English

If sec θ = 13/5, show that (2 sin θ – 3 cos θ)/(4 sin θ – 9 cosθ) = 3.

Advertisements
Advertisements

Question

If `sec θ = 13/5`, show that `(2 sin θ - 3 cos θ)/(4 sin θ - 9 cos θ) = 3`.

Sum
Advertisements

Solution 1


Given: `sec θ = 13/5`

We know that,

sec θ = `"Hypotenuse"/"Adjacent Side"`

sec θ = `13/5 = "AC"/"BC"`

Let AC = 13k and BC = 5k

In ΔABC, ∠B = 90°

By Pythagoras theorem,

AC2 = AB2 + BC2

(13k)2 = AB2 + (5k)2

AB2 = 169k2 – 25k2

AB2 = 144k2

AB = 12k

sin θ = `"AB"/"AC" = "12k"/"13k" = 12/13`

cos θ = `"BC"/"AC" = "5k"/"13k" = 5/13`

LHS = `(2 sin θ - 3 cos θ)/(4 sin θ - 9 cos θ)`

LHS = `[2 × (12/13) - 3 × (5/13)]/[4 × (12/13) - 9 × (5/13)]`

LHS = `[24/13 - 15/13]/[48/13 + 45/13]`

LHS = `[9/13]/[3/13]`

LHS = `9/(cancel13) × cancel13/3`

LHS = `9/3`  

LHS = 3

RHS = 3

LHS = RHS

`(2 sin θ - 3 cos θ)/(4 sin θ - 9 cos θ) = 3`

Hence proved.

shaalaa.com

Solution 2

Given: sec θ = `13/5`

cos θ = `1/secθ = 5/13`

sin2θ = 1 – cos2θ

sin2θ = `1 - (5/13)^2`

sin2θ = `1 - 25/169`

sin2θ = `(169 − 25)/169`

sin2θ = `144/169`

sin θ = `12/13`

Now, put the values in the equation,

LHS = `(2 sin θ - 3 cos θ)/(4 sin θ - 9 cos θ)`

LHS = `(2 × (12/13) - 3 × (5/13))/(4 × (12/13) - 9 × (5/13))`

LHS = `(24/13 - 15/13)/(48/13 - 45/13)`

LHS = `((24- 15)/cancel13)/((48 - 45)/cancel13)`

LHS = `9/3`

LHS = 3

RHS = 3

LHS = RHS

`(2 sin θ - 3 cos θ)/(4 sin θ - 9 cos θ) = 3`

Hence proved.

shaalaa.com
  Is there an error in this question or solution?
Chapter 17: Trigonometric Ratios - Exercise 17A [Page 360]

APPEARS IN

Nootan Mathematics [English] Class 9 ICSE
Chapter 17 Trigonometric Ratios
Exercise 17A | Q 18. | Page 360
R.D. Sharma Mathematics [English] Class 10
Chapter 10 Trigonometric Ratios
Exercise 10.1 | Q 13 | Page 24

Video TutorialsVIEW ALL [2]

RELATED QUESTIONS

In ΔABC right angled at B, AB = 24 cm, BC = 7 m. Determine:

sin C, cos C


If sin A = `3/4`, calculate cos A and tan A.


If cot θ = `7/8` evaluate `((1+sin θ )(1-sin θ))/((1+cos θ)(1-cos θ))`.


In ΔPQR, right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.


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

The value of tan A is always less than 1.


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

sec A = `12/5` for some value of angle A.


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

cot A is the product of cot and A.


If 4 tan θ = 3, evaluate `((4sin theta - cos theta + 1)/(4sin theta + cos theta - 1))`


In the following, trigonometric ratios are given. Find the values of the other trigonometric ratios.

`sin A = 2/3`


In the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.

`tan alpha = 5/12`


If `tan theta = a/b`, find the value of `(cos theta + sin theta)/(cos theta - sin theta)`


If `tan theta = 1/sqrt7`     `(cosec^2 theta - sec^2 theta)/(cosec^2 theta + sec^2 theta) = 3/4`


If sin θ = `12/13`, Find `(sin^2 θ - cos^2 θ)/(2sin θ cos θ) × 1/(tan^2 θ)`.


if `cos theta = 3/5`, find the value of `(sin theta - 1/(tan theta))/(2 tan theta)`


if `cot theta = 3/4` prove that `sqrt((sec theta - cosec theta)/(sec theta +cosec theta)) = 1/sqrt7`


If `tan theta = 24/7`, find that sin θ + cos θ.


Evaluate the following

cos 60° cos 45° - sin 60° ∙ sin 45°


Evaluate the following:

(cosec2 45° sec2 30°)(sin2 30° + 4 cot2 45° − sec2 60°)


Evaluate the Following

`cot^2 30^@ - 2 cos^2 60^circ- 3/4 sec^2 45^@ - 4 sec^2 30^@`


Evaluate the Following

4(sin4 30° + cos2 60°) − 3(cos2 45° − sin2 90°) − sin2 60°


Find the value of x in the following :

`2 sin  x/2 = 1`


Find the value of x in the following :

`sqrt3 sin x = cos x`


`(1 + tan^2 "A")/(1 + cot^2 "A")` is equal to ______.


If sin A = `1/2`, then the value of cot A is ______.


The value of the expression (sin 80° – cos 80°) is negative.


Prove that: cot θ + tan θ = cosec θ·sec θ

Proof: L.H.S. = cot θ + tan θ

= `square/square + square/square`  ......`[∵ cot θ = square/square, tan θ = square/square]`

= `(square + square)/(square xx square)`  .....`[∵ square + square = 1]`

= `1/(square xx square)`

= `1/square xx 1/square`

= cosec θ·sec θ  ......`[∵ "cosec"  θ = 1/square, sec θ = 1/square]`

= R.H.S.

∴ L.H.S. = R.H.S.

∴ cot θ + tan θ = cosec·sec θ


If f(x) = `3cos(x + (5π)/6) - 5sinx + 2`, then maximum value of f(x) is ______.


If `θ∈[(5π)/2, 3π]` and 2cosθ + sinθ = 1, then the value of 7cosθ + 6sinθ is ______.


Evaluate: 5 cosec2 45° – 3 sin2 90° + 5 cos 0°.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×