If x is a real quantity, what is the minimum value of (16 cos 2x + 9 sec 2 x)?
24
The correct answer is 24.
For positive y, y = \sqrt{b/a}. Substituting this back into the original expression: a\sqrt{b/a} + \frac{b}{\sqrt{b/a}} = a\sqrt{\frac{b}{a}} + b\sqrt{\frac{a}{b}} = \sqrt{a^2 \frac{b}{a}} + \sqrt{b^2 \frac{a}{b}} = \sqrt{ab} + \sqrt{ab} = 2\sqrt{ab}. In our case, a=16 and b=9, so the minimum value is 2\sqrt{16 \cdot 9} = 2\sqrt{144} = 2 \cdot 12 = 24. This confirms the AM-GM result and shows the two methods are consistent.
If θ = 120° and p = cos θ + sin θ, then :
If sin θ sec 2 θ = 2/3, 0° < θ < 90°, then the value of (tan 2 θ + sin 2 θ) is:
What is the value of \(\frac{{tan9^\circ tan23^\circ tan60^\circ tan67^\circ tan81^\circ }}{{cosec{^2}72^\circ + {{\cos }^2}15^\circ - {{\tan }^2}18^\circ + {{\cos }^2}75^\circ }} \)
If 8 cos 2 θ - 2 cos θ - 3 = 0, 0° < θ < 90° , then what is the value of 3 tan θ + cosec θ + cot θ ?
The value of \(\frac{{(\tan 25^\circ \cot 65^\circ - cose{c^2}\;65^\circ ) + co{t^2}\;61^\circ - se{c^2}\;29^\circ }}{{si{n^2}\;5^\circ + si{n^2}\;7^\circ + si{n^2}\;9^\circ + \cdots + si{n^2}\;85^\circ }}\) is:
If 7 cos 2 θ + 5 sin 2 θ - 6 = 0, (0° < θ < 90°), then what is the value of \(\sqrt {\frac{{\text{cosec} \ θ\ +\ \cot θ }}{{\text{cosec} \ θ\ -\ \cot θ }}}\) ?
If cosec A = \(\frac {5}{4}\) , then what is the value of tan A ?
If 14 sin 2 θ + 3 sin θ - 5 = 0, 0° < θ < 90°, then the value of \(\frac{\sec 2\theta + \cot 2\theta}{\rm cosec 2\theta + \tan 2\theta }\) is:
If \(\frac{{\sec θ + \tan θ }}{{\sec θ - \tan θ }} = \frac{5}{3}\) , 0° < θ < 90°, then what is the value of (cosec θ + cos θ + cot θ)?
If 6 cos 2 θ + cos θ = 2, 0° < θ < 90°, then the value of (cosec θ + cot θ + tan θ) is:
(secθ + tanθ)/(secθ - tanθ) is equal to:
If tan 45°, cot θ then the value of θ, in radians is
ABC is a triangle If sin (A+B)/2 = √3/2, then the value of sin C/2 is
The angles of elevation of the top of a temple, from the foot and the top of a building 30 m high, are 60° and 30° respectively. Then height of the temple is
The value of sin θ and cos θ when the angle θ is 0 ≤ θ ≤ 90°, is ______