Which Trigonometric Expression Is Equivalent To $\sin 30^{\circ}$? Select All That Apply.A. $\sin -210^{\circ}$ B. $\cos 30^{\circ}$ C. $\tan 30^{\circ}$ D. $\cos 120^{\circ}$ E. $\sin

**Which Trigonometric Expression is Equivalent to $\sin 30^{\circ}$? Select All That Apply**

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject in mathematics and has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will explore the trigonometric expression equivalent to $\sin 30^{\circ}$ and discuss the properties of trigonometric functions.

Trigonometric functions are used to describe the relationships between the sides and angles of triangles. The six basic trigonometric functions are:

• Sine (sin)
• Cosine (cos)
• Tangent (tan)
• Cotangent (cot)
• Secant (sec)
• Cosecant (csc)

These functions are defined as follows:

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Trigonometric functions have several properties that are essential to understand. Some of these properties include:

• Periodicity: Trigonometric functions are periodic, meaning they repeat their values at regular intervals. For example, the sine function has a period of $360^{\circ}$.
• Symmetry: Trigonometric functions have symmetry properties. For example, the sine function is symmetric about the origin, while the cosine function is symmetric about the y-axis.
• Identities: Trigonometric functions have various identities that can be used to simplify expressions. For example, the Pythagorean identity states that $\sin^2 \theta + \cos^2 \theta = 1$.

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Now that we have discussed the properties of trigonometric functions, let's explore which expression is equivalent to $\sin 30^{\circ}$. We will examine each option and determine whether it is equivalent to $\sin 30^{\circ}$.

Option A: $\sin -210^{\circ}$

To determine whether $\sin -210^{\circ}$ is equivalent to $\sin 30^{\circ}$, we need to understand the properties of the sine function. The sine function is an odd function, meaning that $\sin (-\theta) = -\sin \theta$. Therefore, $\sin -210^{\circ} = -\sin 210^{\circ}$.

Since $\sin 210^{\circ} = \sin (180^{\circ} + 30^{\circ}) = \sin 30^{\circ}$, we can conclude that $\sin -210^{\circ} = -\sin 30^{\circ}$. Therefore, option A is not equivalent to $\sin 30^{\circ}$.

Option B: $\cos 30^{\circ}$

To determine whether $\cos 30^{\circ}$ is equivalent to $\sin 30^{\circ}$, we need to understand the properties of the cosine function. The cosine function is an even function, meaning that $\cos (-\theta) = \cos \theta$. Therefore, $\cos 30^{\circ} = \cos (-30^{\circ})$.

Since $\cos (-30^{\circ}) = \cos 30^{\circ}$, we can conclude that $\cos 30^{\circ}$ is not equivalent to $\sin 30^{\circ}$.

Option C: $\tan 30^{\circ}$

To determine whether $\tan 30^{\circ}$ is equivalent to $\sin 30^{\circ}$, we need to understand the properties of the tangent function. The tangent function is defined as $\tan \theta = \frac{\sin \theta}{\cos \theta}$. Therefore, $\tan 30^{\circ} = \frac{\sin 30^{\circ}}{\cos 30^{\circ}}$.

Since $\sin 30^{\circ} = \frac{1}{2}$ and $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$, we can conclude that $\tan 30^{\circ} = \frac{1}{\sqrt{3}}$. Therefore, option C is not equivalent to $\sin 30^{\circ}$.

Option D: $\cos 120^{\circ}$

To determine whether $\cos 120^{\circ}$ is equivalent to $\sin 30^{\circ}$, we need to understand the properties of the cosine function. The cosine function is an even function, meaning that $\cos (-\theta) = \cos \theta$. Therefore, $\cos 120^{\circ} = \cos (-120^{\circ})$.

Since $\cos (-120^{\circ}) = \cos 120^{\circ}$, we can conclude that $\cos 120^{\circ}$ is not equivalent to $\sin 30^{\circ}$.

Option E: $\sin 150^{\circ}$

To determine whether $\sin 150^{\circ}$ is equivalent to $\sin 30^{\circ}$, we need to understand the properties of the sine function. The sine function is an odd function, meaning that $\sin (-\theta) = -\sin \theta$. Therefore, $\sin 150^{\circ} = \sin (180^{\circ} - 30^{\circ}) = \sin 30^{\circ}$.

Since $\sin 150^{\circ} = \sin 30^{\circ}$, we can conclude that option E is equivalent to $\sin 30^{\circ}$.

In conclusion, the trigonometric expression equivalent to $\sin 30^{\circ}$ is $\sin 150^{\circ}$. This is because $\sin 150^{\circ} = \sin (180^{\circ} - 30^{\circ}) = \sin 30^{\circ}$.

Q: What is the value of $\sin 30^{\circ}$?

A: The value of $\sin 30^{\circ}$ is $\frac{1}{2}$.

Q: What is the value of $\cos 30^{\circ}$?

A: The value of $\cos 30^{\circ}$ is $\frac{\sqrt{3}}{2}$.

Q: What is the value of $\tan 30^{\circ}$?

A: The value of $\tan 30^{\circ}$ is $\frac{1}{\sqrt{3}}$.

Q: What is the value of $\sin 150^{\circ}$?

A: The value of $\sin 150^{\circ}$ is $\frac{1}{2}$.

Q: What is the value of $\cos 120^{\circ}$?

A: The value of $\cos 120^{\circ}$ is $-\frac{1}{2}$.

Q: What is the value of $\sin -210^{\circ}$?

A: The value of $\sin -210^{\circ}$ is $-\frac{1}{2}$.

• "Trigonometry" by Michael Corral
• "Trigonometry" by I. M. Gelfand
• "Trigonometry" by Charles P. McKeague

• Sine: The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
• Cosine: The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
• Tangent: The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
• Periodicity: A function is periodic if it repeats its values at regular intervals.
• Symmetry: A function is symmetric if it has the same value for a given input and its negative counterpart.
• Identities: Trigonometric identities are equations that relate the values of trigonometric functions.