Question

Find the value of 1+cos theta

Answer 1

The value of 1+cosΘ is 2sin²(Θ/2).

Solution,

1+cosΘ can be written as  = 1 + cos2(Θ/2).

• We know that 1 + cos2Θ = 1 - (cos²Θ-sin²Θ)

Similarly,

⇒      1 + cos2(Θ/2)= 1-(cos²(Θ/2)-sin²(Θ/2)

• Opening the brackets and changing the sign we get,

⇒    1 + cos2(Θ/2)= 1- cos²(Θ/2) +sin²(Θ/2).

• As we know that 1 - cos²Θ = sin²Θ

⇒    1 + cos2(Θ/2)= 2sin²(Θ/2).

Hence, 1+cos(Θ)=2sin²(Θ/2).

Answer 2

Answer:

Concept:

Concepts of Trigonometry in Basic Mathematics..

Explanation:

                           $1+\cos (\theta)=2 \cos ^{2}\left(\frac{\theta}{2}\right)$

Let us revise the basic trigonometry formulas we have

$\begin{gathered}\sin ^{2} \alpha+\cos ^{2} \alpha=1 \\\cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta\end{gathered}$

In the 2nd formula $\alpha=\beta=\theta / 2$ , then

$\begin{aligned}\cos \theta &=\cos \frac{\theta}{2} \cos \frac{\theta}{2}-\sin \frac{\theta}{2} \sin \frac{\theta}{2} \\&=\cos ^{2} \frac{\theta}{2}-\sin ^{2} \frac{\theta}{2} \\&=\cos ^{2} \frac{\theta}{2}-1+\cos ^{2} \frac{\theta}{2} \\&=2 \cos ^{2} \frac{\theta}{2}-1\end{aligned}$

From the image attached, use the Law of Cosines on an isosceles triangle whose equal sides are 1 to find cosθ.

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