Which Expression Is Equivalent To $\cos 120^{\circ}$?A. $\cos 60^{\circ}$ B. $\cos 240^{\circ}$ C. $\cos 300^{\circ}$ D. $\cos 420^{\circ}$

$$

Understanding the Problem

When dealing with trigonometric functions, it's essential to understand the properties and relationships between different angles. In this problem, we're asked to find an equivalent expression for $\cos 120^{\circ}$. To approach this, we need to recall the unit circle and the periodic nature of the cosine function.

The Unit Circle and Periodicity

The unit circle is a fundamental concept in trigonometry, where the cosine function is defined as the x-coordinate of a point on the unit circle. The unit circle has a radius of 1 and is centered at the origin of the coordinate plane. The cosine function is periodic, meaning that its value repeats every $360^{\circ}$ or $2\pi$ radians.

Key Angles and Their Cosine Values

Before we dive into the problem, let's recall the cosine values for some key angles:

• $\cos 0^{\circ} = 1$
• $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$
• $\cos 45^{\circ} = \frac{\sqrt{2}}{2}$
• $\cos 60^{\circ} = \frac{1}{2}$
• $\cos 90^{\circ} = 0$

Finding Equivalent Expressions

Now, let's focus on finding an equivalent expression for $\cos 120^{\circ}$. We can use the periodicity of the cosine function to our advantage. Since the cosine function repeats every $360^{\circ}$, we can add or subtract multiples of $360^{\circ}$ to find equivalent expressions.

Option A: $\cos 60^{\circ}$

Option A suggests that $\cos 120^{\circ}$ is equivalent to $\cos 60^{\circ}$. However, we know that $\cos 60^{\circ} = \frac{1}{2}$, which is not equal to $\cos 120^{\circ}$. Therefore, Option A is incorrect.

Option B: $\cos 240^{\circ}$

Option B suggests that $\cos 120^{\circ}$ is equivalent to $\cos 240^{\circ}$. To verify this, let's use the periodicity of the cosine function. We can add $120^{\circ}$ to $240^{\circ}$ to get $360^{\circ}$, which is equivalent to $0^{\circ}$. Since $\cos 0^{\circ} = 1$, we can conclude that $\cos 240^{\circ} = -1$. However, we know that $\cos 120^{\circ} = -\frac{1}{2}$, which is not equal to $\cos 240^{\circ}$. Therefore, Option B is incorrect.

Option C: $\cos 300^{\circ}$

Option C suggests that $\cos 120^{\circ}$ is equivalent to $\cos 300^{\circ}$. To verify this, let's use the periodicity of the cosine function. We can add $180^{\circ}$ to $120^{\circ}$ to get $300^{\circ}$. Since $\cos 300^{\circ} = \frac{1}{2}$, we can conclude that $\cos 300^{\circ}$ is not equal to $\cos 120^{\circ}$. Therefore, Option C is incorrect.

Option D: $\cos 420^{\circ}$

Option D suggests that $\cos 120^{\circ}$ is equivalent to $\cos 420^{\circ}$. To verify this, let's use the periodicity of the cosine function. We can add $360^{\circ}$ to $420^{\circ}$ to get $780^{\circ}$, which is equivalent to $120^{\circ}$. Since $\cos 120^{\circ} = -\frac{1}{2}$, we can conclude that $\cos 420^{\circ} = -\frac{1}{2}$. Therefore, Option D is correct.

Conclusion

In conclusion, the correct answer is Option D: $\cos 420^{\circ}$. This is because $\cos 420^{\circ}$ is equivalent to $\cos 120^{\circ}$ due to the periodicity of the cosine function.

Final Answer

The final answer is $\boxed{D}$.

Additional Tips and Tricks

When dealing with trigonometric functions, it's essential to understand the properties and relationships between different angles. Here are some additional tips and tricks to help you solve similar problems:

• Use the unit circle to visualize the cosine function and its periodicity.
• Recall the cosine values for key angles, such as $0^{\circ}$, $30^{\circ}$, $45^{\circ}$, $60^{\circ}$, and $90^{\circ}$.
• Use the periodicity of the cosine function to find equivalent expressions.
• Verify your answers by using the unit circle and the periodicity of the cosine function.

By following these tips and tricks, you'll be able to solve similar problems with ease and confidence.

Q: What is Trigonometry?

A: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It involves the study of triangles, particularly right triangles, and the relationships between their angles and side lengths.

Q: What are the Basic Trigonometric Functions?

A: The basic trigonometric functions are:

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

Q: What is the Unit Circle?

A: The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It is used to define the trigonometric functions and their values for different angles.

Q: What is the Periodicity of the Trigonometric Functions?

A: The periodicity of the trigonometric functions refers to the fact that their values repeat every 360 degrees or 2π radians. This means that the value of a trigonometric function at an angle is the same as its value at that angle plus or minus any multiple of 360 degrees or 2π radians.

Q: How Do I Evaluate Trigonometric Functions?

A: To evaluate trigonometric functions, you can use the following methods:

• Use the unit circle to find the values of the trigonometric functions for different angles.
• Use the periodicity of the trigonometric functions to find equivalent expressions.
• Use the trigonometric identities to simplify and evaluate expressions.

Q: What are Trigonometric Identities?

A: Trigonometric identities are equations that relate different trigonometric functions. They are used to simplify and evaluate expressions involving trigonometric functions.

Q: How Do I Use Trigonometric Identities?

A: To use trigonometric identities, you can follow these steps:

• Identify the trigonometric functions involved in the expression.
• Use the trigonometric identities to simplify and rewrite the expression.
• Evaluate the expression using the simplified form.

Q: What are Some Common Trigonometric Identities?

A: Some common trigonometric identities include:

• sin^2(x) + cos^2(x) = 1
• tan(x) = sin(x) / cos(x)
• cot(x) = cos(x) / sin(x)
• sec(x) = 1 / cos(x)
• csc(x) = 1 / sin(x)

Q: How Do I Solve Trigonometric Equations?

A: To solve trigonometric equations, you can follow these steps:

• Simplify the equation using trigonometric identities.
• Isolate the trigonometric function.
• Use the unit circle or periodicity to find the values of the trigonometric function.
• Solve for the variable.

Q: What are Some Common Trigonometric Equations?

A: Some common trigonometric equations include:

• sin(x) = 0
• cos(x) = 0
• tan(x) = 0
• cot(x) = 0
• sec(x) = 0
• csc(x) = 0

Q: How Do I Graph Trigonometric Functions?

A: To graph trigonometric functions, you can follow these steps:

• Use the unit circle to find the values of the trigonometric function for different angles.
• Plot the points on the coordinate plane.
• Connect the points to form the graph.

Q: What are Some Common Graphs of Trigonometric Functions?

A: Some common graphs of trigonometric functions include:

• The sine function: y = sin(x)
• The cosine function: y = cos(x)
• The tangent function: y = tan(x)
• The cotangent function: y = cot(x)
• The secant function: y = sec(x)
• The cosecant function: y = csc(x)

Conclusion

In conclusion, trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It involves the study of triangles, particularly right triangles, and the relationships between their angles and side lengths. The basic trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. The unit circle is used to define the trigonometric functions and their values for different angles. The periodicity of the trigonometric functions refers to the fact that their values repeat every 360 degrees or 2π radians. Trigonometric identities are equations that relate different trigonometric functions. They are used to simplify and evaluate expressions involving trigonometric functions.