Find Sin ⁡ 285 ∘ \sin 285^{\circ} Sin 28 5 ∘ Without Using A Calculator.Use The Angle Addition Or Subtraction Formulas:${ \cos (A \pm B) = \cos A \cos B \mp \sin A \sin B }$ { \sin (A \pm B) = \sin A \cos B \pm \cos A \sin B \} Calculate

$$

Introduction

In this article, we will explore how to find the value of $\sin 285^{\circ}$ without using a calculator. We will utilize the angle addition and subtraction formulas to simplify the problem and arrive at the solution.

Understanding the Angle Addition and Subtraction Formulas

The angle addition and subtraction formulas are essential tools in trigonometry. They allow us to express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles.

The formulas are as follows:

${ \cos (A \pm B) = \cos A \cos B \mp \sin A \sin B }$

${ \sin (A \pm B) = \sin A \cos B \pm \cos A \sin B }$

Breaking Down the Problem

To find $\sin 285^{\circ}$, we can break down the problem into smaller, more manageable parts. We can express $285^{\circ}$ as the sum of two angles: $270^{\circ}$ and $15^{\circ}$.

Using the Angle Addition Formula for Sine

We can use the angle addition formula for sine to express $\sin 285^{\circ}$ as the sum of the sines of $270^{\circ}$ and $15^{\circ}$.

${ \sin 285^{\circ} = \sin (270^{\circ} + 15^{\circ}) }$

Using the angle addition formula for sine, we get:

${ \sin 285^{\circ} = \sin 270^{\circ} \cos 15^{\circ} + \cos 270^{\circ} \sin 15^{\circ} }$

Evaluating the Trigonometric Functions

We know that $\sin 270^{\circ} = -1$ and $\cos 270^{\circ} = 0$. We also know that $\cos 15^{\circ}$ and $\sin 15^{\circ}$ are positive values.

Substituting these values into the equation, we get:

${ \sin 285^{\circ} = (-1) \cos 15^{\circ} + (0) \sin 15^{\circ} }$

Simplifying the Expression

Simplifying the expression, we get:

${ \sin 285^{\circ} = -\cos 15^{\circ} }$

Finding the Value of $\cos 15^{\circ}$

To find the value of $\cos 15^{\circ}$, we can use the angle addition formula for cosine.

${ \cos (A + B) = \cos A \cos B - \sin A \sin B }$

We can express $15^{\circ}$ as the sum of two angles: $45^{\circ}$ and $-30^{\circ}$.

Using the Angle Addition Formula for Cosine

We can use the angle addition formula for cosine to express $\cos 15^{\circ}$ as the difference of the cosines of $45^{\circ}$ and $-30^{\circ}$.

${ \cos 15^{\circ} = \cos (45^{\circ} - 30^{\circ}) }$

Using the angle addition formula for cosine, we get:

${ \cos 15^{\circ} = \cos 45^{\circ} \cos (-30^{\circ}) + \sin 45^{\circ} \sin (-30^{\circ}) }$

Evaluating the Trigonometric Functions

We know that $\cos 45^{\circ} = \frac{\sqrt{2}}{2}$, $\sin 45^{\circ} = \frac{\sqrt{2}}{2}$, $\cos (-30^{\circ}) = \cos 30^{\circ}$, and $\sin (-30^{\circ}) = -\sin 30^{\circ}$.

Substituting these values into the equation, we get:

${ \cos 15^{\circ} = \left(\frac{\sqrt{2}}{2}\right) \cos 30^{\circ} + \left(\frac{\sqrt{2}}{2}\right) (-\sin 30^{\circ}) }$

Simplifying the Expression

Simplifying the expression, we get:

${ \cos 15^{\circ} = \left(\frac{\sqrt{2}}{2}\right) \cos 30^{\circ} - \left(\frac{\sqrt{2}}{2}\right) \sin 30^{\circ} }$

Finding the Value of $\cos 30^{\circ}$ and $\sin 30^{\circ}$

We know that $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$ and $\sin 30^{\circ} = \frac{1}{2}$.

