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Range of Trigonometric Expressions

beginvmatrixs...

Question

$∣ ∣ ∣ ∣ ∣ \begin{matrix} s i n^{2} 23 & s i n^{2} 67 & c o s 180 s i n^{2} 67 & s i n^{2} 23 & c o s^{2} 180 c o s 180 & s i n^{2} 23 & s i n^{2} 67 \end{matrix} ∣ ∣ ∣ ∣ ∣$

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Solution

$∣ ∣ ∣ ∣ ∣ \begin{matrix} s i n^{2} 23 & s i n^{2} 67 & c o s 180 s i n^{2} 67 & s i n^{2} 23 & c o s^{2} 180 c o s 180 & s i n^{2} 23 & s i n^{2} 67 \end{matrix} ∣ ∣ ∣ ∣ ∣$
cos 180=1
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} s i n^{2} 23 & s i n^{2} (90 - 23) & - 1 s i n^{2} 67 & s i n^{2} (90 - 67) & (- 1)^{2} - 1 & s i n^{2} 23 & s i n^{2} (90 - 23) \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} s i n^{2} 23 & c o s^{2} 23 & - 1 s i n^{2} 67 & c o s^{2} 67 & 1 - 1 & s i n^{2} 23 & c o s^{2} 23 \end{matrix} ∣ ∣ ∣ ∣ ∣ C_{1} \to C_{1} + C_{2} + C_{3}$
= $∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 - 1 & C O S^{2} 23 & - 1 2 & c o s^{2} 67 & 1 - 1 + 1 & s i n^{2} 23 & c o s^{2} 23 \end{matrix} ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 0 & c o s^{2} 23 & - 1 2 & c o s^{2} 67 & 1 0 & s i n^{2} 23 & c o s^{2} 23 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= - c o s^{2} 23 (2 c o s^{2} 23 - 0) - 1 (2 s i n^{2} 23) = - 2 c o s^{4} 23 - 2 s i n^{2} 23 = - 2 [c o s^{4} 23 + s i n^{2} 23]$

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Similar questions

\frac{(\sin^{2} 23 ° + \sin^{2} 67 °)}{(\cos^{2} 13 ° + \cos^{2} 77 °)} + \sin^{2} 59 ° + \cos 59 ° \sin 21 ° = ?

(a) 3
(b) 2
(c) 1
(d) 0

Q. Say True or False
The value of the expression

({cos}^{2} 23^{\circ} - {sin}^{2} 67^{\circ})

\frac{\sec^{2} θ - \cot^{2} (90 ° - θ)}{{cosec}^{2} 67 ° - \tan^{2} 23 °} + (\sin^{2} 40 ° + \sin^{2} 50 °)

Q. Without expanding, show that the values of each of the following determinants are zero:
(i)

|\begin{matrix} 8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3 \end{matrix}|

|\begin{matrix} 6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2 \end{matrix}|

|\begin{matrix} 2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12 \end{matrix}|

|\begin{matrix} 1 / a & a^{2} & b c \\ 1 / b & b^{2} & a c \\ 1 / c & c^{2} & a b \end{matrix}|

|\begin{matrix} a + b & 2 a + b & 3 a + b \\ 2 a + b & 3 a + b & 4 a + b \\ 4 a + b & 5 a + b & 6 a + b \end{matrix}|

|\begin{matrix} 1 & a & a^{2} - b c \\ 1 & b & b^{2} - a c \\ 1 & c & c^{2} - a b \end{matrix}|

|\begin{matrix} 49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3 \end{matrix}|

|\begin{matrix} 0 & x & y \\ - x & 0 & z \\ - y & - z & 0 \end{matrix}|

|\begin{matrix} 1 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2 \end{matrix}|

|\begin{matrix} 1^{2} & 2^{2} & 3^{2} & 4^{2} \\ 2^{2} & 3^{2} & 4^{2} & 5^{2} \\ 3^{2} & 4^{2} & 5^{2} & 6^{2} \\ 4^{2} & 5^{2} & 6^{2} & 7^{2} \end{matrix}|

|\begin{matrix} a & b & c \\ a + 2 x & b + 2 y & c + 2 z \\ x & y & z \end{matrix}|

|\begin{matrix} {(2^{x} + 2^{- x})}^{2} & {(2^{x} - 2^{- x})}^{2} & 1 \\ {(3^{x} + 3^{- x})}^{2} & {(3^{x} - 3^{- x})}^{2} & 1 \\ {(4^{x} + 4^{- x})}^{2} & {(4^{x} - 4^{- x})}^{2} & 1 \end{matrix}|

|\begin{matrix} \sin α & \cos α & \cos (α + δ) \\ \sin β & \cos β & \cos (β + δ) \\ \sin γ & \cos γ & \cos (γ + δ) \end{matrix}|

|\begin{matrix} \sin^{2} 23 ° & \sin^{2} 67 ° & \cos 180 ° \\ - \sin^{2} 67 ° & - \sin^{2} 23 ° & \cos^{2} 180 ° \\ \cos 180 ° & \sin^{2} 23 ° & \sin^{2} 67 ° \end{matrix}|

|\begin{matrix} \cos (x + y) & - \sin (x + y) & \cos 2 y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y \end{matrix}|

|\begin{matrix} \sqrt{23} + \sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} + \sqrt{46} & 5 & \sqrt{10} \\ 3 + \sqrt{115} & \sqrt{15} & 5 \end{matrix}|

|\begin{matrix} \sin^{2} A & \cot A & 1 \\ \sin^{2} B & \cot B & 1 \\ \sin^{2} C & \cot C & 1 \end{matrix}|, where A, B, C are the angles of ∆ A B C .

{sin}^{2} 66 \frac{1^{o}}{2} - {sin}^{2} 23 \frac{1^{o}}{2}

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