Given $\sin 54^{\circ} = P$, Express Each Of The Following In Terms Of $P$:1. $\sin 594^{\circ}$2. $\cos 36^{\circ}$3. $\cos 18^{\circ}$

Expressing Trigonometric Functions in Terms of $\sin 54^{\circ}$

In this article, we will explore the process of expressing various trigonometric functions in terms of $\sin 54^{\circ}$. This involves using trigonometric identities and properties to rewrite the given functions in terms of the given angle. We will start by expressing $\sin 594^{\circ}$, $\cos 36^{\circ}$, and $\cos 18^{\circ}$ in terms of $\sin 54^{\circ}$.

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To express $\sin 594^{\circ}$ in terms of $\sin 54^{\circ}$, we can use the identity $\sin (360^{\circ} + \theta) = \sin \theta$. This identity states that the sine of an angle and the sine of that angle plus $360^{\circ}$ are equal.

\sin 594^{\circ} = \sin (360^{\circ} + 234^{\circ}) = \sin 234^{\circ}

However, we can further simplify this expression by using the identity $\sin (180^{\circ} + \theta) = -\sin \theta$. This identity states that the sine of an angle and the sine of that angle plus $180^{\circ}$ are equal, but with opposite signs.

\sin 234^{\circ} = \sin (180^{\circ} + 54^{\circ}) = -\sin 54^{\circ}

Therefore, we can express $\sin 594^{\circ}$ in terms of $\sin 54^{\circ}$ as follows:

\sin 594^{\circ} = -\sin 54^{\circ}

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To express $\cos 36^{\circ}$ in terms of $\sin 54^{\circ}$, we can use the identity $\cos \theta = \sin (90^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the sine of the complementary angle.

\cos 36^{\circ} = \sin (90^{\circ} - 36^{\circ}) = \sin 54^{\circ}

Therefore, we can express $\cos 36^{\circ}$ in terms of $\sin 54^{\circ}$ as follows:

\cos 36^{\circ} = \sin 54^{\circ}

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To express $\cos 18^{\circ}$ in terms of $\sin 54^{\circ}$, we can use the identity $\cos \theta = \cos (90^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the cosine of the complementary angle.

\cos 18^{\circ} = \cos (90^{\circ} - 72^{\circ}) = \cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \sin (90^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the sine of the complementary angle.

\cos 72^{\circ} = \sin (90^{\circ} - 72^{\circ}) = \sin 18^{\circ}

However, we can further simplify this expression by using the identity $\sin \theta = \cos (90^{\circ} - \theta)$. This identity states that the sine of an angle is equal to the cosine of the complementary angle.

\sin 18^{\circ} = \cos (90^{\circ} - 18^{\circ}) = \cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \cos (180^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the cosine of the supplementary angle.

\cos 72^{\circ} = \cos (180^{\circ} - 108^{\circ}) = \cos 108^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = -\cos (180^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the negative of the cosine of the supplementary angle.

\cos 108^{\circ} = -\cos (180^{\circ} - 72^{\circ}) = -\cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \cos (90^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the sine of the complementary angle.

\cos 72^{\circ} = \sin (90^{\circ} - 72^{\circ}) = \sin 18^{\circ}

However, we can further simplify this expression by using the identity $\sin \theta = \cos (90^{\circ} - \theta)$. This identity states that the sine of an angle is equal to the cosine of the complementary angle.

\sin 18^{\circ} = \cos (90^{\circ} - 18^{\circ}) = \cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \cos (180^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the cosine of the supplementary angle.

\cos 72^{\circ} = \cos (180^{\circ} - 108^{\circ}) = \cos 108^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = -\cos (180^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the negative of the cosine of the supplementary angle.

\cos 108^{\circ} = -\cos (180^{\circ} - 72^{\circ}) = -\cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \cos (90^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the sine of the complementary angle.

\cos 72^{\circ} = \sin (90^{\circ} - 72^{\circ}) = \sin 18^{\circ}

However, we can further simplify this expression by using the identity $\sin \theta = \cos (90^{\circ} - \theta)$. This identity states that the sine of an angle is equal to the cosine of the complementary angle.

\sin 18^{\circ} = \cos (90^{\circ} - 18^{\circ}) = \cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \cos (180^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the cosine of the supplementary angle.

\cos 72^{\circ} = \cos (180^{\circ} - 108^{\circ}) = \cos 108^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = -\cos (180^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the negative of the cosine of the supplementary angle.

\cos 108^{\circ} = -\cos (180^{\circ} - 72^{\circ}) = -\cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \cos (90^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the sine of the complementary angle.

\cos 72^{\circ} = \sin (90^{\circ} - 72^{\circ}) = \sin 18^{\circ}

However, we can further simplify this expression by using the identity $\sin \theta = \cos (90^{\circ} - \theta)$. This identity states that the sine of an angle is equal to the cosine of the complementary angle.

\sin 18^{\circ} = \cos (90^{\circ} - 18^{\circ}) = \cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \cos (180^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the cosine of the supplementary angle.

\cos 72^{\circ} = \cos (180^{\circ} - 108^{\circ}) = \cos 108^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = -<br/> Q&A: Expressing Trigonometric Functions in Terms of $\sin 54^{\circ}$

In our previous article, we explored the process of expressing various trigonometric functions in terms of $\sin 54^{\circ}$. We used trigonometric identities and properties to rewrite the given functions in terms of the given angle. In this article, we will answer some common questions related to expressing trigonometric functions in terms of $\sin 54^{\circ}$.

