Consider the system of equations $$\left\{ \begin{array} { l }
\cos 2 \alpha + \cos 2 \beta = \frac { 4 } { 15 } \\
\cos \alpha \cos \beta = - \frac { 2 \sqrt { 15 } } { 15 }
\end{array} \right.$$ where $0 \leqq \alpha \leqq \pi , \quad 0 \leqq \beta \leqq \pi$, and $\alpha < \beta$ with $$| \cos \alpha | \geqq | \cos \beta |$$ We want to find the values of $\cos \alpha$ and $\cos \beta$. Using the double angle formula, from (1) we obtain $$\cos ^ { 2 } \alpha + \cos ^ { 2 } \beta = \frac { \text { アイ } } { \text { ウエ } }$$ Also, from (2), $\cos ^ { 2 } \alpha \cos ^ { 2 } \beta = \frac { \square } { 15 }$. Therefore, using condition (3), $$\cos ^ { 2 } \alpha = \frac { \boldsymbol { \text { カ } }} { \boldsymbol { y } } , \quad \cos ^ { 2 } \beta = \frac { \boldsymbol { \text { ク } }} { \boldsymbol { \text { ケ } } }$$ From (2) and the conditions $0 \leqq \alpha \leqq \pi , 0 \leqq \beta \leqq \pi , \alpha < \beta$, [2] On the coordinate plane, take point $\mathrm { A } \left( 0 , \frac { 3 } { 2 } \right)$, and on the graph of the function $y = \log _ { 2 } x$, take two points $\mathrm { B } \left( p , \log _ { 2 } p \right), \mathrm { C } \left( q , \log _ { 2 } q \right)$. When point C divides segment AB internally in the ratio $1 : 2$, we want to find the values of $p$ and $q$. By the condition on the domain of the logarithm, $p >$ タ, $q >$ タ. Here, for a logarithm $\log _ { a } b$, $a$ is called the base and $b$ is called the argument. The coordinates of the point that divides segment AB internally in the ratio $1 : 2$ are expressed in terms of $p$ as Since this coincides with the coordinates of C, holds. Equation (5) can be rewritten as $$p = \frac { \square } { \square } q \text { 衣 }$$ Solving the system of equations formed by (4) and (6), and noting that $p >$ □タ, $q >$ □タ, we have $$p = \square \sqrt { \square } , \quad q = \square \sqrt { \square }$$ The $y$-coordinate of C is $\log _ { 2 }$ (□ヒ $\sqrt { }$ □フ). When this value is rounded to one decimal place (rounding the second decimal place), we get □ヘ. Choose the correct answer for □ヘ from the following options (0) through (b). Here, $\log _ { 10 } 2 = 0.3010$ , $\log _ { 10 } 3 = 0.4771 , \log _ { 10 } 7 = 0.8451$. (0) 0.3 (1) 0.6 (2) 0.9 (3) 1.3 (4) 1.6 (5) 1.9 (6) 2.3 (7) 2.6 (8) 2.9 (9) 3.3 (a) 3.6 (b) 3.9
& 11 & 1
Consider the system of equations
$$\left\{ \begin{array} { l }
\cos 2 \alpha + \cos 2 \beta = \frac { 4 } { 15 } \\
\cos \alpha \cos \beta = - \frac { 2 \sqrt { 15 } } { 15 }
\end{array} \right.$$
where $0 \leqq \alpha \leqq \pi , \quad 0 \leqq \beta \leqq \pi$, and $\alpha < \beta$ with
$$| \cos \alpha | \geqq | \cos \beta |$$
We want to find the values of $\cos \alpha$ and $\cos \beta$.
Using the double angle formula, from (1) we obtain
$$\cos ^ { 2 } \alpha + \cos ^ { 2 } \beta = \frac { \text { アイ } } { \text { ウエ } }$$
Also, from (2), $\cos ^ { 2 } \alpha \cos ^ { 2 } \beta = \frac { \square } { 15 }$.
Therefore, using condition (3),
$$\cos ^ { 2 } \alpha = \frac { \boldsymbol { \text { カ } }} { \boldsymbol { y } } , \quad \cos ^ { 2 } \beta = \frac { \boldsymbol { \text { ク } }} { \boldsymbol { \text { ケ } } }$$
From (2) and the conditions $0 \leqq \alpha \leqq \pi , 0 \leqq \beta \leqq \pi , \alpha < \beta$,
[2] On the coordinate plane, take point $\mathrm { A } \left( 0 , \frac { 3 } { 2 } \right)$, and on the graph of the function $y = \log _ { 2 } x$, take two points $\mathrm { B } \left( p , \log _ { 2 } p \right), \mathrm { C } \left( q , \log _ { 2 } q \right)$. When point C divides segment AB internally in the ratio $1 : 2$, we want to find the values of $p$ and $q$.
By the condition on the domain of the logarithm, $p >$ タ, $q >$ タ. Here, for a logarithm $\log _ { a } b$, $a$ is called the base and $b$ is called the argument.
The coordinates of the point that divides segment AB internally in the ratio $1 : 2$ are expressed in terms of $p$ as
Since this coincides with the coordinates of C,
holds.
Equation (5) can be rewritten as
$$p = \frac { \square } { \square } q \text { 衣 }$$
Solving the system of equations formed by (4) and (6), and noting that $p >$ □タ, $q >$ □タ, we have
$$p = \square \sqrt { \square } , \quad q = \square \sqrt { \square }$$
The $y$-coordinate of C is $\log _ { 2 }$ (□ヒ $\sqrt { }$ □フ). When this value is rounded to one decimal place (rounding the second decimal place), we get □ヘ. Choose the correct answer for □ヘ from the following options (0) through (b). Here, $\log _ { 10 } 2 = 0.3010$ , $\log _ { 10 } 3 = 0.4771 , \log _ { 10 } 7 = 0.8451$.
(0) 0.3
(1) 0.6
(2) 0.9
(3) 1.3
(4) 1.6
(5) 1.9
(6) 2.3
(7) 2.6
(8) 2.9
(9) 3.3
(a) 3.6
(b) 3.9