Trigonometric Ratios & Identities for Competitive Exams

Trigonometric Ratios & Identities

Q. If cos A = 1/2, what is the value of A in degrees?

A. 30
B. 60
C. 90
D. 120

Solution

The angle A for which cos A = 1/2 is 60°.

Correct Answer: A — 30

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Q. If cos B = 1/2, what is the value of sin B?

A. √3/2
B. 1/2
C. 0
D. √2/2

Solution

Using the identity sin^2 B + cos^2 B = 1, we have sin B = sqrt(1 - (1/2)^2) = sqrt(3/4) = √3/2.

Correct Answer: A — √3/2

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Q. If cos(x) = 0, what are the possible values of x?

A. 90° + k*180°
B. k*90°
C. k*180°
D. 0° + k*360°

Solution

cos(x) = 0 at x = 90° + k*180°, where k is any integer.

Correct Answer: A — 90° + k*180°

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Q. If cos(x) = 0, what is the value of tan(x)?

A. 0
B. 1
C. undefined
D. ∞

Solution

tan(x) = sin(x)/cos(x) is undefined when cos(x) = 0.

Correct Answer: C — undefined

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Q. If cos(x) = 0, what is the value of x?

A. π/2 + nπ
B. nπ
C. 0
D. π

Solution

cos(x) = 0 at x = π/2 + nπ, where n is any integer.

Correct Answer: A — π/2 + nπ

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Q. If cos(x) = 1/2, what is the value of sin(x)?

A. √3/2
B. 1/2
C. 0
D. √2/2

Solution

Using the identity sin^2(x) + cos^2(x) = 1, we have sin^2(x) = 1 - (1/2)^2 = 1 - 1/4 = 3/4. Therefore, sin(x) = √(3/4) = √3/2.

Correct Answer: A — √3/2

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Q. If cos(θ) = 1/2, what is the value of sin(θ)?

A. √3/2
B. 1/2
C. 0
D. √2/2

Solution

Using the identity sin^2(θ) + cos^2(θ) = 1, we have sin^2(θ) = 1 - (1/2)^2 = 1 - 1/4 = 3/4. Thus, sin(θ) = ±√3/2.

Correct Answer: A — √3/2

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Q. If cot θ = 3/4, what is the value of sin θ?

A. 4/5
B. 3/5
C. 5/4
D. 3/4

Solution

Using the identity cot θ = cos θ / sin θ, we find sin θ = 4/5.

Correct Answer: A — 4/5

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Q. If cot(x) = 5/12, what is sin(x)?

A. 12/13
B. 5/13
C. 13/12
D. 5/12

Solution

Using the identity cot(x) = cos(x)/sin(x), we can find sin(x) = 12/13 using the Pythagorean theorem.

Correct Answer: A — 12/13

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Q. If sec A = 2, what is the value of cos A?

A. 1/2
B. 2
C. √2
D. 1

Solution

Since sec A = 1/cos A, we have cos A = 1/2.

Correct Answer: A — 1/2

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Q. If sec θ = 2, what is the value of cos θ?

A. 1/2
B. 2
C. 1
D. 0

Solution

Since sec θ = 1/cos θ, if sec θ = 2, then cos θ = 1/2.

Correct Answer: A — 1/2

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Q. If sec(x) = 2, what is the value of cos(x)?

A. 1/2
B. 2
C. √3/2
D. 1

Solution

sec(x) = 1/cos(x), so cos(x) = 1/2.

Correct Answer: A — 1/2

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Q. If sin A = 0.6, what is the value of tan A?

A. 0.8
B. 1.2
C. 0.75
D. 1.5

Solution

Using the identity tan A = sin A / cos A, we find cos A = sqrt(1 - (0.6)^2) = 0.8, thus tan A = 0.6 / 0.8 = 0.75.

Correct Answer: B — 1.2

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Q. If sin A = 1/2, what is the value of A in degrees?