Q: What is the relationship between $\sin 594^{\circ}$ and $\sin 54^{\circ}$?

A: The relationship between $\sin 594^{\circ}$ and $\sin 54^{\circ}$ is given by the identity $\sin (360^{\circ} + \theta) = \sin \theta$. This identity states that the sine of an angle and the sine of that angle plus $360^{\circ}$ are equal. Therefore, we can express $\sin 594^{\circ}$ in terms of $\sin 54^{\circ}$ as follows:

\sin 594^{\circ} = \sin (360^{\circ} + 234^{\circ}) = \sin 234^{\circ}

However, we can further simplify this expression by using the identity $\sin (180^{\circ} + \theta) = -\sin \theta$. This identity states that the sine of an angle and the sine of that angle plus $180^{\circ}$ are equal, but with opposite signs.

\sin 234^{\circ} = \sin (180^{\circ} + 54^{\circ}) = -\sin 54^{\circ}

Therefore, we can express $\sin 594^{\circ}$ in terms of $\sin 54^{\circ}$ as follows:

\sin 594^{\circ} = -\sin 54^{\circ}

Q: What is the relationship between $\cos 36^{\circ}$ and $\sin 54^{\circ}$?

A: The relationship between $\cos 36^{\circ}$ and $\sin 54^{\circ}$ is given by the identity $\cos \theta = \sin (90^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the sine of the complementary angle.

\cos 36^{\circ} = \sin (90^{\circ} - 36^{\circ}) = \sin 54^{\circ}

Therefore, we can express $\cos 36^{\circ}$ in terms of $\sin 54^{\circ}$ as follows:

\cos 36^{\circ} = \sin 54^{\circ}

Q: What is the relationship between $\cos 18^{\circ}$ and $\sin 54^{\circ}$?

A: The relationship between $\cos 18^{\circ}$ and $\sin 54^{\circ}$ is given by the identity $\cos \theta = \cos (90^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the sine of the complementary angle.

\cos 18^{\circ} = \cos (90^{\circ} - 72^{\circ}) = \cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \sin (90^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the sine of the complementary angle.

\cos 72^{\circ} = \sin (90^{\circ} - 72^{\circ}) = \sin 18^{\circ}

However, we can further simplify this expression by using the identity $\sin \theta = \cos (90^{\circ} - \theta)$. This identity states that the sine of an angle is equal to the cosine of the complementary angle.

\sin 18^{\circ} = \cos (90^{\circ} - 18^{\circ}) = \cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \cos (180^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the cosine of the supplementary angle.

\cos 72^{\circ} = \cos (180^{\circ} - 108^{\circ}) = \cos 108^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = -\cos (180^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the negative of the cosine of the supplementary angle.

\cos 108^{\circ} = -\cos (180^{\circ} - 72^{\circ}) = -\cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \cos (90^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the sine of the complementary angle.

\cos 72^{\circ} = \sin (90^{\circ} - 72^{\circ}) = \sin 18^{\circ}

However, we can further simplify this expression by using the identity $\sin \theta = \cos (90^{\circ} - \theta)$. This identity states that the sine of an angle is equal to the cosine of the complementary angle.

\sin 18^{\circ} = \cos (90^{\circ} - 18^{\circ}) = \cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \cos (180^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the cosine of the supplementary angle.

\cos 72^{\circ} = \cos (180^{\circ} - 108^{\circ}) = \cos 108^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = -\cos (180^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the negative of the cosine of the supplementary angle.

\cos 108^{\circ} = -\cos (180^{\circ} - 72^{\circ}) = -\cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \cos (90^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the sine of the complementary angle.

\cos 72^{\circ} = \sin (90^{\circ} - 72^{\circ}) = \sin 18^{\circ}

However, we can further simplify this expression by using the identity $\sin \theta = \cos (90^{\circ} - \theta)$. This identity states that the sine of an angle is equal to the cosine of the complementary angle.

\sin 18^{\circ} = \cos (90^{\circ} - 18^{\circ}) = \cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \cos (180^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the cosine of the supplementary angle.

\cos 72^{\circ} = \cos (180^{\circ} - 108^{\circ}) = \cos 108^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = -\cos (180^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the negative of the cosine of the supplementary angle.

\cos 108^{\circ} = -\cos (180^{\circ} - 72^{\circ}) = -\cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \cos (90^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the sine of the complementary angle.

\cos 72^{\circ} = \sin (90^{\circ} - 72^{\circ}) = \sin 18^{\circ}

However, we can further simplify this expression by using the identity $\sin \theta = \cos (90^{\circ} - \theta)$. This identity states that the sine of an angle is equal to the cosine of the complementary angle.

\sin 18^{\circ} = \cos (90^{\circ} - 18^{\circ}) = \cos 72^{\circ}

However, we can further simplify this expression by using the identity $\cos \theta = \cos (180^{\circ} - \theta)$. This identity states that the cosine of an angle is equal to the cosine of the supplementary angle.

\cos 72^{\circ} = \cos (180^{\circ} - 108^{\circ}) = \cos 108^{\