Evaluate { \cos 15^ \circ}$}$.Using The Half-angle Identity ${ \cos \frac{30^{\circ }{2} = \pm \sqrt{\frac{1+\cos 30^{\circ}}{2}} }$

Introduction

Trigonometric functions are a fundamental aspect of mathematics, and their evaluation is crucial in various mathematical and real-world applications. In this article, we will focus on evaluating the trigonometric function $\cos 15^{\circ}$ using the half-angle identity. The half-angle identity is a powerful tool for evaluating trigonometric functions, and it is essential to understand its application in solving various mathematical problems.

Understanding the Half-Angle Identity

The half-angle identity is a mathematical formula that relates the cosine of an angle to the cosine of its half-angle. The formula is given by:

$$\cos \frac{\theta}{2} = \pm \sqrt{\frac{1+\cos \theta}{2}}$$

where $\theta$ is the angle, and the $\pm$ sign indicates that the cosine of the half-angle can be either positive or negative.

Applying the Half-Angle Identity to Evaluate $\cos 15^{\circ}$

To evaluate $\cos 15^{\circ}$ using the half-angle identity, we need to first find the value of $\cos 30^{\circ}$. We know that $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$.

Now, we can substitute this value into the half-angle identity formula:

$$\cos \frac{30^{\circ}}{2} = \pm \sqrt{\frac{1+\cos 30^{\circ}}{2}}$$

$$\cos 15^{\circ} = \pm \sqrt{\frac{1+\frac{\sqrt{3}}{2}}{2}}$$

To simplify this expression, we can first find the value of $1+\frac{\sqrt{3}}{2}$:

$$1+\frac{\sqrt{3}}{2} = \frac{2+\sqrt{3}}{2}$$

Now, we can substitute this value back into the expression:

$$\cos 15^{\circ} = \pm \sqrt{\frac{\frac{2+\sqrt{3}}{2}}{2}}$$

$$\cos 15^{\circ} = \pm \sqrt{\frac{2+\sqrt{3}}{4}}$$

To simplify this expression further, we can rationalize the denominator by multiplying both the numerator and denominator by $\sqrt{4}$:

$$\cos 15^{\circ} = \pm \sqrt{\frac{(2+\sqrt{3})\sqrt{4}}{4\sqrt{4}}}$$

$$\cos 15^{\circ} = \pm \sqrt{\frac{(2+\sqrt{3})\cdot2}{4\cdot2}}$$

$$\cos 15^{\circ} = \pm \sqrt{\frac{4+2\sqrt{3}}{8}}$$

$$\cos 15^{\circ} = \pm \frac{\sqrt{4+2\sqrt{3}}}{2\sqrt{2}}$$

$$\cos 15^{\circ} = \pm \frac{\sqrt{2(2+\sqrt{3})}}{2\sqrt{2}}$$

$$\cos 15^{\circ} = \pm \frac{\sqrt{2}\sqrt{2+\sqrt{3}}}{2\sqrt{2}}$$

$$\cos 15^{\circ} = \pm \frac{\sqrt{2+\sqrt{3}}}{2}$$

Conclusion

In this article, we have evaluated the trigonometric function $\cos 15^{\circ}$ using the half-angle identity. We have shown that the value of $\cos 15^{\circ}$ is $\pm \frac{\sqrt{2+\sqrt{3}}}{2}$. The half-angle identity is a powerful tool for evaluating trigonometric functions, and it is essential to understand its application in solving various mathematical problems.

Final Answer

The final answer is $\boxed{\pm \frac{\sqrt{2+\sqrt{3}}}{2}}$.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Additional Resources

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Calculus
• [3] Wolfram Alpha: Trigonometric Functions
  Evaluating Trigonometric Functions: A Comprehensive Guide ===========================================================

Q&A: Evaluating Trigonometric Functions

Q: What is the half-angle identity?

A: The half-angle identity is a mathematical formula that relates the cosine of an angle to the cosine of its half-angle. The formula is given by:

$$\cos \frac{\theta}{2} = \pm \sqrt{\frac{1+\cos \theta}{2}}$$

Q: How do I apply the half-angle identity to evaluate a trigonometric function?

A: To apply the half-angle identity, you need to first find the value of the cosine of the angle you want to evaluate. Then, you can substitute this value into the half-angle identity formula and simplify the expression.

Q: What is the value of $\cos 15^{\circ}$ using the half-angle identity?

A: The value of $\cos 15^{\circ}$ using the half-angle identity is $\pm \frac{\sqrt{2+\sqrt{3}}}{2}$.

Q: Can I use the half-angle identity to evaluate other trigonometric functions?

A: Yes, you can use the half-angle identity to evaluate other trigonometric functions. The half-angle identity can be applied to any trigonometric function, including sine, cosine, and tangent.

Q: What are some common applications of the half-angle identity?

A: The half-angle identity has many applications in mathematics and real-world problems. Some common applications include:

• Evaluating trigonometric functions in right triangles
• Solving trigonometric equations
• Finding the area and perimeter of triangles
• Calculating the height and distance of objects

Q: How do I simplify expressions involving the half-angle identity?

A: To simplify expressions involving the half-angle identity, you can use algebraic manipulations, such as multiplying and dividing by the same value, and using the properties of square roots.

Q: Can I use the half-angle identity to evaluate trigonometric functions with negative angles?

A: Yes, you can use the half-angle identity to evaluate trigonometric functions with negative angles. The half-angle identity can be applied to any angle, including negative angles.

Q: What are some common mistakes to avoid when using the half-angle identity?

A: Some common mistakes to avoid when using the half-angle identity include:

• Not simplifying the expression correctly
• Not using the correct value of the cosine of the angle
• Not considering the sign of the result

Conclusion

In this article, we have provided a comprehensive guide to evaluating trigonometric functions using the half-angle identity. We have answered some common questions and provided tips and tricks for simplifying expressions involving the half-angle identity.

Final Answer

The final answer is $\boxed{\pm \frac{\sqrt{2+\sqrt{3}}}{2}}$.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Additional Resources

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Calculus
• [3] Wolfram Alpha: Trigonometric Functions