Evaluate Each Expression Without A Calculator.4) ${ 2 \cos 2\left(67.5 {\circ}\right) - 1 = \square }$5) ${ 2 \sin\left(112.5^{\circ}\right) \cos\left(112.5^{\circ}\right) = \square }$6) $[ \frac{2

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will evaluate three trigonometric expressions without the use of a calculator.

Expression 1: $2 \cos^2\left(67.5^{\circ}\right) - 1$

To evaluate this expression, we can use the Pythagorean identity, which states that $\sin^2(x) + \cos^2(x) = 1$. Since we are given $\cos^2(67.5^{\circ})$, we can rewrite the expression as follows:

$$2 \cos^2\left(67.5^{\circ}\right) - 1 = 2 \left(1 - \sin^2\left(67.5^{\circ}\right)\right) - 1$$

Using the Pythagorean identity, we can simplify the expression further:

$$2 \left(1 - \sin^2\left(67.5^{\circ}\right)\right) - 1 = 2 - 2 \sin^2\left(67.5^{\circ}\right) - 1$$

$$2 - 2 \sin^2\left(67.5^{\circ}\right) - 1 = 1 - 2 \sin^2\left(67.5^{\circ}\right)$$

Now, we need to find the value of $\sin(67.5^{\circ})$. Since $67.5^{\circ}$ is not a standard angle, we can use the fact that $\sin(67.5^{\circ}) = \cos(22.5^{\circ})$. Using the unit circle or a trigonometric table, we can find that $\cos(22.5^{\circ}) = \frac{\sqrt{2} + \sqrt{6}}{4}$.

Substituting this value into the expression, we get:

$$1 - 2 \sin^2\left(67.5^{\circ}\right) = 1 - 2 \left(\frac{\sqrt{2} + \sqrt{6}}{4}\right)^2$$

Simplifying the expression further, we get:

$$1 - 2 \left(\frac{\sqrt{2} + \sqrt{6}}{4}\right)^2 = 1 - \frac{2\left(\sqrt{2} + \sqrt{6}\right)^2}{16}$$

$$1 - \frac{2\left(\sqrt{2} + \sqrt{6}\right)^2}{16} = 1 - \frac{2\left(2 + 2\sqrt{3} + 6\right)}{16}$$

$$1 - \frac{2\left(2 + 2\sqrt{3} + 6\right)}{16} = 1 - \frac{2\left(8 + 2\sqrt{3}\right)}{16}$$

$$1 - \frac{2\left(8 + 2\sqrt{3}\right)}{16} = 1 - \frac{16 + 4\sqrt{3}}{16}$$

$$1 - \frac{16 + 4\sqrt{3}}{16} = \frac{16 - 16 - 4\sqrt{3}}{16}$$

$$\frac{16 - 16 - 4\sqrt{3}}{16} = \frac{-4\sqrt{3}}{16}$$

$$\frac{-4\sqrt{3}}{16} = \frac{-\sqrt{3}}{4}$$

Therefore, the value of the expression $2 \cos^2\left(67.5^{\circ}\right) - 1$ is $\frac{-\sqrt{3}}{4}$.

Expression 2: $2 \sin\left(112.5^{\circ}\right) \cos\left(112.5^{\circ}\right)$

To evaluate this expression, we can use the angle addition formula, which states that $\sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b)$. Since we are given $\sin(112.5^{\circ})$ and $\cos(112.5^{\circ})$, we can rewrite the expression as follows:

$$2 \sin\left(112.5^{\circ}\right) \cos\left(112.5^{\circ}\right) = 2 \sin\left(45^{\circ} + 67.5^{\circ}\right)$$

Using the angle addition formula, we can simplify the expression further:

$$2 \sin\left(45^{\circ} + 67.5^{\circ}\right) = 2 \left(\sin(45^{\circ}) \cos(67.5^{\circ}) + \cos(45^{\circ}) \sin(67.5^{\circ})\right)$$

Using the unit circle or a trigonometric table, we can find that $\sin(45^{\circ}) = \frac{\sqrt{2}}{2}$, $\cos(45^{\circ}) = \frac{\sqrt{2}}{2}$, $\sin(67.5^{\circ}) = \frac{\sqrt{2} + \sqrt{6}}{4}$, and $\cos(67.5^{\circ}) = \frac{\sqrt{2} - \sqrt{6}}{4}$.

Substituting these values into the expression, we get:

$$2 \left(\sin(45^{\circ}) \cos(67.5^{\circ}) + \cos(45^{\circ}) \sin(67.5^{\circ})\right) = 2 \left(\frac{\sqrt{2}}{2} \frac{\sqrt{2} - \sqrt{6}}{4} + \frac{\sqrt{2}}{2} \frac{\sqrt{2} + \sqrt{6}}{4}\right)$$

