Explore topic-wise InterviewSolutions in .

Correct option is (d) \( \frac {1}{\SQRT 3}\)

The explanation: (SEC 30°) (cos 60°) = \(( \frac {2}{\sqrt 3}) ( \frac {1}{2} )\)

= \( \frac {1}{\sqrt 3}\)

Correct CHOICE is (C) 2

For explanation I WOULD say: sec^2 A + (1 + TAN A) (1 – tan A) = sec^2 A + 1^2 – tan^2 A

= (sec^2 A – tan^2 A) + 1

= 1 + 1

= 2

The CORRECT choice is (a) 1

For explanation I would say: COS (30° + 60°) = cos (90°) = 1

The VALUE of cos (90°) is 1

Right choice is (a) 8 UNITS

To explain: From PYTHAGORAS theorem, (Hypotenuse)^2 = (OPPOSITE side)^2 + (Adjacent side)^2

(Adjacent side)^2 = (Hypotenuse)^2 – (Opposite side)^2

Adjacent side = \(\sqrt {289 – 225}\) = 8 units

Correct option is (b) Tan^2 θ / SEC ⁡θ

The EXPLANATION is: Sec θ – \(\frac {1}{sec \, \theta } = \frac {1}{sec \, \theta }\)(Sec^2 θ – 1)

= \(\frac {1}{sec \, \theta }\)(Tan^2 θ)

= Tan^2 θ / Sec ⁡θ

Correct option is (c) \(\FRAC {5}{4}\)

The explanation: SIN B = \(\frac {Opposite \, side}{Hypotenuse} = \frac {3}{5}\)

From Pythagoras theorem, (Hypotenuse)^2 = (Opposite side)^2 + (Adjacent side)^2

5^2 = 3^2 + (Adjacent side)^2

(Adjacent side)^2 = 5^2 – 3^2

Adjacent side = √16 = 4

Sec B = \(\frac {Hypotenuse}{Adjacent \, side} = \frac {5}{4}\)

The correct answer is (B) SEC, Cos

The best I can EXPLAIN: A plane is divided into four INFINITE quadrants. The trigonometric ratios that are positive in the fourth quadrant are secant, COSINE and the rest of all trigonometric ratios are negative in this quadrant.

The correct CHOICE is (B) 1

The EXPLANATION: (cosec θ – cot θ) (cosec θ – cot θ) = cosec^2θ – cot^2θ

= 1

The identity used here is cosec^2θ – cot^2θ = 1

Right option is (C) \(\frac {1}{8}\)

BEST explanation: COS 2θ = 2cos θ^2 – 1

cos θ = \(\frac {3}{4}\)

cos 2θ = 2(\(\frac {3}{4}\))^2 – 1

= \(\frac {1}{8}\)

Correct answer is (B) 0

For EXPLANATION: (1 + cosec⁡ θ) (1 – cosec θ) + cot^2 ⁡θ = (1 – cosec^2 θ) + cot^2 θ

= -cot^2⁡ θ + cot^2⁡ θ(∵ cosec^2 A – cot^2 A = 1)

= 0

The correct answer is (a) True

The BEST explanation: ∠P + ∠Q + ∠R = 180° ( ∵ The SUM of angles in a triangle is 180°)

∠P + 90° + ∠R = 180°

∠P + ∠R = 180° – 90°

∠P + ∠R = 90°

Two angles are SAID to be complementary angles if the sum of these two angles is 90°.

The correct choice is (b) 1

Explanation: = (SEC θ – TAN θ) (sec θ + tan θ) = sec^2 θ – tan^2 θ

= 1

The identity USED here is sec^2 θ – tan^2 θ = 1

Correct CHOICE is (d) 3

Best explanation: SIN 90° + cos 0° + √2 cos 45° = 1 + 1 + √2 (\(\frac {1}{\SQRT {2}}\))

= 1 + 1 + 1

= 3

Right option is (b) False

Explanation: Consider, two triangles ABC and DEF

In ∆ABC,

sin B = \(\frac {AC}{AB} = \frac {10}{20} = \frac {1}{2}\) i.e. B = 30°

Now, in ∆DEF,

sin F = \(\frac {DE}{DF} = \frac {20}{40} = \frac {1}{2}\)i.e. F = 30°

From these examples it is evident that the VALUE of the trigonometric RATIOS depends on their angle and not on their lengths.

Correct answer is (b) 1

Easy explanation: Let (X1, y1) = (sin θ, 0) and (x2, y2) = (0, COS θ)

Distance between TWO points = √((x2 – x1)^2 + (y2 – y1)^2)

= √((0 – sin θ)^2 + (cos θ – 0)^2)

= √(sin^2 θ + cos^2 θ)

= √1

= 1

Right choice is (c) \(\frac {17}{89}\)

The explanation is: Tan A and cot A are RECIPROCAL ratios and cot is INVERSE of tan.