A. 30
B. 45
C. 60
D. 90

Solution

sin A = 1/2 corresponds to A = 30°.

Correct Answer: A — 30

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Q. If sin A = 4/5, what is the value of tan A?

A. 3/4
B. 4/3
C. 5/4
D. 5/3

Solution

Using the identity tan A = sin A / cos A, we find cos A = 3/5, thus tan A = (4/5) / (3/5) = 4/3.

Correct Answer: B — 4/3

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Q. If sin(x) = 0, what are the possible values of x?

A. nπ
B. nπ/2
C. nπ + π/2
D. nπ + π

Solution

sin(x) = 0 at x = nπ, where n is any integer.

Correct Answer: A — nπ

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Q. If sin(x) = 0, what is the value of cos(x)?

A. 1
B. 0
C. -1
D. undefined

Solution

If sin(x) = 0, then cos(x) can be either 1 or -1 depending on the angle x.

Correct Answer: A — 1

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Q. If sin(x) = 0, what is the value of tan(x)?

A. 0
B. 1
C. undefined
D. ∞

Solution

tan(x) = sin(x)/cos(x). If sin(x) = 0, then tan(x) is undefined when cos(x) = 0.

Correct Answer: C — undefined

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Q. If sin(x) = 0, what is the value of x?

A. 0
B. π
C. 2π
D. All of the above

Solution

sin(x) = 0 at x = nπ, where n is any integer, hence all of the above.

Correct Answer: D — All of the above

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Q. If sin(x) = 1/2, what are the possible values of x in [0, 2π]?

A. π/6, 5π/6
B. π/4, 3π/4
C. 0, π
D. π/3, 2π/3

Solution

sin(x) = 1/2 at x = π/6 and x = 5π/6 in the interval [0, 2π].

Correct Answer: A — π/6, 5π/6

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Q. If sin(x) = 1/2, what is the value of x in degrees?

A. 30
B. 45
C. 60
D. 90

Solution

sin(30°) = 1/2, so x = 30°.

Correct Answer: A — 30

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Q. If sin(x) = 1/√2, what is cos(x)?

A. 1/√2
B. 0
C. √2/2
D. 1

Solution

Using the identity sin^2(x) + cos^2(x) = 1, we have cos^2(x) = 1 - (1/√2)^2 = 1 - 1/2 = 1/2. Thus, cos(x) = ±1/√2.

Correct Answer: A — 1/√2

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Q. If sin(x) = 1/√2, what is tan(x)?

A. 1
B. √2
C. √3
D. 0

Solution

tan(x) = sin(x)/cos(x) = (1/√2)/(1/√2) = 1.

Correct Answer: B — √2

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Q. If sin(x) = 1/√2, what is the value of cos(x)?

A. 1/√2
B. 0
C. √2/2
D. 1

Solution

Using the identity sin^2(x) + cos^2(x) = 1, we have cos^2(x) = 1 - (1/√2)^2 = 1 - 1/2 = 1/2. Therefore, cos(x) = 1/√2.

Correct Answer: A — 1/√2

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Q. If sin(x) = 3/5, what is cos(x)?

A. 4/5
B. 3/5
C. 5/4
D. 1/5

Solution

Using the identity sin^2(x) + cos^2(x) = 1, we have cos^2(x) = 1 - (3/5)^2 = 1 - 9/25 = 16/25. Therefore, cos(x) = ±4/5. The positive value is taken as x is in the first quadrant.

Correct Answer: A — 4/5

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Q. If sin(α) = 0.6, what is the value of cos(α) using the identity?

A. 0.8
B. 0.6
C. 0.4
D. 0.2

Solution

Using sin^2(α) + cos^2(α) = 1, we find cos(α) = √(1 - 0.6^2) = √(1 - 0.36) = √0.64 = 0.8.

Correct Answer: A — 0.8

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Q. If sin(θ) = 1/√2, what is the value of cos(θ)?

