Introduction to Trigonometry | Exercise 8.1

Question 11

State whether the following are true or false. Justify your answer.

(i) The value of tan⁡A\tan A is always less than 1.

(ii) sec⁡A=125\sec A = \dfrac{12}{5} for some value of angle A.

(iii) cos⁡A\cos A is the abbreviation used for the cosecant of angle A.

(iv) cot⁡A\cot A is the product of cot⁡\cot and AA.

(v) sin⁡θ=43\sin \theta = \dfrac{4}{3} for some angle θ\theta.

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Solution
Understand the Question
  • In a right-angled triangle, the hypotenuse is always the longest side.
  • Trigonometric ratios are defined as ratios of sides:
    • sin⁡θ=OppositeHypotenuse≤1\sin \theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}} \le 1
    • cos⁡θ=AdjacentHypotenuse≤1\cos \theta = \dfrac{\text{Adjacent}}{\text{Hypotenuse}} \le 1
    • sec⁡θ=HypotenuseAdjacent≥1\sec \theta = \dfrac{\text{Hypotenuse}}{\text{Adjacent}} \ge 1
    • tan⁡θ=OppositeAdjacent\tan \theta = \dfrac{\text{Opposite}}{\text{Adjacent}} (can take any positive value)
  • Trigonometric notations like sin⁡A,cos⁡A,cot⁡A\sin A, \cos A, \cot A represent functions of angle AA, not algebraic products.

(i) The value of tan⁡A\tan A is always less than 1.

Step 1 · Check Range of tan⁡A\tan A

By definition tan⁡A=Opposite sideAdjacent side\tan A = \dfrac{\text{Opposite side}}{\text{Adjacent side}}

In a right triangle, the side opposite to ∠A\angle A can be longer than, equal to, or shorter than the adjacent side.

For example, if Opposite=3\text{Opposite} = 3 and Adjacent=2\text{Adjacent} = 2

tan⁡A=32=1.5>1\begin{aligned} \tan A &= \dfrac{3}{2} \\[0.6em] &= 1.5 > 1 \end{aligned}

Therefore, the value of tan⁡A\tan A is not always less than 11.

Answer

(i) False

(ii) sec⁡A=125\sec A = \dfrac{12}{5} for some value of angle A.

Step 1 · Check Validity of sec⁡A\sec A

By definition sec⁡A=HypotenuseAdjacent side\sec A = \dfrac{\text{Hypotenuse}}{\text{Adjacent side}}

In a right-angled triangle, the hypotenuse is the longest side, so Hypotenuse>Adjacent side\text{Hypotenuse} > \text{Adjacent side}.

sec⁡A>1\sec A > 1

Given value sec⁡A=125=2.4>1\sec A = \dfrac{12}{5} = 2.4 > 1

Since 12>512 > 5, such a right triangle can exist.

Answer

(ii) True

(iii) cos⁡A\cos A is the abbreviation used for the cosecant of angle A.

Step 1 · Check Definition of Abbreviations

  • cos⁡A\cos A is the abbreviation used for the cosine of angle AA.
  • The abbreviation for the cosecant of angle AA is cosec A\text{cosec } A (or csc⁡A\csc A).
Answer

(iii) False

(iv) cot⁡A\cot A is the product of cot⁡\cot and AA.

Step 1 · Examine Meaning of Trigonometric Notation

cot⁡A\cot A represents the cotangent of angle AA. It is a single trigonometric function and not a product of cot⁡\cot and AA. The term cot⁡\cot separated from AA has no mathematical meaning.

Answer

(iv) False

(v) sin⁡θ=43\sin \theta = \dfrac{4}{3} for some angle θ\theta.

Step 1 · Check Range of sin⁡θ\sin \theta

By definition sin⁡θ=Opposite sideHypotenuse\sin \theta = \dfrac{\text{Opposite side}}{\text{Hypotenuse}}

Since the hypotenuse is always the longest side in a right triangle, Opposite side≤Hypotenuse\text{Opposite side} \le \text{Hypotenuse}, which means sin⁡θ≤1\sin \theta \le 1.

