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

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

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

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

(iv) cot A is the product of cot and A.

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

Asked by Pragya Singh | 1 year ago | 66

**(i) Right answer is False**

**Explanation:-**

The value of tan A is always less than 1.

Proof: In ΔMNC in which ∠N = 90∘,

MN = 3, NC = 4 and MC = 5

Value of tan M = \( \dfrac{4}{3}\) which is greater than.

The triangle can be formed with sides equal to 3, 4 and hypotenuse

= 5 as it will follow the Pythagoras theorem.

MC^{2}=MN^{2}+NC^{2}

5^{2}=3^{2}+4^{2}

25=9+16

25^{ }=^{ }25

**(ii)** Right answer is **True**

**Explanation:-**

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

Let a ΔMNC in which ∠N = 90º,

MC=12k and MB=5k, where k is a positive real number.

By Pythagoras theorem we get,

MC^{2}=MN^{2}+NC^{2}

(12k)^{2}=(5k)^{2}+NC^{2}

NC^{2}+25k^{2}=144k^{2}

NC^{2}=119k^{2}

Such a triangle is possible as it will follow the Pythagoras theorem.

**(iii)** Right answer is **False**

**Explanation:-**

cos A is the abbreviation used for the cosecant of angle A.

Abbreviation used for cosecant of angle M is cosec M.

cos M is the abbreviation used for cosine of angle M.

**(iv)** Right answer is **False**

**Explanation:-**

cot A is the product of cot and A.

cot M is not the product of cot and M. It is the cotangent of ∠M.

**(v)** Right answer is ** False**

**Explanation:-**

sin θ = \( \dfrac{4}{3}\) for some angle θ.

sin θ = \(\dfrac{Opposite}{Hypotenuse}\)

We know that in a right angled triangle, Hypotenuse is the longest side.

sin θ will always less than 1 and it can never be \( \dfrac{4}{3}\) for any value of θ.

Answered by Abhisek | 1 year agoProve the sinθ sin (90° – θ) – cos θ cos (90° – θ) = 0

Prove that sin 48° sec 48° + cos 48° cosec 42° = 2

Prove that tan 20° tan 35° tan 45° tan 55° tan 70° = 1

Evalute the following:

**(i) **\( \dfrac{sin 20°}{ cos 70°}\)

**(ii)** \(\dfrac{ cos 19°}{ sin 71°}\)

**(iii)** \( \dfrac{sin 21°}{ cos 69°}\)

**(iv) **\( \dfrac{tan 10°}{ cot 80°}\)

**(v) **\( \dfrac{sec 11°}{ cosec 79°}\)