Tan A = \(\frac {1}{Cot A} = \frac {1}{89/17}\)

= \(\frac {17}{89}\)

The correct CHOICE is (a) 0

The best I can explain: \( \FRAC {1 – COT \, 45^{\circ}}{1+cot \, 45^{\circ}} = \frac {1-1}{1+1}\)

= \( \frac {0}{1}\)

= 0

The CORRECT option is (c) COT 60°

The best explanation: \(\frac {2 cot \, ⁡⁡60^{\circ }}{1 + cot \, ⁡⁡45^{\circ }} = \frac {2 (\frac {1}{\sqrt 3})}{1 + 1}\)

= \(\frac {\frac {2}{\sqrt {3}}}{2}\)

= \(\frac {1}{\sqrt 3}\)

= Cot 60°

Right option is (a) \(\frac {3}{4}\)

EASY EXPLANATION: SIN A = \(\frac {OPPOSITE \, side}{Hypotenuse} = \frac {3}{5}\)

From Pythagoras theorem, (Hypotenuse)^2 = (Opposite side)^2 + (Adjacent side)^2

5^2 = 3^2 + (Adjacent side)^2

(Adjacent side)^2 = 5^2 – 3^2

Adjacent side = √16 = 4

Tan A = \(\frac {Opposite \, side}{Adjacent \, side} = \frac {3}{4}\)

Right OPTION is (d) Tan θ, Cot θ

To explain: Tan θ, Cot θ are the RECIPROCAL ratios because they are INVERSE to each other.

Tanθ = \(\frac {1}{Cot \theta }\)

Correct CHOICE is (a) True

To explain I would say: The word TRIGONOMETRY is from the language Greek where ‘TRI’ means three and ‘Goria’ means ‘angle’ and ‘Metron’ means measure which gives the MEANING as three ANGLES measure.

Correct answer is (C) sine, cosine and tangent

To ELABORATE: The BASIC trigonometric RATIOS are sine, cosine and tangent and the other three trigonometric ratios secant, cosecant, cotangent are derived from these basic trigonometric ratios.

Correct ANSWER is (a) cos^2 θ = 1 – sin^2 θ

The explanation is: The APPROPRIATE TRIGONOMETRIC identity used here is sin^2 θ + cos^2 θ = 1.

cos^2 θ = 1 – sin^2 θ

The correct answer is (c) Sine

Easiest explanation: The INVERSE of cosecant is sine.

Cosec θ = \(\frac {1}{SIN \THETA }\)

The correct CHOICE is (C) Cosec 15° + Sec 25°

Best EXPLANATION: Sec 65° + Cosec 75° = Sec (90° – 25°) + Cosec (90° – 15°)

= Cosec 25° + sec 15°

Right CHOICE is (b) COT 45°

Explanation: sec 60° – \( \frac {1}{cosec \, 90^{\CIRC }}\) = 2 – \( \frac {1}{1}\)

= 1

= cot 45°

The correct option is (a) \( \frac {1}{\sqrt 3}\)

The explanation: Tan θ = \( \frac {SIN \, \THETA}{Cos \, \theta} = \frac {Sin \, 30^{\circ }}{Cos \, 30^{\circ }} \)

= \( \frac {1/2}{\sqrt 3/2}\)

= \( \frac {1}{2} \times \frac {2}{\sqrt 3}\)

= \( \frac {1}{\sqrt 3}\)

The correct ANSWER is (d) \(\FRAC {1}{\SQRT {2}}\)

For EXPLANATION: sin30°cos15° + cos30°sin15° = sin⁡45° = \(\frac {1}{\sqrt {2}}\)

Right ANSWER is (b) True

The best explanation: YES, an equation is an identity. If this identity is true for all VALUES of the variables in an equation. For example (a – b)^2 = a^2 + b^2 – 2ab is an identity.

Right choice is (a) Cosec, Sin

The EXPLANATION is: A plane is DIVIDED into four infinite quadrants. The TRIGONOMETRIC ratios that are positive in the SECOND quadrant are sine, cosecant and the rest of all trigonometric ratios are NEGATIVE in this quadrant.

Right ANSWER is (a) True

Easiest EXPLANATION: The inverse of cosecant is sine.

Sin θ = \(\FRAC {1}{COSEC \THETA }\)

Sin θ.Cosec θ = 1

The correct answer is (c) 1

The best explanation: Since the triangle is right angles at C,

The SUM of the remaining TWO angles will be 90

∴ cosec(A + B) = Cosec 90° = 1

The correct ANSWER is (b) -cosec of angle A

The best I can explain: (180° – a) refers to the SECOND quadrant which LIES in the range from 90° to 180°. Trigonometric ratios SINE and cosec are only positive in the second quadrant and remaining all the trigonometric ratios are negative.

So, Cot (180° – a) = -Cot A

Right ANSWER is (a) COS x

The best I can explain: (90° – x) refers to the first QUADRANT which lies in the range from 0° to 90°. All trigonometric RATIOS are POSITIVE in the first quadrant and sine changes to cosine when it is 90° or 270°.

The correct CHOICE is (a) Hipparchus

Easiest explanation: Hipparchus was a Greek mathematician and is considered as the ‘Father of TRIGONOMETRY’. Hipparchus not only contributed to MATHEMATICS but also to ASTRONOMY.

Correct answer is (a) TAN θ

For explanation I would say: SIN θ = \(\frac {Length \, of \, opposite \, side \, of \, the \, triangle}{Length \, of \, HYPOTENUSE \, of \, the \, triangle}\)

Cos θ = \(\frac {Length \, of \, adjacent \, side \, of \, the \, triangle}{Length \, of \, hypotenuse \, of \, the \, triangle}\)

\(\frac {Sin \THETA }{Cos \theta} = \frac {Length \, of \, opposite \, side \, of \, the \, triangle}{Length \, of \, adjacent \, of \, the \, triangle}\)

= Tan θ