Given value sin⁡θ=43≈1.33>1\sin \theta = \dfrac{4}{3} \approx 1.33 > 1

Since sin⁡θ\sin \theta cannot exceed 11, this value is impossible.

Answer

(v) False

Common Mistakes
  • Treating Trig Functions as Multiplication: Assuming cot⁡A=cot⁡×A\cot A = \cot \times A. Trigonometric functions operate on an angle; they are not multiplying variables.
  • Confusing Sine and Secant Bounds: For acute angles, sin⁡θ≤1\sin \theta \le 1 and cos⁡θ≤1\cos \theta \le 1 because the hypotenuse is in the denominator. Conversely, sec⁡θ≥1\sec \theta \ge 1 and csc⁡θ≥1\csc \theta \ge 1 because the hypotenuse is in the numerator.
  • Abbreviation Mix-up: Confusing cos⁡A\cos A (cosine) with cosec A\text{cosec } A (cosecant).

More questions in Exercise 8.1

Q1

In ΔABC\Delta \text{ABC}, right-angled at B, AB=24 cm\text{AB} = 24\text{ cm}, BC=7 cm\text{BC} = 7\text{ cm}. Determine :

(i) sin⁡A\sin A, cos⁡A\cos A

(ii) sin⁡C\sin C, cos⁡C\cos C

Q2

In Fig. 8.13, find tan⁡P−cot⁡R\tan P - \cot R.

Q3

If sin⁡A=34\sin A = \dfrac{3}{4}, calculate cos⁡A\cos A and tan⁡A\tan A.

Q4

Given 15cot⁡A=815 \cot A = 8, find sin⁡A\sin A and sec⁡A\sec A.

Q5

Given sec⁡θ=1312\sec \theta = \dfrac{13}{12}, calculate all other trigonometric ratios.

Q6

If ∠A\angle A and ∠B\angle B are acute angles such that cos⁡A=cos⁡B\cos A = \cos B, then show that ∠A=∠B\angle A = \angle B.

Q7

If cot⁡θ=78\cot \theta = \dfrac{7}{8}, evaluate :

(i) (1+sin⁡θ)(1−sin⁡θ)(1+cos⁡θ)(1−cos⁡θ)\dfrac{(1 + \sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(1 - \cos \theta)}

(ii) cot⁡2θ\cot^2 \theta

Q8

If 3cot⁡A=43 \cot A = 4, check whether 1−tan⁡2A1+tan⁡2A=cos⁡2A−sin⁡2A\dfrac{1 - \tan^2 A}{1 + \tan^2 A} = \cos^2 A - \sin^2 A or not.

Q9

In triangle ABC, right-angled at B, if tan⁡A=13\tan A = \dfrac{1}{\sqrt{3}}, find the value of:

(i) sin⁡Acos⁡C+cos⁡Asin⁡C\sin A \cos C + \cos A \sin C

(ii) cos⁡Acos⁡C−sin⁡Asin⁡C\cos A \cos C - \sin A \sin C

Q10

In ΔPQR\Delta \text{PQR}, right-angled at Q, PR+QR=25 cm\text{PR} + \text{QR} = 25\text{ cm} and PQ=5 cm\text{PQ} = 5\text{ cm}. Determine the values of sin⁡P\sin P, cos⁡P\cos P and tan⁡P\tan P.

Q11

State whether the following are true or false. Justify your answer.

(i) The value of tan⁡A\tan A is always less than 1.

(ii) sec⁡A=125\sec A = \dfrac{12}{5} for some value of angle A.

(iii) cos⁡A\cos A is the abbreviation used for the cosecant of angle A.

(iv) cot⁡A\cot A is the product of cot⁡\cot and AA.

(v) sin⁡θ=43\sin \theta = \dfrac{4}{3} for some angle θ\theta.

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