Use A Half-angle Identity To Find The Exact Value Of Sin ⁡ 157.5 ∘ \sin 157.5^{\circ} Sin 157. 5 ∘ .Which Half-angle Formula Should Be Used To Find The Exact Value Of Sin ⁡ 157.5 ∘ \sin 157.5^{\circ} Sin 157. 5 ∘ ? Select The Correct Choice Below And Fill In The Answer Boxes To

Introduction

Trigonometric identities are a crucial part of mathematics, and solving them requires a deep understanding of the underlying concepts. In this article, we will focus on using half-angle identities to find the exact value of $\sin 157.5^{\circ}$. We will explore the different half-angle formulas and determine which one should be used to solve this problem.

Understanding Half-Angle Identities

Half-angle identities are a set of trigonometric identities that relate the values of sine and cosine functions at half the angle to the values of these functions at the full angle. These identities are essential in solving trigonometric equations and are used extensively in various mathematical applications.

There are two main half-angle identities:

• Half-Angle Formula for Sine: $\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}$
• Half-Angle Formula for Cosine: $\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}$

Choosing the Correct Half-Angle Formula

To determine which half-angle formula should be used to find the exact value of $\sin 157.5^{\circ}$, we need to consider the given angle. The angle $157.5^{\circ}$ is greater than $90^{\circ}$, which means it is in the second quadrant of the unit circle.

In the second quadrant, the sine function is positive, and the cosine function is negative. Therefore, we need to use the half-angle formula for sine, which is:

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

Applying the Half-Angle Formula

Now that we have chosen the correct half-angle formula, we can apply it to find the exact value of $\sin 157.5^{\circ}$. We will use the formula:

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

Substituting $\theta = 157.5^{\circ}$, we get:

$\sin \frac{157.5^{\circ}}{2} = \pm \sqrt{\frac{1 - \cos 157.5^{\circ}}{2}}$

To evaluate this expression, we need to find the value of $\cos 157.5^{\circ}$. Using the unit circle or a calculator, we find that:

$\cos 157.5^{\circ} = -\frac{\sqrt{3}}{2}$

Substituting this value into the expression, we get:

$\sin \frac{157.5^{\circ}}{2} = \pm \sqrt{\frac{1 - (-\frac{\sqrt{3}}{2})}{2}}$

Simplifying this expression, we get:

$\sin \frac{157.5^{\circ}}{2} = \pm \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}}$

$\sin \frac{157.5^{\circ}}{2} = \pm \sqrt{\frac{2 + \sqrt{3}}{4}}$

$\sin \frac{157.5^{\circ}}{2} = \pm \frac{\sqrt{2 + \sqrt{3}}}{2}$

Simplifying the Expression

Now that we have evaluated the expression, we need to simplify it to find the exact value of $\sin 157.5^{\circ}$. We can do this by rationalizing the denominator.

Multiplying the numerator and denominator by $\sqrt{2 - \sqrt{3}}$, we get:

$\sin \frac{157.5^{\circ}}{2} = \pm \frac{\sqrt{2 + \sqrt{3}} \cdot \sqrt{2 - \sqrt{3}}}{2 \cdot \sqrt{2 - \sqrt{3}}}$

Simplifying this expression, we get:

$\sin \frac{157.5^{\circ}}{2} = \pm \frac{\sqrt{4 - 3}}{2 \cdot \sqrt{2 - \sqrt{3}}}$

$\sin \frac{157.5^{\circ}}{2} = \pm \frac{1}{2 \cdot \sqrt{2 - \sqrt{3}}}$

Rationalizing the Denominator

To rationalize the denominator, we need to multiply the numerator and denominator by $\sqrt{2 + \sqrt{3}}$. This will eliminate the radical in the denominator.

