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 general formula for the Madhava series for the cosine function?

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

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

Multiple choice

The equation (\sin^2\theta + \cos^2\theta = 1) is known as:

  1. Pythagorean theorem

  2. Euler's formula

  3. Trigonometric identity

  4. Law of cosines
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation (\sin^2\theta + \cos^2\theta = 1) is known as a trigonometric identity, which is an equation that is true for all values of the variable.

Multiple choice

The equation (\frac{d}{dx} \sin x = \cos x) is known as:

  1. Chain rule

  2. Product rule

  3. Quotient rule

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

The equation (\frac{d}{dx} \sin x = \cos x) is known as the chain rule, which is used to find the derivative of a composite function.

Multiple choice

What is the solution to the equation sin(x) = 1/2?

  1. x = π/6

  2. x = π/3

  3. x = π/4

  4. x = π/2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To solve the equation sin(x) = 1/2, we can use the inverse sine function (also known as arcsin). The inverse sine of 1/2 is π/6. Therefore, the solution to the equation sin(x) = 1/2 is x = π/6.

Multiple choice

The formula $\sin^2 \theta + \cos^2 \theta = 1$ is known as the:

  1. Pythagorean identity

  2. Euler's formula

  3. Trigonometric identity

  4. Brahmagupta's formula
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The formula $\sin^2 \theta + \cos^2 \theta = 1$ is a fundamental trigonometric identity that expresses the relationship between the sine and cosine functions of an angle.

Multiple choice

The formula $\sin 2\theta = 2 \sin \theta \cos \theta$ is known as the:

  1. Double-angle formula

  2. Half-angle formula

  3. Sum-to-product formula

  4. Product-to-sum formula
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula $\sin 2\theta = 2 \sin \theta \cos \theta$ is a double-angle formula in trigonometry, which expresses the sine of twice an angle in terms of the sine and cosine of the original angle.

Multiple choice

The formula $\cos \theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta}$ is known as the:

  1. Pythagorean identity

  2. Euler's formula

  3. Trigonometric identity

  4. Brahmagupta's formula
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The formula $\cos \theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta}$ is a trigonometric identity that expresses the relationship between the cosine and tangent functions of an angle.

Multiple choice

The formula $\tan \theta = \frac{\sin \theta}{\cos \theta}$ is known as the:

  1. Pythagorean identity

  2. Euler's formula

  3. Trigonometric identity

  4. Brahmagupta's formula
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The formula $\tan \theta = \frac{\sin \theta}{\cos \theta}$ is a fundamental trigonometric identity that expresses the relationship between the tangent, sine, and cosine functions of an angle.

Multiple choice

The formula $\sin (A + B) = \sin A \cos B + \cos A \sin B$ is known as the:

  1. Pythagorean identity

  2. Euler's formula

  3. Sum-to-product formula

  4. Product-to-sum formula
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The formula $\sin (A + B) = \sin A \cos B + \cos A \sin B$ is a sum-to-product formula in trigonometry, which expresses the sine of the sum of two angles in terms of the sines and cosines of the individual angles.

Multiple choice

The formula $\cos (A + B) = \cos A \cos B - \sin A \sin B$ is known as the:

  1. Pythagorean identity

  2. Euler's formula

  3. Sum-to-product formula

  4. Product-to-sum formula
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The formula $\cos (A + B) = \cos A \cos B - \sin A \sin B$ is a sum-to-product formula in trigonometry, which expresses the cosine of the sum of two angles in terms of the cosines and sines of the individual angles.

Multiple choice

Which trigonometric function is used to calculate the longitude of a place?

  1. Sine

  2. Cosine

  3. Tangent

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

The tangent function is used to calculate the longitude of a place, using the formula (\lambda = \tan^{-1}(\frac{\sin t}{\cos t \sin \phi_0 - \tan \delta \cos \phi_0})).

Multiple choice

Which trigonometric function is used to calculate the azimuth of a celestial body?

  1. Sine

  2. Cosine

  3. Tangent

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

The tangent function is used to calculate the azimuth of a celestial body, using the formula (\alpha = \tan^{-1}(\frac{\sin A}{\cos A \sin \phi - \tan \delta \cos \phi})).

Multiple choice

What is the formula for calculating the longitude of a ship using trigonometry?

  1. Longitude = arcsin(cos(altitude) / cos(declination))

  2. Longitude = arccos(sin(altitude) / sin(declination))

  3. Longitude = arctan(tan(altitude) / tan(declination))

  4. Longitude = arccot(cot(altitude) / cot(declination))
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The formula for calculating the longitude of a ship using trigonometry is Longitude = arccos(sin(altitude) / sin(declination)).

Multiple choice

Which trigonometric function is the reciprocal of the sine function?

  1. Cosine

  2. Tangent

  3. Cosecant

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

The cosecant function is defined as the reciprocal of the sine function, meaning $$cosec(x) = 1 / sin(x)$$.

Multiple choice

What is the relationship between the sine and cosine functions in terms of their graphs?

  1. They are perpendicular to each other.

  2. They have the same amplitude.

  3. They have the same period.

  4. They are reflections of each other across the x-axis.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The graphs of the sine and cosine functions are perpendicular to each other, meaning that when one function is at its maximum value, the other function is at its minimum value.