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 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

The tangent of an angle is defined as the ratio of the:

  1. Opposite side to the adjacent side

  2. Adjacent side to the opposite side

  3. Hypotenuse to the opposite side

  4. Hypotenuse to the adjacent side
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The tangent of an angle is calculated by dividing the opposite side by the adjacent side.

Multiple choice

What is the name of the trigonometric function that is the reciprocal of the tangent function?

  1. Sine

  2. Cosine

  3. Tangent

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

The cotangent function is defined as the reciprocal of the tangent function.

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.

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 the tangent of 60 degrees?

  1. 0.5

  2. 0.707

  3. 1

  4. 1.414
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

The tangent of 60 degrees is 1.732.

Multiple choice

What is the value of the arc tangent of 1?

  1. 30 degrees

  2. 45 degrees

  3. 60 degrees

  4. 90 degrees
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The arc tangent of 1 is 45 degrees.

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

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 $cos(\frac{x}{2}) = \sqrt{\frac{1 + cos(x)}{2}}$.