Multiplying the numerator and denominator by $\sqrt{2 + \sqrt{3}}$, we get:

$\sin \frac{157.5^{\circ}}{2} = \pm \frac{1 \cdot \sqrt{2 + \sqrt{3}}}{2 \cdot \sqrt{2 - \sqrt{3}} \cdot \sqrt{2 + \sqrt{3}}}$

Simplifying this expression, we get:

$\sin \frac{157.5^{\circ}}{2} = \pm \frac{\sqrt{2 + \sqrt{3}}}{2 \cdot (2 - \sqrt{3})}$

$\sin \frac{157.5^{\circ}}{2} = \pm \frac{\sqrt{2 + \sqrt{3}}}{4 - 2 \sqrt{3}}$

Simplifying the Expression Further

To simplify the expression further, we can multiply the numerator and denominator by the conjugate of the denominator. The conjugate of $4 - 2 \sqrt{3}$ is $4 + 2 \sqrt{3}$.

Multiplying the numerator and denominator by $4 + 2 \sqrt{3}$, we get:

$\sin \frac{157.5^{\circ}}{2} = \pm \frac{\sqrt{2 + \sqrt{3}} \cdot (4 + 2 \sqrt{3})}{(4 - 2 \sqrt{3}) \cdot (4 + 2 \sqrt{3})}$

Simplifying this expression, we get:

$\sin \frac{157.5^{\circ}}{2} = \pm \frac{(4 + 2 \sqrt{3}) \sqrt{2 + \sqrt{3}}}{16 - 12}$

$\sin \frac{157.5^{\circ}}{2} = \pm \frac{(4 + 2 \sqrt{3}) \sqrt{2 + \sqrt{3}}}{4}$

Final Answer

$$

Q: What is the half-angle identity for sine?

A: The half-angle identity for sine is $\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}$.

Q: What is the half-angle identity for cosine?

A: The half-angle identity for cosine is $\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}$.

Q: How do I choose the correct half-angle formula?

A: To choose the correct half-angle formula, you need to consider the given angle. If the angle is in the second quadrant, you should use the half-angle formula for sine. If the angle is in the first or fourth quadrant, you should use the half-angle formula for cosine.

Q: What is the value of $\cos 157.5^{\circ}$?

A: The value of $\cos 157.5^{\circ}$ is $-\frac{\sqrt{3}}{2}$.

Q: How do I simplify the expression $\sin \frac{157.5^{\circ}}{2} = \pm \frac{\sqrt{2 + \sqrt{3}}}{2}$?

A: To simplify the expression, you can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2 - \sqrt{3}}$. This will eliminate the radical in the denominator.

Q: What is the final answer for $\sin 157.5^{\circ}$?

A: The final answer for $\sin 157.5^{\circ}$ is $\boxed{\frac{(4 + 2 \sqrt{3}) \sqrt{2 + \sqrt{3}}}{4}}$.

Q: Can I use the half-angle identity to find the exact value of $\cos 157.5^{\circ}$?

A: Yes, you can use the half-angle identity to find the exact value of $\cos 157.5^{\circ}$. However, you will need to use the half-angle formula for cosine, which is $\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}$.

Q: How do I find the exact value of $\cos 157.5^{\circ}$ using the half-angle identity?

A: To find the exact value of $\cos 157.5^{\circ}$ using the half-angle identity, you will need to substitute $\theta = 157.5^{\circ}$ into the half-angle formula for cosine. This will give you an expression in terms of $\cos 157.5^{\circ}$, which you can then simplify to find the exact value.

Q: What is the final answer for $\cos 157.5^{\circ}$?

A: The final answer for $\cos 157.5^{\circ}$ is $\boxed{-\frac{\sqrt{3}}{2}}$.

Q: Can I use the half-angle identity to find the exact value of $\sin 157.5^{\circ}$ and $\cos 157.5^{\circ}$?

A: Yes, you can use the half-angle identity to find the exact values of both $\sin 157.5^{\circ}$ and $\cos 157.5^{\circ}$. However, you will need to use the half-angle formulas for both sine and cosine, and then simplify the resulting expressions to find the exact values.