Answer :
Sure, let's go through the solutions step by step.
### Problem 2:
Find the value of [tex]\(\theta\)[/tex] when [tex]\(\tan \theta = \cot \theta\)[/tex].
To solve this, we need to recall the definitions of the tangent and cotangent functions:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
[tex]\[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{1}{\tan \theta} \][/tex]
Setting the two equal gives us:
[tex]\[ \tan \theta = \frac{1}{\tan \theta} \][/tex]
This results in:
[tex]\[ \tan^2 \theta = 1 \][/tex]
Taking the square root of both sides, we get:
[tex]\[ \tan \theta = \pm 1 \][/tex]
The values of [tex]\(\theta\)[/tex] that satisfy [tex]\(\tan \theta = 1\)[/tex] or [tex]\(\tan \theta = -1\)[/tex] within the standard range of [tex]\(0^\circ\)[/tex] to [tex]\(360^\circ\)[/tex] are:
[tex]\[ \theta = 45^\circ, 225^\circ \quad (\text{for } \tan \theta = 1) \][/tex]
[tex]\[ \theta = 135^\circ, 315^\circ \quad (\text{for } \tan \theta = -1) \][/tex]
However, since we are provided with specific options:
[tex]\[ (a) 0^\circ, (b) 30^\circ, (c) 45^\circ, (d) 90^\circ \][/tex]
Comparing these options, the only value that fits our solutions is:
[tex]\[ 45^\circ \][/tex]
Therefore, the correct answer is:
[tex]\[ (c) 45^\circ \][/tex]
### Problem 3:
Find the dot product [tex]\(\vec{a} \cdot \vec{b}\)[/tex] given that [tex]\(\vec{a} \cdot (3) = \vec{b} = \left(\begin{array}{c}-2 \\ \end{array}\right)\)[/tex].
Here, we are given that vector [tex]\(\vec{b}\)[/tex] is defined as a scalar multiplication of [tex]\(-2\)[/tex] with a vector. This can be interpreted as the vector formula:
[tex]\[ \vec{b} = 3 \vec{a} \rightarrow \vec{a} = \frac{\vec{b}}{3} \][/tex]
Therefore, [tex]\(\vec{a} \cdot \vec{b}\)[/tex] is calculated as:
[tex]\[ \vec{a} = \left(\begin{array}{c}\frac{-2}{3} \\ \end{array}\right) \][/tex]
[tex]\[ \vec{b} = \left(\begin{array}{c}-2 \\ \end{array}\right) \][/tex]
The dot product [tex]\(\vec{a} \cdot \vec{b}\)[/tex] is computed as:
[tex]\[ \vec{a} \cdot \vec{b} = \left(\frac{-2}{3}\right)(-2) = \frac{4}{3} \][/tex]
Therefore, the value of [tex]\(\vec{a} \cdot \vec{b}\)[/tex] is:
[tex]\[ -6 \][/tex]
### Problem 2:
Find the value of [tex]\(\theta\)[/tex] when [tex]\(\tan \theta = \cot \theta\)[/tex].
To solve this, we need to recall the definitions of the tangent and cotangent functions:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
[tex]\[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{1}{\tan \theta} \][/tex]
Setting the two equal gives us:
[tex]\[ \tan \theta = \frac{1}{\tan \theta} \][/tex]
This results in:
[tex]\[ \tan^2 \theta = 1 \][/tex]
Taking the square root of both sides, we get:
[tex]\[ \tan \theta = \pm 1 \][/tex]
The values of [tex]\(\theta\)[/tex] that satisfy [tex]\(\tan \theta = 1\)[/tex] or [tex]\(\tan \theta = -1\)[/tex] within the standard range of [tex]\(0^\circ\)[/tex] to [tex]\(360^\circ\)[/tex] are:
[tex]\[ \theta = 45^\circ, 225^\circ \quad (\text{for } \tan \theta = 1) \][/tex]
[tex]\[ \theta = 135^\circ, 315^\circ \quad (\text{for } \tan \theta = -1) \][/tex]
However, since we are provided with specific options:
[tex]\[ (a) 0^\circ, (b) 30^\circ, (c) 45^\circ, (d) 90^\circ \][/tex]
Comparing these options, the only value that fits our solutions is:
[tex]\[ 45^\circ \][/tex]
Therefore, the correct answer is:
[tex]\[ (c) 45^\circ \][/tex]
### Problem 3:
Find the dot product [tex]\(\vec{a} \cdot \vec{b}\)[/tex] given that [tex]\(\vec{a} \cdot (3) = \vec{b} = \left(\begin{array}{c}-2 \\ \end{array}\right)\)[/tex].
Here, we are given that vector [tex]\(\vec{b}\)[/tex] is defined as a scalar multiplication of [tex]\(-2\)[/tex] with a vector. This can be interpreted as the vector formula:
[tex]\[ \vec{b} = 3 \vec{a} \rightarrow \vec{a} = \frac{\vec{b}}{3} \][/tex]
Therefore, [tex]\(\vec{a} \cdot \vec{b}\)[/tex] is calculated as:
[tex]\[ \vec{a} = \left(\begin{array}{c}\frac{-2}{3} \\ \end{array}\right) \][/tex]
[tex]\[ \vec{b} = \left(\begin{array}{c}-2 \\ \end{array}\right) \][/tex]
The dot product [tex]\(\vec{a} \cdot \vec{b}\)[/tex] is computed as:
[tex]\[ \vec{a} \cdot \vec{b} = \left(\frac{-2}{3}\right)(-2) = \frac{4}{3} \][/tex]
Therefore, the value of [tex]\(\vec{a} \cdot \vec{b}\)[/tex] is:
[tex]\[ -6 \][/tex]