A. 1/√2
B. 0
C. √2/2
D. 1

Solution

Using the identity sin^2(θ) + cos^2(θ) = 1, we have cos^2(θ) = 1 - (1/√2)^2 = 1 - 1/2 = 1/2. Thus, cos(θ) = ±1/√2.

Correct Answer: A — 1/√2

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Q. If sin(θ) = 4/5, what is the value of tan(θ)?

A. 3/4
B. 4/3
C. 5/4
D. 5/3

Solution

Using the identity tan(θ) = sin(θ)/cos(θ) and cos(θ) = √(1 - sin^2(θ)), we find cos(θ) = 3/5. Thus, tan(θ) = (4/5)/(3/5) = 4/3.

Correct Answer: A — 3/4

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Q. If tan θ = 1, what is the value of sin θ?

A. 1/√2
B. 1
C. 0
D. √2

Solution

Since tan θ = sin θ / cos θ and tan θ = 1, we have sin θ = cos θ. For θ = 45°, sin θ = 1/√2.

Correct Answer: A — 1/√2

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Q. If tan(x) = 1, what is the value of sin(x) + cos(x)?

A. √2
B. 1
C. 0
D. 2

Solution

If tan(x) = 1, then sin(x) = cos(x). Therefore, sin(x) + cos(x) = 2sin(x) = 2(1/√2) = √2.

Correct Answer: A — √2

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Trigonometric Ratios & Identities MCQ & Objective Questions

Understanding Trigonometric Ratios & Identities is crucial for students preparing for school exams and competitive tests. Mastering these concepts not only enhances your mathematical skills but also boosts your confidence in tackling objective questions. Regular practice with MCQs helps in identifying important questions and reinforces your exam preparation strategy.

What You Will Practise Here

Fundamental Trigonometric Ratios: Sine, Cosine, Tangent, and their reciprocals.
Key Trigonometric Identities: Pythagorean identities, sum and difference formulas.
Applications of Trigonometric Ratios in real-world problems.
Graphs of Trigonometric Functions: Understanding their behavior and properties.
Solving Trigonometric Equations: Techniques and strategies.
Common Angles: Values of trigonometric ratios for 0°, 30°, 45°, 60°, and 90°.
Inverse Trigonometric Functions: Definitions and applications.

Exam Relevance

Trigonometric Ratios & Identities are integral to the mathematics syllabus across various boards, including CBSE and State Boards. This topic frequently appears in NEET, JEE, and other competitive exams, often in the form of direct questions, application-based problems, and conceptual MCQs. Students can expect to encounter questions that require both theoretical understanding and practical application of these concepts.

Common Mistakes Students Make

Confusing the definitions of sine, cosine, and tangent ratios.
Overlooking the importance of angle measurement (degrees vs. radians).
Misapplying trigonometric identities during problem-solving.
Failing to simplify expressions correctly before solving equations.
Ignoring the graphical representation of trigonometric functions, which can lead to misunderstanding their properties.

FAQs

Question: What are the basic trigonometric ratios?
Answer: The basic trigonometric ratios are sine, cosine, and tangent, which relate the angles of a right triangle to the ratios of its sides.

Question: How can I remember the trigonometric identities?
Answer: Practice regularly and use mnemonic devices to help memorize the key identities, such as SOH-CAH-TOA for the basic ratios.

Question: Why are trigonometric ratios important for competitive exams?
Answer: They form the foundation for many advanced topics in mathematics and are frequently tested in various competitive exams, making them essential for success.

Now is the time to enhance your understanding of Trigonometric Ratios & Identities. Dive into our practice MCQs and test your knowledge to excel in your exams!

Trigonometric Ratios & Identities

Trigonometric Ratios & Identities MCQ & Objective Questions

What You Will Practise Here

Exam Relevance

Common Mistakes Students Make

FAQs

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