Express the ratios cos A, tan A and sec A in terms of sin A. - Mathematics

Advertisements
Advertisements
Sum

Express the ratios cos A, tan A and sec A in terms of sin A.

Advertisements

Solution

Since cos2A + sin2A = 1, therefore,

cos2A = 1 – sin2A, i.e., cos A = ` \pm \sqrt{1-\sin ^{2}A`

This gives

`cos A = \sqrt{1-\sin^{2}A) `

Hence,

`\tan A=\frac{\sin A}{\cos A}=\frac{\sin A}{\sqrt{1-\sin^{2}}A}\text{ and}`

`\sec A=\frac{1}{\cos A}=\frac{1}{\sqrt{1-\sin ^{2}A}}`

  Is there an error in this question or solution?

RELATED QUESTIONS

Prove that:

sec2θ + cosec2θ = sec2θ x cosec2θ


Prove the following trigonometric identities.

`sin theta/(1 - cos theta) =  cosec theta + cot theta`


Prove the following trigonometric identities.

`tan^2 theta - sin^2 theta tan^2 theta sin^2 theta`


Prove the following trigonometric identities.

`(1 + sin theta)/cos theta + cos theta/(1 + sin theta) = 2 sec theta`


Prove the following trigonometric identities

`((1 + sin theta)^2 + (1 + sin theta)^2)/(2cos^2 theta) =  (1 + sin^2 theta)/(1 - sin^2 theta)`


Prove the following trigonometric identities.

`cos A/(1 - tan A) + sin A/(1 - cot A)  = sin A + cos A`


Prove the following trigonometric identities.

`1/(sec A + tan A) - 1/cos A = 1/cos A - 1/(sec A - tan A)`


Prove the following trigonometric identities.

`(1/(sec^2 theta - cos theta) + 1/(cosec^2 theta - sin^2 theta)) sin^2 theta cos^2 theta = (1 - sin^2 theta cos^2 theta)/(2 + sin^2 theta + cos^2 theta)`


Prove the following trigonometric identities.

`((1 + sin theta - cos theta)/(1 + sin theta + cos theta))^2 = (1 - cos theta)/(1 + cos theta)`


Prove the following trigonometric identities.

(sec A + tan A − 1) (sec A − tan A + 1) = 2 tan A


Prove the following trigonometric identities.

`tan A/(1 + tan^2  A)^2 + cot A/((1 + cot^2 A)) = sin A  cos A`


Prove.
`cosecA+cotA=1/(cosecA-cotA)`


Prove that

`1/(sinA-cosA)-1/(sinA+cosA)=(2cosA)/(2sin^2A-1)`


`cot^2 theta - 1/(sin^2 theta ) = -1`a


`1+((tan^2 theta) cot theta)/(cosec^2 theta) = tan theta`


` (sin theta - cos theta) / ( sin theta + cos theta ) + ( sin theta + cos theta ) / ( sin theta - cos theta ) = 2/ ((2 sin^2 theta -1))`


Write the value of `(sin^2 theta 1/(1+tan^2 theta))`. 


If 5x = sec ` theta and 5/x = tan theta , " find the value of 5 "( x^2 - 1/( x^2))`


What is the value of (1 + cot2 θ) sin2 θ?


If \[\cos A = \frac{7}{25}\]  find the value of tan A + cot A. 


If sin2 θ cos2 θ (1 + tan2 θ) (1 + cot2 θ) = λ, then find the value of λ. 


Prove the following identity :

 ( 1 + cotθ - cosecθ) ( 1 + tanθ + secθ) 


Prove the following identity : 

`sqrt((secq - 1)/(secq + 1)) + sqrt((secq + 1)/(secq - 1))` = 2 cosesq


Prove the following identity : 

`(sinA + cosA)/(sinA - cosA) + (sinA - cosA)/(sinA + cosA) = 2/(2sin^2A - 1)`


Prove the following identity : 

`(1 + tan^2A) + (1 + 1/tan^2A) = 1/(sin^2A - sin^4A)`


If tan θ = 2, where θ is an acute angle, find the value of cos θ. 


Prove that `sqrt((1 + sin A)/(1 - sin A))` = sec A + tan A. 


