Mathematics
Trigonometric Identities
87 Questions
Trigonometric identities focus on solving equations using tangent, sine, and cosine properties. Questions involve half angle formulas and slope calculations. This topic is a staple in the quantitative aptitude section of major competitive examinations.
Tangent propertiesAngle formulasTrigonometric equationsHalf angle identitiesSlope calculations
Trigonometric Identities Questions
What is the value of (\tan 45^\circ) according to Aryabhata?
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\(\frac{1}{2}\)
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\(\frac{\sqrt{3}}{2}\)
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\(\frac{1}{\sqrt{2}}\)
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\(\frac{\sqrt{2}}{2}\)
Correct answer
Explanation
Aryabhata used the definition of (\tan \theta) as (\frac{\sin \theta}{\cos \theta}) to calculate the values of (\tan \theta). According to his table, (\tan 45^\circ) is equal to (1).
What is the value of (\tan 75^\circ) according to Aryabhata?
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\(\frac{1}{2}\)
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\(\frac{\sqrt{3}}{2}\)
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\(\frac{1}{\sqrt{2}}\)
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\(\frac{\sqrt{2}}{2}\)
Correct answer
Explanation
Aryabhata used the definition of (\tan \theta) as (\frac{\sin \theta}{\cos \theta}) to calculate the values of (\tan \theta) for angles greater than (45^\circ). According to his table, (\tan 75^\circ) is equal to (\frac{\sqrt{6 + \sqrt{3}}}{\sqrt{6 - \sqrt{3}}}).
What is the value of (\tan 90^\circ) according to Aryabhata?
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\(\frac{1}{2}\)
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\(\frac{\sqrt{3}}{2}\)
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\(\frac{1}{\sqrt{2}}\)
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\(\text{undefined}\)
D
Correct answer
Explanation
Aryabhata did not define the value of (\tan 90^\circ). This is because (\tan \theta) is the ratio of the opposite side to the adjacent side, and in a right triangle with an angle of (90^\circ), the adjacent side is equal to (0). Therefore, (\tan 90^\circ) is undefined.
What is the Pythagorean identity?
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sin^2(x) + cos^2(x) = 1
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sin(x) + cos(x) = 1
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tan(x) = sin(x) / cos(x)
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cot(x) = cos(x) / sin(x)
A
Correct answer
Explanation
The Pythagorean identity is a fundamental trigonometric identity that states that the square of the sine of an angle plus the square of the cosine of the same angle is equal to 1.
What does the symbol (\tan) represent?
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Sine
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Cosine
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Tangent
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Cosecant
C
Correct answer
Explanation
(\tan) is a mathematical symbol that represents the tangent of an angle. It is used to find the ratio of the length of the opposite side to the length of the adjacent side of a right triangle.
Which of the following is an example of a series expansion used by Madhava of Sangamagrama for the arctangent function?
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$$\tan^{-1} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$
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$$\tan^{-1} x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
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$$\tan^{-1} x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \cdots$$
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None of the above
A
Correct answer
Explanation
Madhava of Sangamagrama developed a series expansion for the arctangent function, which is given by $$\tan^{-1} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$. This series converges for (|x| \leq 1).
Which of the following is an example of a series expansion used by Madhava of Sangamagrama for the tangent function?
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$$\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \cdots$$
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$$\tan x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
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$$\tan x = 1 + \frac{x^2}{2} + \frac{x^4}{4} + \frac{x^6}{6} + \cdots$$
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None of the above
A
Correct answer
Explanation
Madhava of Sangamagrama developed a series expansion for the tangent function, which is given by $$\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \cdots$$. This series converges for (|x| \leq \frac{\pi}{4}).
What is the value of the expression tan(π/4)?
B
Correct answer
Explanation
The expression tan(π/4) means the tangent of the angle π/4. Since the tangent of an angle is the ratio of the opposite side to the adjacent side of a right triangle, and the opposite side and the adjacent side of a right triangle with an angle of π/4 are equal, the value of the expression is 1.
What is the value of the expression tan(45°)?
A
Correct answer
Explanation
The value of the expression tan(45°) is equal to 1.
The equation (\frac{d}{dx} \tan x = \sec^2 x) is known as:
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Chain rule
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Product rule
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Quotient rule
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Trigonometric rule
D
Correct answer
Explanation
The equation (\frac{d}{dx} \tan x = \sec^2 x) is known as the trigonometric rule, which is used to find the derivative of a trigonometric function.
The equation (\frac{d}{dx} \cot x = -\csc^2 x) is known as:
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Chain rule
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Product rule
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Quotient rule
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Trigonometric rule
D
Correct answer
Explanation
The equation (\frac{d}{dx} \cot x = -\csc^2 x) is known as the trigonometric rule, which is used to find the derivative of a trigonometric function.
The formula $\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ is known as the:
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Pythagorean identity
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Euler's formula
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Sum-to-product formula
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Product-to-sum formula
C
Correct answer
Explanation
The formula $\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ is a sum-to-product formula in trigonometry, which expresses the tangent of the sum of two angles in terms of the tangents of the individual angles.
What is the Pythagorean identity that relates the sine, cosine, and tangent functions?
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$$sin^2(x) + cos^2(x) = 1$$
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$$tan^2(x) + 1 = sec^2(x)$$
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$$sin(x) + cos(x) = 1$$
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$$tan(x) = sin(x) / cos(x)$$
A
Correct answer
Explanation
The Pythagorean identity $$sin^2(x) + cos^2(x) = 1$$ establishes a fundamental relationship between the sine and cosine functions, stating that the sum of the squares of the sine and cosine of an angle is always equal to 1.
What is the value of (\tan 45^\circ) in Indian trigonometry?
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0
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1
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\(\frac{1}{2}\)
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\(\frac{\sqrt{2}}{2}\)
B
Correct answer
Explanation
In Indian trigonometry, the value of (\tan 45^\circ) is 1.
What is the integrating factor for the differential equation $y' + y \tan x = \cos x$?
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$\cos x$
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$\sin x$
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$\sec x$
C
Correct answer
Explanation
The integrating factor for the given differential equation can be found by multiplying both sides of the equation by a suitable function that makes the left-hand side exact.