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Find the mid-point (M) of AB and find cos theta in the form a*sqrt(b), given sin theta = 1/sqrt(5) and theta is acute.

Mathematics

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2 parts
9

(−14)\begin{pmatrix} \text{} -1 \text{} \\ \text{} 4 \text{} \end{pmatrix}

a

255\text{}\frac{2}{5}\sqrt{5}\text{}

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Part 9· 2 steps

Find the mid-point (M) of AB.

To find the position vector of the mid-point M of a line segment AB, we use the formula OM⃗=12(OA⃗+OB⃗)\vec{OM} = \frac{1}{2}(\vec{OA} + \vec{OB}).

  1. Add the position vectors OA⃗\vec{OA} and OB⃗\vec{OB}.

    OA⃗+OB⃗=(23)+(−45)=(2+(−4)3+5)=(−28)\vec{OA} + \vec{OB} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} -4 \\ 5 \end{pmatrix} = \begin{pmatrix} 2 + (-4) \\ 3 + 5 \end{pmatrix} = \begin{pmatrix} -2 \\ 8 \end{pmatrix}

  2. Divide the resulting vector by 2 to find the mid-point M.

    OM⃗=12(−28)=(−2282)=(−14)\vec{OM} = \frac{1}{2} \begin{pmatrix} -2 \\ 8 \end{pmatrix} = \begin{pmatrix} \frac{-2}{2} \\ \frac{8}{2} \end{pmatrix} = \boxed{\begin{pmatrix} \text{} -1 \text{} \\ \text{} 4 \text{} \end{pmatrix}}

Part a

255\text{}\frac{2}{5}\sqrt{5}\text{}

Find cos⁡θ\cos \theta in the form aba\sqrt{b}.

Given sin⁡θ=15\sin \theta = \frac{1}{\sqrt{5}} and θ\theta is an acute angle.

  1. Use the trigonometric identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.

    Since θ\theta is acute, cos⁡θ\cos \theta will be positive.

    cos⁡2θ=1−sin⁡2θ\cos^2 \theta = 1 - \sin^2 \theta cos⁡2θ=1−(15)2\cos^2 \theta = 1 - \left(\frac{1}{\sqrt{5}}\right)^2 cos⁡2θ=1−15\cos^2 \theta = 1 - \frac{1}{5}

  2. Calculate the value of cos⁡2θ\cos^2 \theta.

    cos⁡2θ=55−15=45\cos^2 \theta = \frac{5}{5} - \frac{1}{5} = \frac{4}{5}

  3. Take the square root to find cos⁡θ\cos \theta.

    cos⁡θ=45=45=25\cos \theta = \sqrt{\frac{4}{5}} = \frac{\sqrt{4}}{\sqrt{5}} = \frac{2}{\sqrt{5}}

  4. Rationalize the denominator to express in the form aba\sqrt{b}.

    cos⁡θ=25×55=255\cos \theta = \frac{2}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{5}}{5} cos⁡θ=255\cos \theta = \boxed{\text{}\frac{2}{5}\sqrt{5}\text{}}
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