Simplifying the expression further, we get:

$$2 \left(\frac{\sqrt{2}}{2} \frac{\sqrt{2} - \sqrt{6}}{4} + \frac{\sqrt{2}}{2} \frac{\sqrt{2} + \sqrt{6}}{4}\right) = 2 \left(\frac{\sqrt{2}(\sqrt{2} - \sqrt{6}) + \sqrt{2}(\sqrt{2} + \sqrt{6})}{8}\right)$$

$$2 \left(\frac{\sqrt{2}(\sqrt{2} - \sqrt{6}) + \sqrt{2}(\sqrt{2} + \sqrt{6})}{8}\right) = 2 \left(\frac{2\sqrt{2} - 2\sqrt{6} + 2\sqrt{2} + 2\sqrt{6}}{8}\right)$$

$$2 \left(\frac{2\sqrt{2} - 2\sqrt{6} + 2\sqrt{2} + 2\sqrt{6}}{8}\right) = 2 \left(\frac{4\sqrt{2}}{8}\right)$$

$$2 \left(\frac{4\sqrt{2}}{8}\right) = \frac{4\sqrt{2}}{4}$$

$$\frac{4\sqrt{2}}{4} = \sqrt{2}$$

Therefore, the value of the expression $2 \sin\left(112.5^{\circ}\right) \cos\left(112.5^{\circ}\right)$ is $\sqrt{2}$.

Expression 3: $\frac{2 \sin\left(67.5^{\circ}\right) \cos\left(67.5^{\circ}\right)}{\sin\left(112.5^{\circ}\right)}$

To evaluate this expression, we can use the angle addition formula, which states that $\sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b)$. Since we are given $\sin(67.5^{\circ})$, $\cos(67.5^{\circ})$, and $\sin(112.5^{\circ})$, we can rewrite the expression as follows:

$$\frac{2 \sin\left(67.5^{\circ}\right) \cos\left(67.5^{\circ}\right)}{\sin\left(112.5^{\circ}\right)} = \frac{2 \sin\left(45^{\circ} + 22.5^{\circ}\right)}{\sin\left(45^{\circ} + 67.5^{\circ}\right)}$$

Using the angle addition formula, we can simplify the expression further:

\frac{2 \sin\left(45^{\circ} + 22.5^{\circ}\right)}{\sin\left(45^{\circ} + 67.5^{\circ}\right)} = \frac{2 \left(\sin(45^{\circ}) \cos(22.5^{\circ}) + \cos<br/> **Evaluating Trigonometric Expressions without a Calculator: Q&A** ===========================================================

Introduction

In our previous article, we evaluated three trigonometric expressions without the use of a calculator. In this article, we will answer some common questions related to these expressions and provide additional insights.

Q: What is the value of $2 \cos^2\left(67.5^{\circ}\right) - 1$?

A: The value of $2 \cos^2\left(67.5^{\circ}\right) - 1$ is $\frac{-\sqrt{3}}{4}$.

Q: How did you simplify the expression $2 \cos^2\left(67.5^{\circ}\right) - 1$?

A: We used the Pythagorean identity, which states that $\sin^2(x) + \cos^2(x) = 1$. We rewrote the expression as $2 \left(1 - \sin^2\left(67.5^{\circ}\right)\right) - 1$ and simplified it further using the Pythagorean identity.

Q: What is the value of $2 \sin\left(112.5^{\circ}\right) \cos\left(112.5^{\circ}\right)$?

A: The value of $2 \sin\left(112.5^{\circ}\right) \cos\left(112.5^{\circ}\right)$ is $\sqrt{2}$.

Q: How did you simplify the expression $2 \sin\left(112.5^{\circ}\right) \cos\left(112.5^{\circ}\right)$?

A: We used the angle addition formula, which states that $\sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b)$. We rewrote the expression as $2 \sin\left(45^{\circ} + 67.5^{\circ}\right)$ and simplified it further using the angle addition formula.

Q: What is the value of $\frac{2 \sin\left(67.5^{\circ}\right) \cos\left(67.5^{\circ}\right)}{\sin\left(112.5^{\circ}\right)}$?

A: The value of $\frac{2 \sin\left(67.5^{\circ}\right) \cos\left(67.5^{\circ}\right)}{\sin\left(112.5^{\circ}\right)}$ is $\sqrt{2}$.

Q: How did you simplify the expression $\frac{2 \sin\left(67.5^{\circ}\right) \cos\left(67.5^{\circ}\right)}{\sin\left(112.5^{\circ}\right)}$?

A: We used the angle addition formula, which states that $\sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b)$. We rewrote the expression as $\frac{2 \sin\left(45^{\circ} + 22.5^{\circ}\right)}{\sin\left(45^{\circ} + 67.5^{\circ}\right)}$ and simplified it further using the angle addition formula.

Conclusion

In this article, we answered some common questions related to the evaluation of trigonometric expressions without a calculator. We provided additional insights and simplified expressions using various trigonometric identities. We hope that this article has been helpful in understanding the concepts of trigonometry and how to evaluate expressions without a calculator.

Additional Resources

For more information on trigonometry and how to evaluate expressions without a calculator, please refer to the following resources:

• Trigonometry textbooks and online resources
• Online calculators and trigonometric software
• Math forums and discussion groups

Final Thoughts

Evaluating trigonometric expressions without a calculator requires a deep understanding of trigonometric identities and formulas. By practicing and simplifying expressions using various identities, you can become proficient in evaluating trigonometric expressions without a calculator. We hope that this article has been helpful in your journey to learn trigonometry and how to evaluate expressions without a calculator.