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 the sine of 30 degrees according to the Surya Siddhanta?

  1. 0.5

  2. 0.75

  3. 1.0

  4. 1.5
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The value of the sine of 30 degrees according to the Surya Siddhanta is 0.5.

Multiple choice

What is the double-angle identity 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 identity 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 identity 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 identity 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 is the half-angle identity for sine?

  1. sin(x/2) = sqrt((1 - cos(x))/2)

  2. sin(x/2) = sqrt((1 + cos(x))/2)

  3. sin(x/2) = (1 - cos(x))/2

  4. sin(x/2) = (1 + cos(x))/2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The half-angle identity for sine states that the sine of half an angle is equal to the square root of one minus the cosine of the angle divided by two.

Multiple choice

What is the half-angle identity for cosine?

  1. cos(x/2) = sqrt((1 + cos(x))/2)

  2. cos(x/2) = sqrt((1 - cos(x))/2)

  3. cos(x/2) = (1 + cos(x))/2

  4. cos(x/2) = (1 - cos(x))/2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The half-angle identity for cosine states that the cosine of half an angle is equal to the square root of one plus the cosine of the angle divided by two.

Multiple choice

What is the sum-to-product identity for sine?

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

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

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

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

The sum-to-product identity for sine states that the sum of two sines is equal to twice the sine of half the sum of the angles multiplied by the cosine of half the difference of the angles.

Multiple choice

What is the sum-to-product identity for cosine?

  1. cos(x) + cos(y) = 2cos((x+y)/2)cos((x-y)/2)

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

  3. cos(x) - cos(y) = 2cos((x+y)/2)cos((x-y)/2)

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

The sum-to-product identity for cosine states that the sum of two cosines is equal to twice the cosine of half the sum of the angles multiplied by the cosine of half the difference of the angles.

Multiple choice

What is the product-to-sum identity for sine?

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

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

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

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

The product-to-sum identity for sine states that the product of two sines is equal to half the difference of the cosines of the sum and difference of the angles.

Multiple choice

What is the product-to-sum identity for cosine?

  1. cos(x)cos(y) = (cos(x-y) + cos(x+y))/2

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

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

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

The product-to-sum identity for cosine states that the product of two cosines is equal to half the sum of the cosines of the sum and difference of the angles.

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.