SECTION E: Function values of $$ \left(k \cdot 360^{ heta}+ heta ight), k \in \mathbb{Z} $$ Investigation: Reduction formulas for function values of $$ \left(k \cdot 360^{ heta}+0 ight), k \in \mathbb{Z} $$. It is also possible to get an angle greater than $$ 360^{ heta} $$ on the Cartesian plane. The angle first completes one or more revolutions from the positive x-axis and then rotates further to form an angle of $$ heta $$. For the acute angle $$ heta $$ (in the first quadrant), the angle $$ \left(k \cdot 360^{ heta}+ heta ight) $$ where $$ k \in \mathbb{Z} $$, will again lie in the first quadrant. Now use the results obtained in 12 above and circle the correct relationship: $$ 5.1 \sin \left(k \cdot 360^{ heta}+ heta ight)=\sin heta $$ or $$ -\sin heta $$, where $$ k \in \mathbb{Z} $$ $$ 5.2 \cos \left(k \cdot 360^{ heta}+ heta ight)=\cos heta $$ or $$ -\cos heta $$, where $$ k \in \mathbb{Z} $$ $$ 5.3 an \left(k \cdot 360^{ heta}+ heta ight)= an heta $$ or $$ - an heta $$, where $$ k \in \mathbb{Z} $$

Question

SECTION E: Function values of $$ \left(k \cdot 360^{	heta}+	hetaight), k \in \mathbb{Z} $$ Investigation: Reduction formulas for function values of $$ \left(k \cdot 360^{	heta}+0ight), k \in \mathbb{Z} $$. It is also possible to get an angle greater than $$ 360^{	heta} $$ on the Cartesian plane. The angle first completes one or more revolutions from the positive x-axis and then rotates further to form an angle of $$ 	heta $$. For the acute angle $$ 	heta $$ (in the first quadrant), the angle $$ \left(k \cdot 360^{	heta}+	hetaight) $$ where $$ k \in \mathbb{Z} $$, will again lie in the first quadrant. Now use the results obtained in 12 above and circle the correct relationship: $$ 5.1 \sin \left(k \cdot 360^{	heta}+	hetaight)=\sin 	heta $$ or $$ -\sin 	heta $$, where $$ k \in \mathbb{Z} $$ $$ 5.2 \cos \left(k \cdot 360^{	heta}+	hetaight)=\cos 	heta $$ or $$ -\cos 	heta $$, where $$ k \in \mathbb{Z} $$ $$ 5.3 	an \left(k \cdot 360^{	heta}+	hetaight)=	an 	heta $$ or $$ -	an 	heta $$, where $$ k \in \mathbb{Z} $$

UpStudy AI · Accepted Answer

$$
\textbf{Step 1: Understanding the Periodicity}
$$

The sine, cosine, and tangent functions are periodic with period $$360^\circ$$. This means that for any angle $$\theta$$ and any integer $$k$$,

$$
\sin\left(\theta+360^\circ k\right) = \sin \theta,
$$
$$
\cos\left(\theta+360^\circ k\right) = \cos \theta,
$$
$$
\tan\left(\theta+360^\circ k\right) = \tan \theta.
$$

$$
\textbf{Step 2: Applying the Periodicity to the Given Functions}
$$

We are given the expressions:

$$
5.1 \; \sin\left(k \cdot 360^\circ + \theta\right),
$$
$$
5.2 \; \cos\left(k \cdot 360^\circ + \theta\right),
$$
$$
5.3 \; \tan\left(k \cdot 360^\circ + \theta\right),
$$

for $$k \in \mathbb{Z}$$ and an acute angle $$\theta$$ (which means $$0^\circ < \theta < 90^\circ$$).

Using the periodic properties from Step 1, we have:

$$
\sin\left(k \cdot 360^\circ + \theta\right) = \sin \theta,
$$
$$
\cos\left(k \cdot 360^\circ + \theta\right) = \cos \theta,
$$
$$
\tan\left(k \cdot 360^\circ + \theta\right) = \tan \theta.
$$

$$
\textbf{Step 3: Choosing the Correct Relationships}
$$

Now, comparing with the options given:

- For $$5.1$$: $$\sin\left(k \cdot 360^\circ + \theta\right)$$ equals $$\sin \theta$$ (not $$-\sin \theta$$).
- For $$5.2$$: $$\cos\left(k \cdot 360^\circ + \theta\right)$$ equals $$\cos \theta$$ (not $$-\cos \theta$$).
- For $$5.3$$: $$\tan\left(k \cdot 360^\circ + \theta\right)$$ equals $$\tan \theta$$ (not $$-\tan \theta$$).

$$
\textbf{Final Answer:}
$$

$$
\begin{aligned}
5.1 \quad & \sin\left(k \cdot 360^\circ+\theta\right)=\sin \theta,\[5mm]
5.2 \quad & \cos\left(k \cdot 360^\circ+\theta\right)=\cos \theta,\[5mm]
5.3 \quad & \tan\left(k \cdot 360^\circ+\theta\right)=\tan \theta.
\end{aligned}
$$
$$ \begin{aligned} 5.1 \quad & \sin\left(k \cdot 360^\circ+\theta\right)=\sin \theta,\[5mm] 5.2 \quad & \cos\left(k \cdot 360^\circ+\theta\right)=\cos \theta,\[5mm] 5.3 \quad & \tan\left(k \cdot 360^\circ+\theta\right)=\tan \theta. \end{aligned} $$

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