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
What is the value of the expression sin(π/2)?
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.
What is the value of the expression cos(π)?
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.
What is the value of the expression cot(π/4)?
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.
What is the name of the trigonometric function that calculates the reciprocal of the sine function?
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Sine
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Cosine
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Tangent
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Cosecant
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.
What is the name of the trigonometric function that calculates the reciprocal of the cosine function?
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Sine
-
Cosine
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Tangent
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Secant
D
Correct answer
Explanation
The secant function is a trigonometric function that calculates the reciprocal of the cosine function. It is widely used in various mathematical and scientific applications.
What is the name of the trigonometric function that calculates the reciprocal of the tangent function?
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Sine
-
Cosine
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Tangent
-
Cotangent
D
Correct answer
Explanation
The cotangent function is a trigonometric function that calculates the reciprocal of the tangent function. It is widely used in various mathematical and scientific applications.
Bhaskara I's formula for the sine of an angle is given by:
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$$sin(x) = rac{opposite}{hypotenuse}$$
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$$sin(x) = rac{adjacent}{hypotenuse}$$
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$$sin(x) = rac{opposite}{adjacent}$$
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$$sin(x) = rac{hypotenuse}{opposite}$$
A
Correct answer
Explanation
Bhaskara I's formula for the sine of an angle is $$sin(x) = rac{opposite}{hypotenuse}$$, where x is the angle and opposite and hypotenuse are the lengths of the opposite and hypotenuse sides of the right triangle, respectively.
What is the value of the expression sin(30°) + cos(60°)?
B
Correct answer
Explanation
The value of the expression sin(30°) + cos(60°) is equal to (1/2) + (1/2) = 1.
What is the value of the expression cot(45°)?
A
Correct answer
Explanation
The value of the expression cot(45°) is equal to 1.
What is the value of the expression sin(θ) + cos(θ) if sin(θ) = 3/5 and cos(θ) = 4/5?
B
Correct answer
Explanation
The value of the expression sin(θ) + cos(θ) if sin(θ) = 3/5 and cos(θ) = 4/5 is equal to (3/5) + (4/5) = 7/5.
What is the value of the expression tan(θ) if sin(θ) = 3/5 and cos(θ) = 4/5?
A
Correct answer
Explanation
The value of the expression tan(θ) if sin(θ) = 3/5 and cos(θ) = 4/5 is equal to (3/5) / (4/5) = 3/4.
What is the general formula for the Madhava series?
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$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
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$$\sin x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$
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$$\sin x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$$
-
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
A,D
Correct answer
Explanation
The general formula for the Madhava series is $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$.
What is the general formula for the Madhava series for the cosine function?
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$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$$
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$$\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{4} - \frac{x^6}{6} + \cdots$$
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$$\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{4} - \frac{x^6}{6} + \cdots$$
-
$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$$
A,D
Correct answer
Explanation
The general formula for the Madhava series for the cosine function is $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$$.
What is the law of sines?
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$a/sin(A) = b/sin(B) = c/sin(C)$
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$a^2 + b^2 = c^2$
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$A = \frac{1}{2}bh$
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$P = a + b + c$
A
Correct answer
Explanation
The law of sines states that in a triangle, the ratio of the length of a side to the sine of the opposite angle is the same for all three sides.
What is the law of cosines?
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$a^2 + b^2 = c^2$
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$a/sin(A) = b/sin(B) = c/sin(C)$
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$A = \frac{1}{2}bh$
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$P = a + b + c$
Correct answer
Explanation
The law of cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of those two sides and the cosine of the angle between them.