If cosθ = `5/13`, then find sinθ. 


Verify that the points A(–2, 2), B(2, 2) and C(2, 7) are the vertices of a right-angled triangle. 


If sec θ + tan θ = m, show that `(m^2 - 1)/(m^2 + 1) = sin theta`


Prove that `(cot "A" + "cosec A" - 1)/(cot "A" - "cosec A" + 1) = (1 + cos "A")/sin "A"`


Prove that the following identities:
Sec A( 1 + sin A)( sec A - tan A) = 1.


Prove the following identities.

`sqrt((1 + sin theta)/(1 - sin theta)` = sec θ + tan θ


Prove the following identities.

`sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta))` = 2 sec θ


Prove the following identities.

sec6 θ = tan6 θ + 3 tan2 θ sec2 θ + 1


If sin θ + cos θ = `sqrt(3)`, then prove that tan θ + cot θ = 1


If cot θ + tan θ = x and sec θ – cos θ = y, then prove that `(x^2y)^(2/3) – (xy^2)^(2/3)` = 1


If `cos theta/(1 + sin theta) = 1/"a"`, then prove that `("a"^2 - 1)/("a"^2 + 1)` = sin θ


If x = a tan θ and y = b sec θ then


a cot θ + b cosec θ = p and b cot θ + a cosec θ = q then p2 – q2 is equal to


Prove that `cot^2 "A" [(sec "A" - 1)/(1 + sin "A")] + sec^2 "A" [(sin"A" - 1)/(1 + sec"A")]` = 0


Prove that `(tan^2 theta - 1)/(tan^2 theta + 1)` = 1 – 2 cos2θ


If sec θ = `25/7`, find the value of tan θ.

Solution:

1 + tan2 θ = sec2 θ

∴ 1 + tan2 θ = `(25/7)^square`

∴ tan2 θ = `625/49 - square`

= `(625 - 49)/49`

= `square/49`

∴ tan θ = `square/7` ........(by taking square roots)


If sin θ + sin2 θ = 1 show that: cos2 θ + cos4 θ = 1


Choose the correct alternative:

cos θ. sec θ = ?


Choose the correct alternative:

1 + cot2θ = ? 


Choose the correct alternative:

sec2θ – tan2θ =?


If 1 – cos2θ = `1/4`, then θ = ?


Prove that `(cos(90 - "A"))/(sin "A") = (sin(90 - "A"))/(cos "A")`


If 3 sin θ = 4 cos θ, then sec θ = ?


Prove that sec2θ − cos2θ = tan2θ + sin2θ


If tan θ = `7/24`, then to find value of cos θ complete the activity given below.

Activity:

sec2θ = 1 + `square`    ......[Fundamental tri. identity]

sec2θ = 1 + `square^2`

sec2θ = 1 + `square/576`

sec2θ = `square/576`

sec θ = `square` 

cos θ = `square`     .......`[cos theta = 1/sectheta]`


If sec θ = `41/40`, then find values of sin θ, cot θ, cosec θ


Prove that `(tan(90 - theta) + cot(90 - theta))/("cosec"  theta)` = sec θ


Prove that `(1 + sintheta)/(1 - sin theta)` = (sec θ + tan θ)2 


Prove that 2(sin6A + cos6A) – 3(sin4A + cos4A) + 1 = 0


If cosec A – sin A = p and sec A – cos A = q, then prove that `("p"^2"q")^(2/3) + ("pq"^2)^(2/3)` = 1


If tan α + cot α = 2, then tan20α + cot20α = ______.


tan θ × `sqrt(1 - sin^2 θ)` is equal to:


Let x1, x2, x3 be the solutions of `tan^-1((2x + 1)/(x + 1)) + tan^-1((2x - 1)/(x - 1))` = 2tan–1(x + 1) where x1 < x2 < x3 then 2x1 + x2 + x32 is equal to ______.


If 2 cos θ + sin θ = `1(θ ≠ π/2)`, then 7 cos θ + 6 sin θ is equal to ______.


Let α, β be such that π < α – β < 3π. If sin α + sin β = `-21/65` and cos α + cos β = `-27/65`, then the value of `cos  (α - β)/2` is ______.


Share
Notifications



      Forgot password?
Use app×