Advertisements
Advertisements
Question
If 5 sec θ – 12 cosec θ = 0, then find values of sin θ, sec θ
Advertisements
Solution
5 sec θ – 12 cosec θ = 0 ......[Given]
∴ 5 sec θ = 12 cosec θ
∴ `5/costheta = 12/sintheta` ......`[because sectheta = 1/costheta, "cosec" theta = 1/sintheta]`
∴ `sintheta/costheta = 12/5`
∴ tan θ = `12/5`
We know that,
1 + tan2θ = sec2θ
∴ `1 + (12/5)^2` = sec2θ
∴ `1 + 144/25` = sec2θ
∴ `(25 + 144)/25` = sec2θ
∴ sec2θ = `169/25`
∴ secθ = `13/5` ......[Taking square root of both sides]
Now, cos θ = `1/sectheta`
= `1/((13/5))`
∴ cos θ = `5/13`
We know that,
sin2θ + cos2θ = 1
∴ `sin^2theta + (5/13)^2` = 1
∴ `sin^2theta + 25/169` = 1
∴ sec2θ = `1 - 25/169`
∴ sec2θ = `(169 - 25)/169`
∴ sec2θ = `144/169`
∴ sin θ = `12/13` ......[Taking square root of both sides]
∴ sin θ = `12/13`, sec θ = `13/5`.
APPEARS IN
RELATED QUESTIONS
`"If "\frac{\cos \alpha }{\cos \beta }=m\text{ and }\frac{\cos \alpha }{\sin \beta }=n " show that " (m^2 + n^2 ) cos^2 β = n^2`
Prove the following trigonometric identities.
`(cos theta)/(cosec theta + 1) + (cos theta)/(cosec theta - 1) = 2 tan theta`
If sin θ + cos θ = x, prove that `sin^6 theta + cos^6 theta = (4- 3(x^2 - 1)^2)/4`
Show that : tan 10° tan 15° tan 75° tan 80° = 1
`(sec^2 theta-1) cot ^2 theta=1`
`(cot ^theta)/((cosec theta+1)) + ((cosec theta + 1))/cot theta = 2 sec theta`
If tan A =` 5/12` , find the value of (sin A+ cos A) sec A.
If tanθ `= 3/4` then find the value of secθ.
The value of sin2 29° + sin2 61° is
Prove the following identity :
`(cotA + tanB)/(cotB + tanA) = cotAtanB`
Prove the following identity :
`1/(cosA + sinA - 1) + 2/(cosA + sinA + 1) = cosecA + secA`
If `x/(a cosθ) = y/(b sinθ) "and" (ax)/cosθ - (by)/sinθ = a^2 - b^2 , "prove that" x^2/a^2 + y^2/b^2 = 1`
Without using trigonometric identity , show that :
`sin(50^circ + θ) - cos(40^circ - θ) = 0`
Prove that:
tan (55° + x) = cot (35° – x)
Prove that ( 1 + tan A)2 + (1 - tan A)2 = 2 sec2A
If `(cos alpha)/(cos beta)` = m and `(cos alpha)/(sin beta)` = n, then prove that (m2 + n2) cos2 β = n2
Prove that `"cot A"/(1 - cot"A") + "tan A"/(1 - tan "A")` = – 1
Prove that
`(cot "A" + "cosec A" - 1)/(cot"A" - "cosec A" + 1) = (1 + cos "A")/"sin A"`
If 4 tanβ = 3, then `(4sinbeta-3cosbeta)/(4sinbeta+3cosbeta)=` ______.
Prove the following:
`1 + (cot^2 alpha)/(1 + "cosec" alpha)` = cosec α
