Advertisements
Advertisements
Question
“The inequality `2^sintheta + 2^costheta ≥ 2^(1/sqrt(2))` holds for all real values of θ”
Options
True
False
Advertisements
Solution
This statement is True.
Explanation:
Since `2sin^theta` and `2^costheta` are positive real numbers, so A.M. (Arithmetic Mean) of these two numbers is greater or equal to their G.M. (Geometric Mean)
Hence `(2^sintheta + 2^costheta)/2 ≥ sqrt(2^sintheta xx 2^costheta)`
= `sqrt(2^(sintheta + costheta))`
`≥ 2 ^((sintheta + costheta)/2) = 2^(1/sqrt(2)(1/sqrt(2) sintheta + 1/sqrt(2) cos theta))`
`≥ 2^(1/sqrt(2) sin(pi/4 + theta))`
Since, `-1 ≤ sin(pi/4 + theta) ≤ 1`
We have `(2^sintheta + 2^costheta)/2 ≥ 2^((-1)/sqrt(2))`
⇒ `2^sintheta + 2^costheta ≥ 2^(1 - 1/sqrt(2))`
APPEARS IN
RELATED QUESTIONS
Find the radian measure corresponding to the following degree measure:
25°
Find the radian measure corresponding to the following degree measure:
520°
Find the degree measure corresponding to the following radian measure (Use `pi = 22/7`)
-4
In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.
Find the angle in radian through which a pendulum swings if its length is 75 cm and the tip describes an arc of length
21 cm
Find the degree measure corresponding to the following radian measure:
\[\frac{9\pi}{5}\]
Find the degree measure corresponding to the following radian measure:
\[\left( \frac{18\pi}{5} \right)\]
Find the degree measure corresponding to the following radian measure:
(−3)c
Find the degree measure corresponding to the following radian measure:
1c
Find the radian measure corresponding to the following degree measure:
300°
Find the radian measure corresponding to the following degree measure: 35°
Find the radian measure corresponding to the following degree measure: −56°
The difference between the two acute angles of a right-angled triangle is \[\frac{2\pi}{5}\] radians. Express the angles in degrees.
One angle of a triangle \[\frac{2}{3}\] x grades and another is \[\frac{3}{2}\] x degrees while the third is \[\frac{\pi x}{75}\] radians. Express all the angles in degrees.
Find the magnitude, in radians and degrees, of the interior angle of a regular octagon.
Find the magnitude, in radians and degrees, of the interior angle of a regular heptagon.
Find the magnitude, in radians and degrees, of the interior angle of a regular duodecagon.
The angles of a triangle are in A.P. and the number of degrees in the least angle is to the number of degrees in the mean angle as 1 : 120. Find the angles in radians.
Find the length which at a distance of 5280 m will subtend an angle of 1' at the eye.
A railway train is travelling on a circular curve of 1500 metres radius at the rate of 66 km/hr. Through what angle has it turned in 10 seconds?
Find the distance from the eye at which a coin of 2 cm diameter should be held so as to conceal the full moon whose angular diameter is 31'.
Find the diameter of the sun in km supposing that it subtends an angle of 32' at the eye of an observer. Given that the distance of the sun is 91 × 106 km.
Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm.
If the angles of a triangle are in A.P., then the measures of one of the angles in radians is
If the arcs of the same length in two circles subtend angles 65° and 110° at the centre, than the ratio of the radii of the circles is
A circular wire of radius 7 cm is cut and bent again into an arc of a circle of radius 12 cm. The angle subtended by the arc at the centre is
Find the value of `sqrt(3)` cosec 20° – sec 20°
If θ lies in the second quadrant, then show that `sqrt((1 - sin theta)/(1 + sin theta)) + sqrt((1 + sin theta)/(1 - sin theta))` = −2sec θ
Find the value of tan 9° – tan 27° – tan 63° + tan 81°
If tan θ = `(-4)/3`, then sin θ is ______.
The value of tan1° tan2° tan3° ... tan89° is ______.