Substituting these values into the equation, we get:

${ \cos 15^{\circ} = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right) }$

Simplifying the Expression

Simplifying the expression, we get:

${ \cos 15^{\circ} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} }$

Finding the Value of $\sin 285^{\circ}$

Now that we have found the value of $\cos 15^{\circ}$, we can substitute it into the equation for $\sin 285^{\circ}$.

${ \sin 285^{\circ} = -\cos 15^{\circ} }$

Substituting the value of $\cos 15^{\circ}$, we get:

${ \sin 285^{\circ} = -\left(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\right) }$

Simplifying the Expression

Simplifying the expression, we get:

${ \sin 285^{\circ} = -\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} }$

Conclusion

In this article, we have found the value of $\sin 285^{\circ}$ without using a calculator. We utilized the angle addition and subtraction formulas to simplify the problem and arrive at the solution. The final answer is:

${ \sin 285^{\circ} = -\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} }$

$$

Introduction

In our previous article, we explored how to find the value of $\sin 285^{\circ}$ without using a calculator. We utilized the angle addition and subtraction formulas to simplify the problem and arrive at the solution. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the angle addition formula for sine?

A: The angle addition formula for sine is:

${ \sin (A + B) = \sin A \cos B + \cos A \sin B }$

Q: What is the angle addition formula for cosine?

A: The angle addition formula for cosine is:

${ \cos (A + B) = \cos A \cos B - \sin A \sin B }$

Q: How do I use the angle addition and subtraction formulas to find $\sin 285^{\circ}$?

A: To find $\sin 285^{\circ}$, you can break down the problem into smaller, more manageable parts. You can express $285^{\circ}$ as the sum of two angles: $270^{\circ}$ and $15^{\circ}$. Then, you can use the angle addition formula for sine to express $\sin 285^{\circ}$ as the sum of the sines of $270^{\circ}$ and $15^{\circ}$.

Q: What is the value of $\cos 15^{\circ}$?

A: To find the value of $\cos 15^{\circ}$, you can use the angle addition formula for cosine. You can express $15^{\circ}$ as the sum of two angles: $45^{\circ}$ and $-30^{\circ}$. Then, you can use the angle addition formula for cosine to express $\cos 15^{\circ}$ as the difference of the cosines of $45^{\circ}$ and $-30^{\circ}$.

Q: What is the value of $\sin 285^{\circ}$?

A: To find the value of $\sin 285^{\circ}$, you can substitute the value of $\cos 15^{\circ}$ into the equation for $\sin 285^{\circ}$. The final answer is:

${ \sin 285^{\circ} = -\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} }$

Q: Why do we need to use the angle addition and subtraction formulas to find $\sin 285^{\circ}$?

A: We need to use the angle addition and subtraction formulas to find $\sin 285^{\circ}$ because it allows us to simplify the problem and arrive at the solution. By breaking down the problem into smaller, more manageable parts, we can use the formulas to express $\sin 285^{\circ}$ in terms of the sines and cosines of smaller angles.

Q: Can I use a calculator to find $\sin 285^{\circ}$?

A: Yes, you can use a calculator to find $\sin 285^{\circ}$. However, the purpose of this article is to show you how to find the value of $\sin 285^{\circ}$ without using a calculator.

Q: What are some other applications of the angle addition and subtraction formulas?

A: The angle addition and subtraction formulas have many applications in trigonometry. They can be used to find the values of sine and cosine of angles that are not easily expressed in terms of the unit circle. They can also be used to solve problems involving the sum and difference of angles.

Conclusion

In this article, we have answered some frequently asked questions related to finding $\sin 285^{\circ}$ without using a calculator. We have also provided a brief overview of the angle addition and subtraction formulas and their applications in trigonometry.