multiple choice

What is the value of the expression cos(π)?

  1. -1

  2. 0

  3. 1

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

The expression cos(π) means the cosine of the angle π. Since the cosine of an angle is the ratio of the adjacent side to the hypotenuse of a right triangle, and the adjacent side of a right triangle with an angle of π is equal to the opposite side, the value of the expression is -1.

Multiple choice

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

  1. 0

  2. 1

  3. √2

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

The expression cot(π/4) means the cotangent of the angle π/4. Since the cotangent of an angle is the ratio of the adjacent side to the opposite side of a right triangle, and the adjacent side and the opposite 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 name of the trigonometric function that calculates the reciprocal of the sine function?

  1. Sine

  2. Cosine

  3. Tangent

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

The cosecant function is a trigonometric function that calculates the reciprocal of the sine function. It is widely used in various mathematical and scientific applications.