Advertisements
Advertisements
Question
If `cot theta = 1/sqrt3` show that `(1 - cos^2 theta)/(2 - sin^2 theta) = 3/5`
Advertisements
Solution
`cot theta = 1/sqrt3 (1 - cos^2 theta)/(2 - sin^2 theta) = 3/5`
`cot theta = "๐๐๐๐๐๐๐๐ก ๐ ๐๐๐"/"๐๐๐๐๐ ๐๐ก๐ ๐ ๐๐๐" = 1/sqrt3`
Let x be the hypotenuse
By applying Pythagoras
๐ด๐ถ2 = ๐ด๐ต2 + ๐ต๐ถ2
`x^2 = (sqrt3)^2 + 1`
`x^2 = 3 + 1`
๐ฅ2 = 3 + 1 ⇒ ๐ฅ = 2
`cos theta = (BC)/(AC) = 1/2`
`sin theta = (AB)/(AC) = sqrt3/2`
`(1 - cos^2 theta)/(2 - sin^2 theta) => (1 - (1/2)^2)/(2 - (sqrt3)/2)^2`
`=> (1 - 1/4)/(2 - 3/4) => (3/4)/(5/4`
`= 3/5`
APPEARS IN
RELATED QUESTIONS
In ΔABC right angled at B, AB = 24 cm, BC = 7 m. Determine:
sin C, cos C
In ΔABC, right angled at B. If tan A = `1/sqrt3` , find the value of
- sin A cos C + cos A sin C
- cos A cos C − sin A sin C
State whether the following are true or false. Justify your answer.
The value of tan A is always less than 1.
In the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.
`cot theta = 12/5`
if `cos theta = 3/5`, find the value of `(sin theta - 1/(tan theta))/(2 tan theta)`
Evaluate the following
sin2 30° + sin2 45° + sin2 60° + sin2 90°
`(1 + tan^2 "A")/(1 + cot^2 "A")` is equal to ______.
If cos A = `4/5`, then the value of tan A is ______.
Prove that sec θ + tan θ = `cos θ/(1 - sin θ)`.
Proof: L.H.S. = sec θ + tan θ
= `1/square + square/square`
= `square/square` ......`(โต sec θ = 1/square, tan θ = square/square)`
= `((1 + sin θ) square)/(cos θ square)` ......[Multiplying `square` with the numerator and denominator]
= `(1^2 - square)/(cos θ square)`
= `square/(cos θ square)`
= `cos θ/(1 - sin θ)` = R.H.S.
∴ L.H.S. = R.H.S.
∴ sec θ + tan θ = `cos θ/(1 - sin θ)`
Find the value of sin 45° + cos 45° + tan 45°.
If sec θ = `1/2`, what will be the value of cos θ?
Prove that `tan θ/(1 - cot θ) + cot θ/(1 - tanθ)` = 1 + sec θ cosec θ
If cos(α + β) = `(3/5)`, sin(α – β) = `5/13` and 0 < α, β < `π/4`, then tan (2α) is equal to ______.
Let f(x) = sinx.cos3x and g(x) = cosx.sin3x, then the value of `7((f(π/7) + g(π/7))/(g((5π)/14) + f((5π)/14)))` is ______.
Let tan9° = `(1 - sqrt((sqrt(5)k)/m))k` where k = `sqrt(5) + 1` then m is equal to ______.
The maximum value of the expression 5cosα + 12sinα – 8 is equal to ______.
In a right triangle PQR, right angled at Q. If tan P = `sqrt(3)`, then evaluate 2 sin P cos P.
