Evaluate $\cos 135^{\circ}$ Without Using A Calculator.A. $\frac{\sqrt{2}}{2}$ B. $\frac{\sqrt{3}}{2}$ C. $-\frac{1}{2}$ D. $-\frac{\sqrt{2}}{2}$

Introduction

In this article, we will explore the concept of evaluating the cosine of an angle without using a calculator. We will focus on the specific angle of $135^{\circ}$ and provide a step-by-step guide on how to calculate its cosine value. This will involve using trigonometric identities and properties to simplify the expression and arrive at the final answer.

Understanding the Angle

Before we dive into the calculation, let's understand the angle $135^{\circ}$. This angle is located in the second quadrant of the unit circle, where the cosine function is negative. The reference angle for $135^{\circ}$ is $45^{\circ}$, which is a special angle with known trigonometric values.

Using Trigonometric Identities

To evaluate $\cos 135^{\circ}$, we can use the trigonometric identity $\cos (A + B) = \cos A \cos B - \sin A \sin B$. We can rewrite $135^{\circ}$ as the sum of $90^{\circ}$ and $45^{\circ}$, which gives us:

$$\cos 135^{\circ} = \cos (90^{\circ} + 45^{\circ})$$

Applying the Cosine Sum Formula

Now, we can apply the cosine sum formula to simplify the expression:

$$\cos (90^{\circ} + 45^{\circ}) = \cos 90^{\circ} \cos 45^{\circ} - \sin 90^{\circ} \sin 45^{\circ}$$

Evaluating the Trigonometric Functions

We know that $\cos 90^{\circ} = 0$, $\sin 90^{\circ} = 1$, and $\cos 45^{\circ} = \frac{\sqrt{2}}{2}$. Substituting these values into the expression, we get:

$$\cos 135^{\circ} = 0 \cdot \frac{\sqrt{2}}{2} - 1 \cdot \frac{\sqrt{2}}{2}$$

Simplifying the Expression

Simplifying the expression, we get:

$$\cos 135^{\circ} = -\frac{\sqrt{2}}{2}$$

Conclusion

In this article, we evaluated the cosine of $135^{\circ}$ without using a calculator. We used trigonometric identities and properties to simplify the expression and arrive at the final answer. The cosine of $135^{\circ}$ is $-\frac{\sqrt{2}}{2}$.

Comparison with Other Options

Let's compare our answer with the other options provided:

• A. $\frac{\sqrt{2}}{2}$: This is the cosine value of $45^{\circ}$, not $135^{\circ}$.
• B. $\frac{\sqrt{3}}{2}$: This is the cosine value of $30^{\circ}$, not $135^{\circ}$.
• C. $-\frac{1}{2}$: This is the cosine value of $120^{\circ}$, not $135^{\circ}$.
• D. $-\frac{\sqrt{2}}{2}$: This is our final answer, which we arrived at through the use of trigonometric identities and properties.

Final Answer

The final answer is $\boxed{-\frac{\sqrt{2}}{2}}$.

Introduction

In our previous article, we explored the concept of evaluating the cosine of an angle without using a calculator. We focused on the specific angle of $135^{\circ}$ and provided a step-by-step guide on how to calculate its cosine value. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the reference angle for $135^{\circ}$?

A: The reference angle for $135^{\circ}$ is $45^{\circ}$.

Q: Why is the cosine function negative in the second quadrant?

A: The cosine function is negative in the second quadrant because the x-coordinate of a point on the unit circle in this quadrant is negative.

Q: How do I use the cosine sum formula to evaluate $\cos 135^{\circ}$?

A: To use the cosine sum formula, you need to rewrite the angle $135^{\circ}$ as the sum of two angles, one of which is $90^{\circ}$. Then, you can apply the formula $\cos (A + B) = \cos A \cos B - \sin A \sin B$ to simplify the expression.

Q: What are the values of $\cos 90^{\circ}$ and $\sin 90^{\circ}$?

A: The value of $\cos 90^{\circ}$ is 0, and the value of $\sin 90^{\circ}$ is 1.

Q: How do I simplify the expression $\cos 135^{\circ} = 0 \cdot \frac{\sqrt{2}}{2} - 1 \cdot \frac{\sqrt{2}}{2}$?

A: To simplify the expression, you can multiply the numbers together and then combine like terms. In this case, the expression simplifies to $-\frac{\sqrt{2}}{2}$.

Q: Why is the final answer $-\frac{\sqrt{2}}{2}$?

A: The final answer $-\frac{\sqrt{2}}{2}$ is the result of simplifying the expression $\cos 135^{\circ} = 0 \cdot \frac{\sqrt{2}}{2} - 1 \cdot \frac{\sqrt{2}}{2}$ using the values of $\cos 90^{\circ}$ and $\sin 90^{\circ}$.

Conclusion

In this article, we answered some frequently asked questions related to evaluating the cosine of an angle without using a calculator. We provided step-by-step explanations and examples to help clarify the concepts. If you have any further questions or need additional clarification, please don't hesitate to ask.

Additional Resources

If you want to learn more about trigonometry and how to evaluate the cosine of an angle without using a calculator, here are some additional resources:

• Trigonometry for Dummies
• Cosine Function
• Unit Circle

Final Answer

The final answer is $\boxed{-\frac{\sqrt{2}}{2}}$.